Numerical investigation and strength prediction of core-enhanced cold-formed steel hexagonal columns

1 Introduction

The application of thin-walled sections has seen an increasing demand across various sectors of the engineering industry, attributed to their lower self-weight, improved performance of structural systems characterized by streamlined fabrication processes and economic advantages in both transportation and assembly. The buckling behaviour of thin-walled columns depends on various features such as cross-sectional geometry, the dimensions and the slenderness ratios (Zhu et al., 2024). A wide range of research work of local buckling behaviour of tubular member of different cross-sectional geometries such as Square Hollow Sections (SHS), Rectangular Hollow Sections (RHS), Circular Hollow Sections (CHS) and Elliptical Hollow Sections (EHS) has been carried out. Hexagonal hollow column sections belong to the typology of polygonal hollow sections. These sections are widely used in different civil engineering applications as transmission structures, substation structures, transmission poles, telegraph towers and lattice structures (Chen et al., 2021). Hollow sections like hexagonal and octagonal shapes, offer a better combination of improved resistance to buckling and the flat surfaces of these hexagonal and octagonal sections, unlike circular or elliptical hollow sections, provide a convenient surface for operators to perform welding or bolted connections using T-stubs or gusset plates. Furthermore, due to their comparatively smaller flat widths than square and rectangular hollow sections, they have higher resistance to local buckling (Liu et al., 2023a). One of the main issues in tubular columns is the buckling of the column walls. This problem is solved in thin wall cold-formed sections by the application of internal stiffeners. These stiffeners give continuous support to the thin walls, thus increasing their resistance to buckling stress (Zhu et al., 2016). Stiffeners divide flat plate elements into smaller sub-elements and increase the critical buckling stress, which delays local buckling. The efficiency of the inner stiffeners depends on their geometry, spacing and material properties. Optimising these parameters improves the structural performance without substantially increasing weight or material usage (Liu et al., 2023a). Studied the hexagonal hollow stub column on high strength steel and performed experimental and numerical investigation and derived the design approaches of Continuous Strength Method (CSM) and Generalised Slenderness Resistance Method (GSRM) which were found to provide more accurate predictions than the Direct Strength Method (DSM) (Chen et al., 2021) carried out a large number of numerical experiments on the behaviour of octagonal hexagonal welded and cold-formed stub columns. The material properties and residual stress distributions of cold-formed hollow structural sections with different aspect ratios were experimentally investigated. The results show that the current limitations in the design standards are inadequate for the safe design of octagonal hollow sections (OctHSs). The assessment findings suggest new cross-sectional slenderness limitations in accordance with applicable design methodologies. The cross-sectional resistance predicted by different design methods were evaluated and compared with the experimental and finite element results. Liu et al. (2022b) conducted an examination of the structural performance of 18 high-strength steel hexagonal hollow section stub columns under axial compression, comparing the results with predictions from recognized codes. The Continuous Strength Method (CSM) provides more accurate results than the Direct Strength Method (DSM) as it accounts for strain hardening effects in metallic materials for stocky sections that fail owing to cross-section yielding. The cross-sectional resistance of narrow sections was overestimated by the DSM, while the CSM offers more accurate predictions than its DSM equivalent. Though the CSM tends to generate results that are more precise than DSM, a more suitable base curve and a relevant material model for the CSM may require further assessment. Zhong et al. (2024) have done extensive research on stainless steel hexagonal hollow sections. An experimental study on the axial compression behaviour of 15 specimens of stainless steel hexagonal hollow section stub columns is presented. The stub column specimens were made by welding two sections, press braked from hot-rolled grade 304 austenitic stainless-steel sheets. The same research team broadened their study to include combined compression and bending (Zhong et al., 2025) thereby offering a more comprehensive results on the behaviour of hexagonal sections subjected to diverse loading conditions. Twelve stub column specimens were fabricated using austenitic stainless steel hexagonal hollow sections with various cross-sectional dimensions and initial eccentricities od loading and tested in eccentric compression. An improved ASCE design interaction curve was presented with increased accuracy and consistency of resistance estimates. The mechanical characteristics of hexagonal multi-cell concrete filled steel tubular stub columns were investigated by (Zhang et al., 2019). The author employed finite element analysis combined with experimental results to explore the effects of geometric and material parameters. The research work concentrated to the composite behaviour with concrete infill which gave valuable insights into the buckling behaviour of hexagonal steel tubes. Hexagonal sections have similar local buckling behaviour to octagonal sections, with individual plate elements fail between corner restraints. The local buckling resistance of larger plate elements is slightly lower than that of octagonal sections of same dimensions. Evirgen et al. (2014) carried out a comparative study of concrete filled steel tubular sections with circular, hexagonal, rectangular and square cross sections under axial compression. Their study involved 16 hollow cold-formed steel tubes and 48 concrete filled specimens, indicated that circular specimens had the highest axial stress and ductility values. Martins et al. (2019) obtained significant results on the local and distortional post-buckling behaviour of thin-walled regular polygonal tubular columns using the Generalised Beam Theory. Their investigation revealed that polygonal sections can present varied buckling modes and emphasised the significant effect of imperfection sensitivity on the ultimate capacity determination. Stub columns with slender plate elements are generally controlled by local buckling. Polygonal sections are flat and deform like a plate, with half-wavelengths on the order of the width of the plate. The corner regions provide the rotational restraint leading to simply supported or partially restrained boundary conditions for the flat plates. More sides in a given overall section size reduce the width of each plate and this usually increases the resistance to local buckling. Liu et al. (2022a) studied the local buckling behaviours and cross-sectional compression capacity of high strength steel Q690 cold-formed irregular octagonal hollow sections (IOctHS) with press-braked corners through experiment and numerical analysis. Sections with aspect ratios varying from 1.0 to 2.0 were studied. The limitations of cross section classifications and the accuracy of strength predictions from primary design standards such as the Direct Strength Method (DSM), Continuous Strength Method (CSM) and their modified variants were compared against experimental results and finite element analyses. The results showed that the current codified slenderness limits cannot be directly applied to high-strength cold-formed IOctHS, and although the modified design methodologies improved the prediction accuracy, further refinement is needed to consider the effects of aspect ratio. Saumiyaa et al. (2024) studied the behaviour and design of cold-formed ultra-high-strength steel (UHSS) hollow section columns by extensive finite element studies. Subsequently, after model validation, a parametric study was carried out on 146 columns with various cross-sectional slenderness. The results indicated that the normalised axial strength and stiffness increased with the increase of plate thickness. The predicted accuracy of the existing Direct Strength Method (DSM) in AISI S100 and the previously proposed DSM and Continuous Strength Method (CSM) methodologies was evaluated. Modified DSM and CSM formulae were proposed for thin UHSS hollow columns in this study which showed better accuracy and reliability than existing design methods.

