Validation of AISI Design of Cold-Formed Steel Beams Using Non-Linear Finite Element Analysis

Abstract

In the local building industry of Pakistan, pre-engineered steel building manufacturers mainly employ their own self-developed software and Excel sheets. These systems are based on empirical formulas mentioned in the AISI manual. Under this scenario, a need was found to validate AISI flow charts using commercial software like CUFSM 5.04 and ABAQUS R2019x. This study presents a validation of the CUFSM software and the American Iron and Steel Institute (AISI) Direct Strength Method (DSM) results of channel section flexural members using the non-linear finite element method employing ABAQUS. In this study, eight standard cold-formed channel-section (C-section) steel members were modeled and analyzed using ABAQUS to simulate realistic behavior under four-point loading conditions. The non-linear finite element models incorporated material and geometric non-linearities to capture the actual response of the steel elements. The results obtained from ABAQUS were compared with those predicted by the CUFSM and DSM, focusing on critical parameters such as nominal strength, buckling modes, and deformation patterns. During this study, it was observed that out of the selected sections, the AISI charts predict conservative and even unsafe flexural capacities in some of the cases concerning other methods, with a maximum difference of 14.03%. The differences obtained using DSM and ABAQUS when compared with the results of the AISI charts varies on both the plus and minus sides. This study will not only affect the industry in terms of innovative designs for efficient structures but also the community in regards to low-budget construction.

1. Introduction

In the past, cold-formed steel (CFS) was not well recognized due to a lack of awareness and familiarity with this product among professionals. Also, the perception of CFS was that it was less durable when compared with traditional hot-rolled steel. Designing with CFS is more complex due to factors such as local, distortional, and torsional buckling. However, the structural analysis and design of CFS sections have undergone significant progress in recent years, mainly due to the development of various analytical and numerical methods. Among them, the Direct Strength Method (DSM) has gained wide acceptance and is included in the North American Specification for the Design of Cold-Formed Steel Structural Members (AISI S100). The DSM offers a more rational and less conservative approach compared to traditional methods because it directly relates the strength of a CFS element to its elastic buckling behavior, which then can be translated to inelastic buckling behavior and finally, to plastic flexural capacity.

CUFSM (Constrained and Unconstrained Finite Strip Method) software, which implements DSM principles, is a powerful tool for determining the elastic buckling characteristics of CFS cross-sections. It enables the elastic analysis of local, distortional, and global buckling modes and provides basic data for the application of DSM in design. While the CUFSM and DSM provide a robust framework for predicting the behavior of CFS members, it is critical to validate these analytical approaches against more comprehensive and detailed methods.

Non-linear finite element analysis (FEA) using software such as ABAQUS offers a high-fidelity approach to modeling the complex behavior of CFS sections under various loading conditions. By incorporating material and geometric non-linearities, FEA can capture the actual behavior of CFS members more accurately than linear analytical methods. Thus, validating CUFSM and DSM predictions using results obtained from non-linear FEA can increase confidence in these methods and ensure their reliability in practical design applications.

Cold-formed steel elements are thin, light, and cost-effective. However, this efficiency comes with complications. Engineers must account for cross-sectional instability (i.e., local and distortional), in addition to the global buckling of the member [1]. The Direct Strength Method does not use effective width or iteration to determine the strength of the element for design. Instead, the designer must calculate the elastic buckling load for three cases of behavior: local, distortional, and global. These data, combined with the load that causes the first yield, are then analyzed and used to “directly” provide strength using simple prediction equations [2]. In another study, the analysis was carried out using the four-point bending method. For a thin-walled beam with an open cross-section, where the load and support act at the shear center, it is not possible to eliminate the torsional moment load because the investigated beams are not perfect [3]. CUFSM is a useful tool for identifying buckling classes, i.e., global, distortional, local, and other deformations, and the constrained finite strip method can provide both modal decomposition and modal identification of the conventional finite strip solution [4]. Schafer and Adany created a cross-sectional model in CUFSM and an eigenvalue analysis was then performed to determine the expected imperfection modes. The CUFSM model was then exported to ABAQUS (via an input file) to run an eigenvalue buckling analysis using the same imperfection modes [5]. In developing the DSM, Yu and Schafer performed two series of bending tests and finite element analyses on different C- and Z-profiles including local buckling [6] and deformational buckling [7]. These series were aimed at isolating the local and biased modes used in DSM validation [8].

Artificial Intelligence techniques were also applied for the prediction of the axial capacity of cold-formed steel channel sections to compare the results with those of EWM and DSM, leading to proposed improvements in the AS/NZS 4600:2018 design equations [9,10,11].

The increasing use of back-to-back built-up cold-formed steel (CFS) channel beams, valued for their cost-effectiveness and higher load capacities, is evaluated. This evaluation involves 12 experimental tests and a non-linear finite element (FE) model. Flexural strength was also affected by screw spacing [12]. The flexural behavior of CFS four-limb built-up beams with both closed and open sections was investigated through experiments and FE modeling. Four-point bending tests reveal that the open-section beams have higher ultimate load capacity than closed-section beams, with local buckling as the failure mode. Validated FE models show that flexural moment capacity increases with thickness and web depth [13]. The buckling mode is a key factor that influences the bending strength of the specimens. Also, the finite element method (FEM) is a reliable tool to obtain quite accurate results in a reasonable amount of time [14] as an optimization framework for CFS members to maximize their flexure capacity through EC3. It was found that for the same amount of material, folded-flange and super-sigma sections have a higher flexural capacity enhancement of approx. 60% and 65%, respectively [15]. A simplified finite element model was defined to analyze stresses in the components. The Bending capacity and flexural behavior were investigated numerically by using FEM and through experimental evaluations. The moment capacity of the beams decreased as the compression flange’s plate slenderness increased. The results were used to evaluate, and propose modifications to, the Direct Strength Method (DSM) for cold-formed steel specialized beams prone to local buckling [16,17,18]. Cold-formed steel (CFS) upright sections were evaluated using finite element analysis and AI/ML techniques to predict normalized ultimate load and deflection. The finite element method (FEM) provided data for analysis, and the “feature selection” method identified key factors affecting flexural strength [19].

Cold-formed thin-walled C steel channel is not only being used as an independent member but also in composite form as a composite floor system [20].

Non-linear FEM results are effectively used to describe all the buckling modes. Key parameters like length, thickness, and geometric imperfections significantly impact the beams’ behavior [21,22].

This study aims to validate the analytical approach of CUFSM and AISI DSM using the non-linear finite element method (FEM), employing ABAQUS for channel section flexural members. By conducting comparative analyses on eight different sections, this research evaluates the accuracy of DSM and CUFSM predictions for CFS beam sections. The findings will contribute to a better understanding of the strengths and limitations of these methods, ultimately leading to their effective application in the design of cold-formed steel structures.

2. Materials and Methods

The design approaches used are described in the following subsections:

2.1. DSM (Direct Strength Method) Analytical Approach

The Direct Strength Method (DSM) is an efficient and modern approach to the design of cold-formed steel structures. It simplifies the analysis and design process by directly calculating the strength of members, considering various limiting states such as local, distortional, and global buckling [23].

The American Iron and Steel Institute (AISI) Specification for the Design of Cold-Formed Steel Structural Members provides a set of formulas, tables, and charts for the design of CFS members [24,25].

2.1.1. Yielding and Lateral–Torsional Buckling

For doubly or singly symmetrical sections bending about a symmetric axis, the following formulas are used.

Elastic flexural buckling about the y-axis is determined as follows:

$${\sigma }_{ey}={\displaystyle \frac{{\pi }^{2}E}{{\left(\frac{K_y{L}_y}{r_y}\right)}^2}}$$

(1)

$${\sigma }_t={\displaystyle \frac{1}{A{r}_o^2}}GJ+{\displaystyle \frac{{\pi }^{2}E{C}_w}{{\left(\frac{K_y{L}_y}{r_y}\right)}^2}}$$

(2)

$$F_{cre}={\displaystyle \frac{C_b{r}_{o}A}{S_f}}\sqrt{{\sigma }_{ey}}{\sigma }_t$$

(3)

For F_cre ≥ 2.78F_y,

F_n = F_y

(4)

For 2.78 F_y > F_cre > 0.56F_y,

$$F_n={\displaystyle \frac{10}{9}}{F}_y\left(1-{\displaystyle \frac{10F_y}{36F_{cre}}}\right)$$

(5)

For F_cre <= 0.56F_y,

F_n = F_cre

(6)

M_y = S_fy F_y

(7)

M_ne = S_f F_n

(8)

M_ne is to be taken as the minimum critical elastic buckling moment in flexural, torsional, or flexural-torsional buckling.

M_ne is allowed to be increased, considering inelastic reserve capacity, only in cases in which M_cre > 2.78 M_y.