The design of cold-formed steel compression member is controlled by several international standards such as the American Iron and Steel Institute Specification (AISI), North American Specification (NAS) (American Iron and Steel Institute, 2016), Australian/New Zealand Standard AS/NZS 4600 (Standards Australia/Standards New Zealand, 2018), Eurocode 3 Part 1-3 (European Committee for Standardization, 1993) and Eurocode 3 Part 1-5 (European Committee for Standardization, 2006). Several studies have investigated the accuracy of current design standards to predict the capacity of cold-formed steel polygonal hollow section columns, and have shown different degrees of conservatism and, in some cases, no conservatism. Several improvements and new methodologies have been proposed by researchers specifically for polygonal hollow sections. The applicability and accuracy of the direct strength method (DSM) of AISI S100 in predicting the compressive strength of columns with closed cross-sections have been examined by a number of researchers (Saumiyaa et al., 2024; Selvaraj and Madhavan, 2023; Zhang and Alam, 2023; Yao et al., 2025) by comparing experimental and finite element results. The Continuous Strength Method (CSM) is an alternative to the effective width methods and has been extended by (Yun and Gardner, 2018) to cold-formed steel non-slender tubular cross-sections. While the available metadata does not provide specific metrics of accuracy, the CSM has shown promise in producing more accurate and less dispersed predictions for stocky portions where strain hardening effects are significant. Chen et al. (2021) proposed new cross-sectional slenderness limits and modifications to existing design methods for octagonal hollow sections under compression. They used parametric slenderness estimations and compared them to existing standards and ended up with design proposals of octagonal geometry. The design equations for global buckling of cold-formed steel octagonal hollow section columns were developed by (Zhu et al., 2024) and verified by both experimental tests and finite element method analyses.

Figure 1 summarizes few of the sections studied under cold-formed steel hollow sections investigated previously. The existing research primarily focuses on hollow and internally stiffened sections. Hexagonal sections demonstrate intermediate behavior between octagonal and square configurations, offering better strength to weight efficiency while remaining susceptible to local buckling owing to the slenderness of their constituent plate elements. Existing research has focused on behavior and design of hollow cold formed steel sections. Consequently, current design provisions have been developed and validated for hollow sections without considering internal stiffener elements or plate restraining configurations. In contrast the incorporation of internal circular core connected through internal stiffeners introduces a structural system, in which the core, stiffeners and outer hexagon resist axial compression. The internal stiffeners provide restraint to the flat plate elements, reduce their effective unsupported width, enhance the load transfer and redistribution of stress between the elements and delay the initiation of local buckling. This alters the stiffness characteristics, buckling mode and post buckling response of the section, resulting in structural behavior that differs significantly from hollow sections. Table 1 summarizes the previous studies carried out in cold-formed hollow sections in terms of geometry, loading condition, analysis approach and limitations. As evident from the summary, hollow sections under axial compression or combined loading have been examined predominantly. In particular the influence of internal core on structural behavior, local buckling behavior needs further investigation. Furthermore, the applicability of existing design methods to CEHS columns needs systematic investigation and development of suitable design provision. Moreover, the use of core-enhanced hexagonal column sections allows for the development of a modular integrated construction, providing a broad spectrum of opportunities for customising the structural configuration and connection options. The use of core-enhanced hexagonal column sections facilitates in the modular and prefabricated construction. This is because the internal core can be incorporated in the fabrication process without any modifications of the outer geometry. Further detailed analysis is required to establish whether the design specifications are suitable for the CFS Core enhanced Hexagon columns. A comprehensive analysis of the interaction between the constituent plates of Core enhanced hexagonal column section is essential for accurately predicting their load carrying capacity and structural behaviour. In this context, the present study presents a comprehensive numerical investigation of the local buckling behaviour of Core enhanced hexagonal columns, providing results in their load carrying capacity and structural behaviour under axial compression.