M_cre = S_f F_cre

(9)

M_p = Z_f F_y

(10)

For M_cre > 2.78 M_y

$$M_{ne}=M_p-(M_p-M_y) {\displaystyle \frac{\sqrt{M_y/M_{cre}}-0.23}{0.37}}<=M_p$$

(11)

For M_cre <= 2.78 M_y

M_ne = S_f F_n

(12)

2.1.2. Local Buckling Combined with Yielding and Global Buckling

For members without holes and having flat webs:

M_crl = Critical Elastic Buckling moment

M_crl = 342 S_f (SI units)

(13)

$${\lambda }_l=\sqrt{{\displaystyle \frac{M_{ne}}{M_{crl}}}}$$

(14)

For λ_l < 0.776,

M_nl = M_ne

(15)

For λ_l > 0.776,

$$M_{nl}=[1-0.15(M_{crl}/M_{ne}\left)0.4\right](M_{crl}/M_{ne})0.4M_{ne}$$

(16)

Revised λ_l for consideration of inelastic reserve capacity,

$${\lambda }_l=\sqrt{{\displaystyle \frac{M_y}{M_{crl}}}}$$

(17)

C_yl = Z_f . F

(18)

M_nl may be revised to include the inelastic reserve capacity for sections symmetric about the axis of bending if λ_l <= 0.776 and M_ne >= M_y, as follows:

M_nl = M_y + (1 − 1/C²_yl)(M_p − M_y)

(19)

2.1.3. Distortional Buckling (AISI)

If the flange has full or partial rotational restraint provided by a brace, panel, or sheathing, the rotational stiffness (K_ɸ) may be added to the solution to employ a more detailed method mentioned in the AISI Commentary Section 2.3.3.3. The simplified method for unrestrained C and Z sections with simple lip stiffeners was used for the present study.

Outer depth, h_o = h + t;

Outer width, b_o = b + t/2 (without lips) and b + t (with lips).

This method is applicable under the following limitations:

50 ≤ h_o/t ≤ 200;
  25 <= b_o/t <= 100;
  6.25 <= D/t <= 50;
 45° <= θ <= 90°;
  2 <= h_o/b_o <= 8;
    0.04 <= DSin θ/b_o <= 0.5.

(20)

β = 1.0 for L_m >= L_cr

(21)

β = 1.0 <= 1 + 0.4 (L/L_m)^0.7 (1 + M1/M2)^0.7 <= 1.3

(22)

F_crd = β K_d {π²E/12(1 − v²)}(t/b_o)²

(23)

M_crd = S_f F_crd = Critical distortion buckling moment

(24)

$${\lambda }_d=\sqrt{{\displaystyle \frac{M_y}{M_{crd}}}}$$

(25)

For λ_d <= 0.673, M_nd = M_y

(26)

For λ_d > 0.673, M_nd = {1 − 0.22 (M_crd/M_y)^0.5} (M_crd/M_y)^0.5 M_y

(27)

Revised λ_d for consideration of in elastic reserve capacity,

$${\lambda }_d=\sqrt{{\displaystyle \frac{M_y}{M_{crd}}}}$$

(28)

$$C_{yd}=\sqrt{{\displaystyle \frac{0.673}{{\lambda }_d}}}<=0.3$$

(29)

M_p = Z_f F_y

(30)

M_nd may be revised to include the inelastic reserve capacity for sections symmetric about the axis of bending if λ_d <= 0.673, as follows:

M_nd = M_y + (1 − 1/C²_yd) (M_p − M_y)

(31)

2.2. CUFSM

CUFSM (Constrained and Unconstrained Finite Strip Method) is a powerful tool used for the analysis and design of cold-formed steel (CFS) structures. Developed by Ben Schafer in 1997, CUFSM is widely recognized for its capabilities in evaluating the stability and performance of thin-walled structural members, which are critical components in modern construction. CUFSM is primarily used to performing following tasks:

• Analyzing the local, distortional, and global buckling behavior of thin-walled sections.
• Determining the elastic buckling loads and modes of CFS members.
• Aiding in the design and optimization of CFS sections by providing insights into their stability characteristics.

CUFSM utilizes the finite strip method (FSM), which is particularly efficient for analyzing prismatic (constant cross-section) members. FSM divides the structure into strips along its length, simplifying the calculation of buckling modes and loads.