TABLE 1. Summary of previous studies on cold formed steel hollow sections.

2 Modelling OF CFS core enhanced hexagon column sections

The present study deals with cold formed hexagonal columns, namely, (a) Unstiffened Hollow hexagon Section (UHS) and (b) Core-enhanced Hexagonal Section (CEHS) as shown in Figure 2. The CEHS column section consists of an outer tubular section with an inner circular core and a stiffening plate. The core and stiffening plate are spot welded together at a constant interval of 50 mm along the length to form a closed cross-section. The geometrical specifications of the analysed sections including the side width, thickness, height, core radius and inner plate dimensions are summarised in Table 2. The selected section width and thickness were based on geometric ranges evaluated in previous studies on cold-formed hollow compression members (Zhu et al., 2016). The internal circular core diameter was selected to provide effective restraint to the sides of hexagon without altering the external geometry of the section, thereby providing load transfer and stress redistribution. The intermediate stiffeners and the spot-weld spacing were selected to ensure structural continuity, efficient load transfer and maintain practical fabrication and assembly requirements. To evaluate the influence of cross-sectional geometry and core enhancement on structural performance, a range of section widths 100 mm–300 mm was studied, and UHS sections without the core were also used for comparative purposes. The identification of the specimen is based on sectional dimensions, and the designation indicates the side width and the thickness of the section. The CEHS specimens with the enhanced configuration are described with additional parameters, such as core radius and inner plate length. The UHS specimens are conventional hollow sections without the internal stiffening. The dataset incorporates variations in geometry to capture the influence of slenderness and section proportions on axial behaviour. The designation “CEHS200T3.2L1000” denotes core enhanced hexagonal section of side width 200 mm and all the elements of thickness 3.2 mm and a column length of 1,000 mm. Similarly “UHS200T3.2L1000” represents the unstiffened hexagonal hollow section of width 200 mm and thickness 3.2 mm and column length of 1,000 mm. The internal core and internal stiffeners and outer hexagon consist of the same thickness for each section respectively.

TABLE 2. Cross sectional details of the proposed cold-formed Core-enhanced Hexagonal Section (CEHS) and Unstiffened Hollow hexagonal Section (UHS).

3 Developement of finite element model

The finite element software (Dassault Systèmes Simulia Corp, 2023) was used to model core-enhanced columns in order to evaluate their compressive behaviour. The column was aligned toe to toe with spot welds on the vertical axis. The axial loading was applied at the centre of gravity of the specimen by displacement-controlled methods. The model was complete and considered both geometric nonlinearity and initial geometric imperfections. The details of the numerical modelling can be summarised as follows.

3.1 Shell element type

Cold-formed steel sections are typically represented as shell elements. FEM software indicates that shell elements are suitable for modelling structures with a thickness much smaller than other dimensions, where stress in that direction is negligible. Shell elements are generally used to model a structure with a thickness less than 1/10 of a typical global structural dimension. S4R is a appropriate 4-noded, reduced-integration, linear, quadrilateral shell element for a wide variety of applications. This element provides accurate results for the simulation of different shell thicknesses and has been used in similar studies (Liu et al., 2022a; Han et al., 2025; Buchanan et al., 2018).

3.2 Material Definition

The engineering stress–strain data obtained from tensile coupon tests (σ and ε), as reported by (Zhu et al., 2016) were utilized as shown in Figure 3 and transformed into real stress and logarithmic plastic strain values σ true and ε true using the following Equation 1 and Equation 2

σ true = σ 1 + ε (1)

ε true = ln 1 + ε − σ true E (2)

The elastic-perfectly plastic model is the simplest bilinear stress-strain model without strain hardening. It is widely used in nonlinear finite element analysis. For cases where more precision is needed, a material model considering strain hardening into account is used. In this study, a stress–strain relationship with strain-hardening effects was applied using the average yield and ultimate stresses given in (Zhu et al., 2016) to match the material response observed in tensile coupon testing. The density of steel was considered to be 7,850 kg/m³ for quasi-static analysis. The outer hexagonal shell, internal circular core and inner stiffeners were assigned to the same material properties. The material properties adopted in the finite element model are Elastic modulus (E) of 201000MPa, Poisson’s ratio (ν) of 0.30, Yield strength (Fy) of 449.8 MPa, Ultimate strength (fu) of 522.4 MPa. Figure 3 shows the adopted true stress-plastic strain data for the FE material properties of the model.