2.3. ABAQUS

2.3.1. General

ABAQUS is a comprehensive and powerful suite of finite element analysis (FEA) software applications, widely used for simulating the physical behavior of structures under various conditions. The finite element (FE) software ABAQUS was employed to create a finite element model (FEM) of CFS (cold-formed steel) channel sections. This tool has been effectively used for linear and non-linear analysis in past. The FEM included the measured dimensions of the specimens, initial geometric imperfections, and non-linear material properties. The analysis of the FEM was carried out in two distinct steps. The first step was used for linear perturbation analysis to obtain the buckling modes (Model-1), whereas during the second step, the “general static” method was applied for calculation of the moment capacity (Model-2).

2.3.2. Geometry Modeling and Material Properties

Linear buckling analysis is performed as the first phase of numerical investigations. For this, eight AISI standard sections were modeled in ABAQUS, as per their geometric properties. The following are the key features of the materials and properties used for this study:

(1)

Material grade, Fy = 380 MPa (55 ksi);

(2)

Modulus of elasticity, E = 200 GPa (29,500 ksi);

(3)

Poison ratio = 0.3;

(4)

Unbraced length for torsional buckling = L/3.

The length for each investigated section is taken from AISI Chart-II-1. The loads are applied at the L/4 distance from each end. To control the torsional buckling, lateral supports are provided at a spacing of L/3, and to control the general torsional effects, the loads are applied at the shear center.

Two shell extruded sections are defined, one for the main channel section and the other for flat plate attached at the back of the channel.

2.3.3. Element Type and Mesh Size

The shell elements are chosen for both the main channel sections and for the plate attached at the back of channel. To initiate the analysis step, eight eigenvalues were requested, with 16 vectors and 300 maximum iterations.

In this model, typically, the S4R element was used for meshing because of its effectiveness in capturing the behavior of structures such as plates, shells, and other thin-walled members. The S4R designation indicates a four-node doubly curved thin or thick shell element with reduced integration and hourglass control.

To control mesh size sensitivity, initially, a coarser mesh size of 1 mm was provided, and finally, 0.25 mm was chosen for convergence with minimum computing effort. To perform the Riks analysis, a minimum of one eigenvalue is required; however, eight eigenvalues were used in this study because this number was sufficient to capture all buckling modes in relatively less time.

2.3.4. Boundary Conditions and Loading Procedure

For this research work, simply supported boundary conditions were applied at both ends relative to the major axis bending. Laterally, roller supports were provided at both ends and at the L/3 distance intermediately.

Two unit point loads were applied at the L/4 distance from the ends. To avoid the twisting moments, loads were applied at the shear center. To achieve this, a flat plate of 5 mm thickness was attached to the back of the main C-section, along the depth of the section. The width of this plate was set equal to the distance from the shear center to the outer edge of the web.

2.3.5. Initial Geometric Imperfections Modeling

After the successful completion of step 1, the first eigenvalue was noted. The same model (Model-1) was copied as Model-2, and the following changes were made to it:

The material properties were modified for step 2 (non-linear analysis), as shown below.

For the plastic material properties, the parameters include:

• Yield stress = 380 MPa (55 ksi); plastic strain = 0;
• Yield stress = 517 MPa (75 ksi); plastic strain = 0.2.

Then, the first eigenvalue noted in Model-1 was applied as a point load in Model-2. The boundary conditions were kept the same for the non-linear analysis; however, this step required the use of the modified Static-Riks method, thus mandating that the boundary conditions be assigned again.

The Static-Riks step in ABAQUS is specialized type of analysis step used for solving non-linear problems, particularly those involving instability phenomena such as buckling or snap-through behavior; it is also used for the post-buckling behavior of structures. This approach is also known as the arc-length method. The Static-Riks step is primarily used for buckling analysis to capture the post-buckling behavior of structures.

In the next step, the Model-2 input file keywords were modified with some additional text to capture the first mode behavior of Model-1. The following texts were added by using Model-2’s Edit Keywords command at a specified location.

*IMPERFACTION, FILE = JOB-1, STEP = 1

1, 0.6

Then, the new job was created and submitted for analysis. Finally, the results were monitored to determine the critical buckling load.

2.3.6. Convergence Criterion

In linear buckling analysis, ABAQUS determines the critical load factors and the associated buckling modes. The process involves performing a linear eigenvalue analysis to find the load multiplier at which buckling occurs. Since this is a linear analysis, there are no iterative convergence criteria such as those used in non-linear analysis.

For non-linear buckling analysis using the Static-Riks method, automatic incrementation is used, and the arc length increments are provided by fixing the minimum and maximum limits for the estimated total arc length to achieve convergence. The values used in the present analysis are as follows:

• Maximum number of increments = 500;
• Arc length increments—initial = 0.01, minimum = 1 × 10⁻⁵, maximum = 1 × 10³⁶;
• Estimated total arc length = 1.