3.3 Modelling of initial imperfections

Cold-formed steel members are susceptible to initial defects because of thin-walled structure. For an accurate finite element analysis, both local and flexural buckling modes must be considered. Eigen buckling analysis was performed on UHS and CEHS columns to determine the local buckling parameters. The lowest, which is the first mode of local buckling and the lowest order global buckling mode were adopted to define the geometric imperfection. To examine the sensitivity of the FE predictions to the assumed local imperfection amplitude, analyses were performed for two representative specimens CHS100T3.2L1000 and CHS200T3.2L1000. The local imperfection amplitude varied from 0.2t to 1.0t while global imperfection of L/1,000 (Schafer et al., 2010) was maintained. The resulting ultimate axial load capacities are presented in Table 3. The load varies over the investigated range of local imperfection amplitudes. The adopted amplitude is 0.34t which has a maximum variation in load was 1% and 1.4% respectively for the sections and hence retained for subsequent parametric analyses. Residual stresses were not incorporated in the finite element model. Previous experimental and numerical investigations on cold-formed steel compression members have shown that the influence of residual stress on ultimate load and buckling behavior is relatively small compared with the effects of geometric imperfections and material non-linearity. Accordingly, consistent with the numerical modelling approaches adopted in previous studies (Anbarasu and Dar, 2020) residual stresses were neglected in the present analysis.

TABLE 3. Sensitivity of ultimate load capacity to variations in local imperfection amplitude.

3.4 Interactions and boundary conditions

The contact pairs were generated to define the surface interactions among the basic component surfaces. The tiny sliding formulation was applied together with the surface-to-surface discretisation approach. The frictional interactions between the surfaces of the relatively smooth steel plates were ignored. Tangential and normal interactions between surfaces are implied by frictionless and rigid contact. Systematic spot welds were applied at regular intervals of 50 mm along the entire length of the column, particularly at all intersections of the internal stiffeners and at the interfaces between the circular core and the surrounding plates. The welds were modelled by applying tie constraints to force uniform deformation among the associated components as shown in Figure 4. Considering the close spacing of spot welds 50 mm, tie constraints provide better approach to load transfer under axial compression which significantly reduces modelling complexity and computationally efficient. (Zhang and Young, 2015).Therefore, normal pressure can be transferred from one surface to another without penetration. The boundary conditions at the column ends were defined to represent the end-restraint conditions adopted in axial compression tests, rather than idealized perfectly fixed-ended conditions. Accordingly, an effective length factor of K = 0.8 was adopted for the calculation of the global slenderness ratio. A reference point was defined and all nodes in the upper plate were connected to this reference point in z-direction displacement as shown in Figure 4. The finite element analysis was carried out by displacement control approach. This reference point was shifted in the z-direction. The bottom plate of lower support was fixed (U_x = 0, U_y = 0, and U_z = 0; θ_x = 0, θ_y = 0, and θ_z = 0). The upper plate of the upper support was constrained simultaneously from rotation about the x-, y- and z-axes (θ_x = 0, θ_y = 0, and θ_z = 0) and translations in both the x- and y-directions (U_x = 0 and U_y = 0) as shown in Figure 4.

3.5 Analysis procedure

The research was initially performed through a two-step manner in which a linear eigenvalue buckling analysis was performed to find the relevant buckling modes. Subsequently, CFS core-enhanced columns were analysed nonlinearly using the Riks method considering both local and global buckling modes. The static Riks method was used with 100 increments of arc length. The first one was set to be one. The maximum and minimum increments of arc length were determined for the calculation of the total projected arc length of unity.

3.6 Mesh convergence analysis

A mesh convergence study was performed to determine appropriate mesh size that provides an optimal balance between prediction accuracy and computational efficiency. The specimen CEHS200T3.2L1000 was selected as the representative model and shell element sizes of 5,10,15,20,25,30 and 40 mm were investigated. The effect of mesh size on the ultimate load and computational time for CEHS200T3.2L1000 specimen is shown in Figure 5. As the mesh size increased from 5 mm to 15mm, the ultimate load increased from 2,447 kN to 2,656 kN, after which only small variations were observed. The difference in ultimate load between 15mm and 20 mm meshes were less than 0.1% indicating the numerical solution had converged. The mesh sizes 25 mm and 30 mm provided ultimate load variation of 2% of the converged solution and therefore considered insignificant. In contrast 40 mm mesh size resulted reduction in ultimate load, proving the loss of accuracy with coarse mesh. Figure 6 Demonstrates that the predicted local buckling pattern and post-buckling deformation remained essentially unchanged for mesh sizes between 15 mm and 30 mm. The deformation is localized in the same regions of the outer hexagonal shell with comparable buckling wavelengths and mode shapes, indicating that the selected mesh adequately captures the governing instability mechanism. In contrast, the 40 mm mesh produced a comparatively coarser deformation pattern due to insufficient discretization of the plate elements. The computational time (CPU TIME) decreased considerably with increasing mesh size reducing from 93 min to 20 min for 15 mm mesh size. Considering the convergence of the ultimate load, the consistency of local buckling mode and post buckling response an element size of 15 mm was adopted for subsequent analysis.

3.7 Validation

The commercial software Abaqus version 2023 (Dassault Systèmes Simulia Corp, 2023) was used to perform FE modeling. Once the FE models were obtained, their accuracy was evaluated by comparing them to ultimate compressive load and failure mode obtained from experimental testing. The model was validated against the experimental study on inner stiffened SHS sections (Zhu et al., 2016). Table 4 compares the experimental and FE ultimate loads and Figure 7 presents the corresponding failure mode comparison. The FE model predicted ultimate load with differences generally less than 8% and the local buckling behavior was successfully captured. The good agreement in ultimate strength and failure mode provides accuracy of the FE modelling approach adopted for the study.

TABLE 4. Comparison of ultimate axial load obtained from experiments and FE analysis.

4 Parametric analyses OF CFS core enhanced hexagon column section

4.1 Effect of variation of width of the section on load capacity

For the parametric investigation, column lengths ranging from 500 mm to 2,300 mm were considered for CEHS with side widths ranging from 100 mm to 300 mm and thickness 3.2 mm and 4 mm. It is noted that changing these parameters consequently varies the cross-sectional area, sectional stiffness and slenderness among the investigated databases. Thus, the results represent the combined response of core-enhancement elements effects as illustrated in Figures 8a,b.

For CEHS columns, increasing width from 100 mm to 150 mm results in notable strength gains (about 26%–46%), with further gradual improvements observed up to 200 mm. At larger widths (225–300 mm), gradual increases (up to 80%–100%) are achieved, as the internal core restrains local buckling and promotes effective stress redistribution, as seen in Figures 8a,b. With 4.0 mm thickness, the response is more stable, with strength increases of about 32%–47% from 100 mm to 150 mm and further enhancements of approximately 35%–60% at larger widths (200–300 mm), as reduced plate slenderness delays local instability and improves post-buckling strength, as reflected in Figure 8b.

In contrast, UHS sections show limited improvement with increasing width. While initial increase is observed from 100 mm to 150 mm (about 10%–20%), further increases lead to marginal or inconsistent behaviour due to early local buckling and reduced effective width, as shown in Figures 8c,d.

Overall, CEHS sections exhibit more consistent increase in axial capacity than the UHS columns presenting the enhancement effect of the internal core.

4.2 Effect of slenderness of the section on load capacity

For CEHS100T3.2, the reduction in strength with increasing slenderness ratio (λ) remains limited which is about 5%–6%, indicating low sensitivity to slenderness, with a slight increase of λ attributed to stable post-buckling response, as observed in Figure 9a. The 4.0 mm thickness sections exhibit even smaller reductions (1%–5%), as reduced plate slenderness and increased stiffness delays the initiation of local buckling and sustain load-carrying capacity over a wider slenderness range.

In the CEHS150 series, thickness 3.2 mm sections show moderate reductions (16%–17%) with increasing λ, reflecting the increasing influence of slenderness, whereas thickness 4 mm sections restrict the decrease to about eight%–13% due to higher stiffness and delayed local instability, as illustrated in Figure 9c. For the CEHS200 sections, an increase in strength is observed at intermediate λ, with thickness 3.2 mm reaching up to 47% and thickness 4 mm showing consistent increases in the range of 37%–40%, indicating effective core confinement and enhanced stress redistribution leading to improved post-buckling resistance as shown in Figure 9f. For intermediate widths (120–180 mm), steady responses are observed with variations of 5%–10%, which can be attributed to optimal interaction between the effects of slenderness and sectional stiffness as shown in Figures 9b,d,e. The increases in strength observed at the larger widths (225 and 300 mm), particularly at a thickness of 4.0 mm (around 30%–60%), suggest that improved stiffness and core restraint reduce the effect of slenderness, leading to improved post-buckling behaviour and higher axial strength, as shown in Figures 9g,h. However, UHS sections Figures 9i–k are more sensitive to slenderness, showing uniform strength reductions and low post-buckling capacity due to the lack of core confinement.

4.3 Effect of core-enhancement of the section

The effect of core enhancement on the behaviour of cold-formed CEHS hexagonal columns was evaluated through finite element analysis by comparing core-enhanced (CEHS) and unstiffened (UHS) sections. The comparison reveals differences in buckling behaviour, deformation characteristic and axial load capacity Failure modes of CEHS100T3.2 columns with lengths from 500 mm to 2,300 mm are shown in Figure 10. Short columns are governed by local buckling, characterized by localized plate deformations, while increasing column length leads to interaction between local and global buckling modes, with deformation extending along the member length. The internal core presence restricts the inward plate deformation resulting in lower local buckling intensity and transition to more stable buckling modes. In terms of ultimate capacity, the CEHS sections exhibited higher ultimate load capacity than the corresponding UHS sections for the analysed database. The core offers an increase in the sectional stiffness and a delay in local buckling. This allows the section to carry greater stresses before becoming unstable. Figures 11a,b shows a comparison between CEHS and UHS sections that indicates that the main cause of buckling in UHS sections is local buckling that leads to a greater intensity of deformation and an earlier onset of instability. In contrast, the CEHS sections exhibit a combined local-global buckling response and scattered deformation patterns. The inner core serves as reinforcement to improve the stress redistribution as well as reduce the local buckling. Furthermore, CEHS sections demonstrate improved post-buckling behaviour, maintaining load-carrying capacity beyond initial buckling, whereas UHS sections show a rapid reduction in strength after local buckling. To isolate the influence of core enhancement from the effect of the additional steel, the ultimate load capacities were normalized as (Pu/Agfy). The resulting normalized axial strength of the CEHS sections were 1.299,1.277 and 1.241 for section widths of 100 mm, 150 mm and 200 mm respectively. Comparison with the corresponding UHS sections on normalized strength provides a direct assessment of the strength. The results indicate that the increase in axial capacity of the CEHS sections is not completely due to additional steel but also reflects the contribution of inner core and stiffening plate. The Figure 12 represents the comparison of normalized strength to the length of the columns.