Automatic incrementation allows ABAQUS to dynamically adjust the arc length increment based on the convergence behavior during the analysis. This method provides greater flexibility and can handle complex non-linearities in a better way.

2.4. Sections Selected for the Study

A typical pre-engineered steel structure building partially utilizing cold formed sections is shown in Figure 1. The sections shown in

Table 1. Standard sections and corresponding investigated lengths.

S/No	Standard Section Designation SI (FPS)	Length (L) from AISI Chart-II-1 mm (Inches)	Braced Length at L/3 mm (Inches)
1	203CS64 × 2.7	1600.0 (63.0)	533.34 (21.00)
	(8CS2.5 × 105)
2	203CS64 × 2.2	1930.4 (76.0)	643.47 (25.34)
	(8CS2.5 × 085)
3	203CS64 × 1.8	2082.8 (82.0)	694.27 (27.34)
	(8CS2.5 × 070)
4	203CS64 × 1.7	2082.8 (82.0)	694.27 (27.34)
	(8CS2.5 × 065)
5	152CS64 × 2.7	1397.0 (55.0)	465.67 (18.34)
	(6CS2.5 × 105)
6	152CS64 × 2.2	1701.8 (67.0)	567.27 (22.34)
	(6CS2.5 × 085)
7	152CS64 × 1.8	1790.7 (70.5)	596.90 (23.50)
	(6CS2.5 × 070)
8	152CS64 × 1.6	1778.0 (70.0)	592.67 (23.34)
	(6CS2.5 × 065)

were selected to evaluate the buckling strengths. The lengths were taken from AISI Chart-II-1. The stress-strain curve for the steel used in this study is given in Figure 2 and the selected sections are graphically shown in Figure 3. These sections are standard AISI sections, while the thicknesses are selected to reflect those used in the local fabrication market of Pakistan.

3. Results

3.1. Analytical

The DSM results were obtained using the expressions of Section 2.1, employing Excel sheets. The results are presented in

Table 2. DSM flexural capacities of selected sections.

S/No	Section	Mcre kN-m (kip-in)	Mcrl kN-mm (kip-in)	Mcrd kN-m (kip-in)		Mne kN-m (kip-in)	Mnl kN-m (kip-in)	Mnd kN-m (kip-in)
				Detailed Method	Approx. Method
1	203CS64 × 2.7	22.32	20.03	39.38	22.68	21.98	18.32	20.98
	(8CS2.5 × 105)	(197.50)	(177.30)	(348.50)	(200.74)	(194.50)	(162.10)	(185.70)
2	203CS64 × 2.2	18.13	16.34	24.11	13.47	18.15	14.90	15.62
	(8CS2.5 × 085)	(160.40)	(144.60)	(213.40)	(119.20)	(160.60)	(131.90)	(138.20)
3	203CS64 × 1.8	15.01	13.50	15.48	8.38	15.01	12.31	11.83
	(8CS2.5 × 070)	(132.80)	(119.50)	(137.00)	(74.20)	(132.80)	(108.90)	(104.70)
4	203CS64 × 1.7	13.94	12.55	13.09	7.01	13.94	11.47	10.63
	(8CS2.5 × 065)	(123.40)	(111.10)	(115.80)	(62.00)	(123.40)	(101.30)	(94.10)
5	152CS64 × 2.7	15.14	13.63	31.87	18.83	15.14	12.43	14.95
	(6CS2.5 × 105)	(134.00)	(120.60)	(282.00)	(166.60)	(134.00)	(110.00)	(132.30)
6	152CS64 × 2.2	12.35	11.11	19.70	11.21	12.35	10.14	11.27
	(6CS2.5 × 085)	(109.30)	(98.30)	(174.30)	(99.20)	(109.30)	(89.70)	(99.70)
7	152CS64 × 1.8	10.23	92.10	12.74	6.99	10.23	8.40	8.61
	(6CS2.5 × 070)	(90.50)	(81.50)	(112.70)	(61.90)	(90.5)	(74.30)	(76.2)
8	152CS64 × 1.6	9.51	8.57	10.79	5.84	9.51	7.81	7.76
	(6CS2.5 × 065)	(84.20)	(75.80)	(95.50)	(51.70)	(84.20)	(69.10)	(68.70)

Bold values indicate the governing or minimum global, local, and distortional buckling strengths.

.

3.2. CUFSM

Linear buckling analysis is performed for the selected eight sections using CUFSM software, and the results are tabulated in

Table 3. CUFSM flexural capacities of selected sections.