To quantitatively assess the ultimate load capacity the mean, standard deviation and coefficient of variation (COV) of the ultimate load capacities were evaluated for all the section widths over the investigated member lengths. The COV values ranged from 0.84% to 14.80%. The lower COV values indicate that the ultimate load capacity indicates relatively small variations in load capacity across the member lengths, whereas the higher COV values indicate high changes to load capacity. This variation is associated with increasing influence of member slenderness and the changes in behavior as the column length increases. This behavior is consistent with previous studies on cold-formed steel closed and stiffened sections, which have identified cross-sectional slenderness and member slenderness as important parameters influencing axial load capacity.

5 Design of core-enhanced hexagonal steel columns

5.1 Effective width method

Core enhancement can significantly improve compressive strength, delay local and distortional buckling and improve ductility. This motivates the present study on the core enhanced design methods of the hexagonal CFS members. AISI S100 (American Iron and Steel Institute, 2016) provides established design procedures for cold-formed members, including the Effective Width Method (EWM) and Continuous Strength Methods. However, the EWM considers sections as consisting of plate elements and requires careful validation of its application to polygonal hollow sections with internal cores or complex distortional modes. A comprehensive numerical database was established through a parametric finite element study covering variations in cross-sectional width, outer plate thickness, and column length. The database consists of 112 finite element models of the core-enhanced hexagonal sections (CEHS) and 42 finite element models of the unstiffened hexagonal sections (UHS), resulting in a total of 154 finite element analyses. This database was subsequently used to evaluate the predictive accuracy of the Effective Width Method (EWM) and to develop and assess the proposed design modifications.

The ultimate load capacities estimated by the Effective Width Method (P_EWM) were calculated using Equation 3

P EWM = A eff F cs (3)

Where F_cs represents the compressive stress; A_eff is the effective area, and F_cs can be calculated using Equations 4, 5

F c s = 0.658 λ C 2 F y s ⇒ λ C ≤ 1.5 (4)

F c s = 0.877 λ C 2 F y s ⇒ λ C > 1.5 (5)

Where F_ys is the yield stress; λ C is the local slenderness ratio, λ C = F y s F e l and F_el is the least applicable elastic global buckling stress and can be calculated using Equation 6

F e l = π 2 E k L ∕ r 2 (6)

Where E is the modulus of elasticity of steel; k is the effective length factor; L is the unbraced length of the member and r is the radius of gyration of the fully unreduced cross section about the axis of buckling.

Then, the effective sectional area of specimen A_eff can be calculated by Equation 7

A e f f = ∑ b e * t (7)

Where t is the thickness of the steel section and b_e is the effective width of the plate and b is the actual width of the plate which can be calculated using the Equations 8, 11

b e = b ⇒ λ ≤ 0.673 (8)

b ⅇ = 1 λ f 1 − 0.22 λ f b ⇒ λ f ≥ 0.673 (9)

λ f = f f c r l (10)

f c r l = k π 2 E 12 1 − μ 2 t b 2 (11)

Where λ f is the flexural rigidity coefficient, f is the maximum compressive edge stress of the plate, f_crl is the critical buckling stress.

The ultimate load capacities estimated by the Effective Width Method (P_EWM) have been compared with the relevant Finite Element Analysis ultimate load capacity results (P_FEA) for the specimens across range of global slenderness. The comparison indicates that the EWM always underestimates the strength, thus showing a moderate approach. For columns with low slenderness ratios, where the behaviour is mainly governed by yielding and local buckling, the difference between P FEA / P EWM is in the range of about 1.05–1.20, thus underestimating by about 5%–20%. The differences are noticeable with the increase of slenderness in the intermediate range where the interaction of local and global buckling occurs. Ratios are normally ranging between 1.20 and 1.60 representing deviations of 20%–60%. For very slender columns which are mainly governed by global buckling, the P_EWM predictions are still conservative with ratios mostly in the range 1.50–1.90 suggesting an underestimation of about 50%–90%. The direct application of the EWM to the present specimens results in conservative strength predictions for all slenderness ranges, with the largest deviation in the intermediate range, confirming the limitations of the method and the need for a revised formulation.

5.2 Proposed EWM cold-formed CEHS columns

The application of the Effective Width Method (EWM) directly following the North American design provisions does not accurately predict the structural response of the assembled section in this study. This is mainly the composite interaction of the constituent elements and the resulting local-global instability effects. The formulation is based on the behaviour of an isolated plate and therefore does not properly consider the stiffening and load redistribution mechanisms that are typical of the built-up configuration. It is proposed to introduce a single modification factor to the theoretical expression for axial compressive capacity to account for the effect of global slenderness.

The modified formulation is expressed as in Equation 12 where γ is the proposed modification factor defined as a function of the global slenderness parameter as in Equation 13

P EWM ‐ M = γ A E F F F c s (12)

γ = 0.0003 λ 2 − 0.0042 λ + 1.5901 (13)

This expression is obtained through regression analysis of the ratio between the finite element results and the corresponding EWM predictions, thereby enabling a systematic correction of the inherent conservatism in the original method.

Figure 13 shows the updated EWM estimates in comparison to the finite element results for the entire range of slenderness. The modification proposed leads to a significant improvement of the accuracy. The mean value was reduced from 1.59 in the original EWM estimates to 1.00 in the revised formulation, indicating a significant reduction in the systematic capacity. The coefficient of variation was reduced from 0.087 to 0.078, indicating a better consistency of predictions for different slenderness levels. The CEHS specimen axial load capacity derived from the Finite Element Analysis (P_FEA), the current EWM (P_EWM), and modified EWM (P_EWM-M) results were compared. Where, λ is the maximum slenderness ratio.

The statistical evaluation of the predictive performance of the original and the modified Effective Width Method are presented in Table 5. The modified EWM achieved mean prediction ratio of 1.00 compared with 1.58 for original EWM. The prediction ratio ranged from 0.91 to 1.10, indicating conservatism and a lower nonconservative prediction rate of 0.23% for the database analyzed. These results demonstrate that the proposed modification provides a more accurate and statistically consistent prediction of the axial compressive strength of the investigated core-enhanced hexagonal sections within the considered parameter range. A formal reliability calibration in accordance with current design specifications is recommended as future work before practical code implementation.

TABLE 5. Statistical evaluation of the predictive performance of the original and the modified Effective Width Method.

5.3 Continuous strength method (CSM)

The CSM, a design method for non-slender stainless-steel sections first introduced by Gardner and Nethercot, this technique is characterised by two key features: the base curve relating the cross-section deformation capacity to the slenderness and the material model of the elastic linear hardening. The CSM has two major advantages over existing design approaches. First the rational strain-hardening is used and second the interactions between the parts within the cross-section are considered. The CSM base curve connects the maximum attainable strain (ε_csm) to the total cross-section slenderness (λ_p), as expressed by Equation 16 where, λ p is given by Equation 14

λ p = f y f c r (14)

Where f_y is the yield stress of steel and f_cr is the elastic buckling stress which can be calculated using Equation 15.

f c r l = k π 2 E 12 1 − μ 2 t b 2 (15)

(Zhao et al., 2017) expanded the CSM method to include slender sections. The proposed base curve for slender cross-sections is as in Equation 16 where εcsm denotes the CSM limiting strain, was calculated from the base curve as a function corresponding to cross-sectional slenderness proposed by (Zhao et al., 2017; Lan et al., 2018) based on the corresponding cross-sectional slenderness. ε_y represents the yield strain, calculated as fy/E, fy is the steel yield stress, E is the steel elastic modulus.

ε c s m ε y = 1 − 0.222 λ p 1.050 1 λ P 1.050 f o r λ p > 0 · 68 (16)

The axial capacity as per CSM for slender cross-sections under compression (P_CSM) is calculated using the gross cross-sectional area. The formula for calculating the axial capacity of a compression member, as indicated by CSM, is presented in Equation 17

P C S M = ε C S M ε y f y A f o r λ p > 0.68 (17)

It is noted that the adopted CSM based curve involves the cross-sectional slenderness and is therefore intended to represent cross-sectional resistance, including the effects of local buckling and inelastic deformation. The formulation does not incorporate the member global slenderness parameter. Accordingly, the CSM formulation in the present study is considered a cross-sectional resistance model. Although the FEA allows the development of global instability, no distinct global buckling failure mode was observed within the investigated range of columns.

Lan et al. (2018) expand the CSM for the design of high-strength steel square and rectangular hollow sections. The overall cross-section slenderness (λp) described in Equation 14 determines the elastic buckling stress (fcr) as specified in Equation 18.

f c r = 4 B 1 H 1 1.7 π 2 E 12 1 − ν 2 t B 1 2 (18)

In this context, E denotes the elastic modulus, v represents the Poisson’s ratio, which is 0.3, t signifies the wall thickness of tubular sections, H₁ indicates the height, and B₁ refers to the width of the central surface of SHS and RHS (Lan et al., 2018). Presented the following base curve for narrow tubular section as in Equation 19

ε c s m ε y = 1 − 0.219 λ p 1.014 1 λ p 1.014 f o r 0.68 < λ p ≤ 2.61 (19)

The predicted ultimate load capacities by the Continuous Strength Method (CSM) formulations proposed by (Zhao et al., 2017; Lan et al., 2018) have been compared with the corresponding finite element results for the current set of specimens as shown in Table 6. The mean values of P FEA / P CSM − Z and P FEA / P CSM − L are 1.08 and 1.07, with corresponding coefficients of variation (COV) of 0.130 and 0.127, respectively shown in Table 7. These results show that both formulations provide rather accurate estimates with small conservative bias. The axial strength in the CSM method is governed by the limiting strain (ε csm which decreases with increasing cross-section slenderness λ p = f y / f c r thus highlighting the growing significance of the local buckling phenomenon. This relationship is representative for the general trend of strength reduction but the high COV values and the presence of non-conservative predictions P FEA / P CSM < 1.0 demonstrate that the existing base curve formulations do not sufficiently account for the variability in post-buckling behaviour, especially in the intermediate slenderness range.

TABLE 6. Comparison of the Finite Element analysis results with the current CSM showing ultimate load capacities P_CSM-Z (Zhao et al., 2017) and P_CSM-L (Lan et al., 2018).

TABLE 7. Comparisons of numerical results with predicted strengths based on different design methods.

On the other hand, the CSM formulation of (Lan et al., 2018) shows a little higher consistency with that of (Zhao et al., 2017) due to its calibration for high strength tubular sections. However, both methods can be improved to increase the reliability of prediction at a wider range of slenderness.

5.4 Proposed CSM cold-formed CEHS columns

The current CSM formulation yields satisfactory estimations for cold-formed Core-enhanced hexagonal column sections (CEHS), modified equations are proposed through the present numerical investigation. The proposed CSM is formulated using regression analysis of the finite element results to more accurately⁠ depict the local buckling response of slender sections⁠. The finite element results were compared with both the original and the modified CSM predictions. As illustrated in Figure 14, the variation of normalized strain ε csm / ε y with cross-sectional slenderness λ p demonstrates that the modified CSM curve closely follows the trend of the FEA results, particularly in the intermediate slenderness range as in Equation 20. ⁠

ε c s m ε y = 1 − 0.25 λ p 0.85 1 λ p 0.85 f or λ p > 0.68 (20)

The original formulation produces a mean ratio of P_FEA/P_CSM-Z and P_FEA/P_CSM-L of 1⁠.0796 and 1.0734 with a coefficient of variation (COV) of 0.129 and0.126, whereas the revised CSM results in a mean value of 1.01 accompanied by⁠ a diminished COV of 0.11 shown in Figure 15. The Comparisons of numerical results with predicted strengths based on different design methods are presented in Table 7. This assessment illustrates that the introduced modification yields a closer alignment with the numerical data. The convergence of the mean value to one and the decrease in COV implies an improvement in prediction. The proposed revised CSM has demonstrated better efficiency in accurately establishing the axial load-carrying capacity of Core-Enhanced Hexagonal column sections (CEHS) columns considering local buckling. The CSM formulation is interpreted as a cross-sectional resistance model. Further investigation incorporating member level global stability effects can be done to extend the proposed approach to columns covering a broader range of global slenderness.

6 Conclusions

The behaviour of cold-formed core enhanced hexagonal sections (CEHS) and unstiffened hexagonal sections (UHS) under axial compression has been thoroughly investigated numerically. The finite element model considered material nonlinearity, geometric imperfections and proper boundary conditions to accurately simulate the structural response. A comprehensive parametric study was conducted for widths ranging from 100 to 300 mm, thicknesses of 3.2 mm and 4.0 mm and a range of slenderness ratio. The study focused on effects of cross-section geometry, slenderness of the member, axial capacity and buckling behavior and on investigation of the applicability of existing design approaches and provide modified design methods.

• The parametric study indicates that the axial capacity of CEHS is significantly influenced by the width and thickness of the section. Increasing section width and thickness increased the load capacity due to combined effect of increased steel cold formed steel area, sectional stiffness and reduced plate slenderness.
• The ultimate load capacity of CEHS was governed by the interaction of local plate buckling, cross sectional deformation and member instability. The internal core restrains the local deformation and improves the post-buckling behavior.
• Failure mode analysis showed that the investigated columns were predominantly failed by local buckling, with the deformation varying with the slenderness of the member. Compared with the UHS, CEHS showed improvements in ultimate load capacity of 20%–40% depending on the cross-sectional arrangement.
• The modified CSM and EWM provided improved agreement with the finite element results and reduced prediction variability in comparison to existing design methods over the investigated parameter range.
• The modified EWM and CSM formulations effectively calculated the axial strength of the investigated CEHS. However, the proposed formulations were calibrated and assessed within the parameter range considered in the study, therefore their applicability to CEHS columns with different geometric and slenderness requires further numerical and experimental validation.

Overall, the results demonstrate that core enhancement improves stiffness characteristics, delays local instability, and increases axial load capacity, providing a more reliable design for cold-formed CEHS columns.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbuil.2026.1933880/full#supplementary-material

Supplementary material Supplementary material at the source

Glossary

F _**cs** Compressive stress A _**eff** Effective area P _**EWM** Ultimate load by Effective Width Method F _**ys** Yield stress of steel λ C Local slenderness ratio F _**el** Elastic Global buckling stress E Elastic modulus of steel K Effective length factor L Unbraced length of the member r Radius of gyration Ag Gross area t Thickness of steel elements v Poisson’s ratio b _**e** Effective width λ p cross-section slenderness ε _**csm** CSM limiting strain ε _**y** Yield strain λ f Flexural rigidity coefficient f c r l Critical buckling stress. B _**1** Width of the central surface H _**1** Height AISI American Iron and Steel Institute CEHS Core Enhanced Hexagonal Section CFS Cold-Formed Steel CSM Continuous Strength Method COV Coefficient of Variation DSM Direct Strength Method EWM Effective Width Method FE Finite Element FEA Finite Element Analysis RHS Rectangular Hollow Section SHS Square Hollow Section UHS Unstiffened Hexagonal Section

Footnotes

Edited by: Carlos Humberto Martins, State University of Maringá, Brazil

Reviewed by: Mohd Syahrul Hisyam Mohd Sani, MARA University of Technology, Malaysia

Chunyang Su, Taiyuan University of Technology, China