S/No	Section	Mcre kN-m (Kip-in)	Unbraced Length mm (Inches)	Mcrl kN-m (Kip-in)	Lcrl for (Mcrl)min mm (Inches)	Mcrd kN-m (Kip-in)	Lcrd for (Mcrd)min mm (Inches)
1	203CS64 × 2.7	36.78	1600.0	58.90	101.6	44.83	533.40
	(8CS2.5 × 105)	(325.50)	(63.0)	(521.14)	(4.0)	(396.70)	(21.00)
2	203CS64 × 2.2	19.84	1930.4	30.18	101.6	26.52	508.00
	(8CS2.5 × 085)	(175.60)	(76.0)	(267.10)	(4.0)	(234.70)	(20.00)
3	203CS64 × 1.8	13.96	2082.8	16.87	101.6	17.45	694.40
	(8CS2.5 × 070)	(123.50)	(82.0)	(149.30)	(4.0)	(154.40)	(27.34)
4	203CS64 × 1.7	13.34	2082.8	13.61	101.6	14.53	694.40
	(8CS2.5 × 065)	(118.00)	(82.0)	(120.40)	(4.0)	(128.60)	(27.34)
5	152CS64 × 2.7	31.30	1397.0	57.32	76.2	35.02	465.80
	(6CS2.5 × 105)	(277.00)	(55.0)	(507.30)	(3.0)	(309.90)	(18.34)
6	152CS64 × 2.2	17.65	1701.8	31.14	76.2	21.79	508.00
	(6CS2.5 × 085)	(156.20)	(67.0)	(275.60)	(3.0)	(192.80)	(20.00)
7	152CS64 × 1.8	12.90	1778.0	17.56	76.2	14.13	592.80
	(6CS2.5 × 070)	(114.10)	(70.0)	(155.40)	(3.0)	(125.00)	(23.34)
8	152CS64 × 1.6	12.43	1778.0	14.75	76.2	12.37	592.80
	(6CS2.5 × 065)	(110.00)	(70.0)	(130.50)	(3.0)	(109.50)	(23.34)

:

3.3. ABAQUS

In ABAQUS, a Load Proportionality Factor (LPF) vs. Arc Length graph is a crucial post-processing tool used in non-linear analysis, particularly for studying the stability and post-buckling behavior of structures. This graph is generated when performing analyses such as the Riks analysis (arc-length method), which is used to trace the equilibrium path of structures through points of instability.

The LPF is a scalar multiplier that scales the applied loads. An LPF of 1 means that the full applied load is active, while values greater than 1 indicate loads greater than the initial applied loads, and values less than 1 indicate partial loading. It helps to understand how the structure responds to different levels of loading and to identify the maximum load-carrying capacity.

Arc length is a measure of the progress of the analysis along the equilibrium path. It is not a physical length but a parameter that controls the progress of the load and displacement in non-linear analyses. It allows the analysis to continue through points where the structure exhibits snap-through or snap-back behavior, ensuring that a complete load-displacement path is traced. At a later stage, the arc length may be scaled to any displacement. The load-deflection curves obtained, along with the failure mode for all samples, are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.

Following are the salient features of these plots:

• X-axis—This represents the vertical deflection under the load point near the roller support. Similarly, LPF may be translated into the load at any stage.
• Y-axis (Load Proportionality Factor)—This represents the scaling of the applied load.
• Ascending branch—This indicates that the structure is carrying an increasing load, without significant deformation. This typically corresponds to the elastic behavior of the structure.
• Peak point—The highest point on the plot represents the maximum load-carrying capacity of the structure. Beyond this point, the structure may undergo buckling or failure.
• Descending branch—If the plot shows a descending branch after the peak, it indicates post-buckling or post-failure behavior. The structure might still carry the load but with significant deformations.
• Post-buckling strength—After very large deformations, some extra post-buckling strength is obtained due to intermediate falsely stable configurations, and the load–deflection curve moves slightly upwards [26,27].

After gathering the LPF data for each investigated section, the nominal moment capacities are calculated and tabulated, as shown in

Table 4. ABAQUS flexural capacities of selected sections.

S/No	Section	Mn N-mm (Kip-in)
1	203CS64 × 2.7	19,413,400
	(8CS2.5 × 105)	(171.8)
2	203CS64 × 2.2	14,546,490
	(8CS2.5 × 085)	(128.7)
3	203CS64 × 1.8	11,243,500
	(8CS2.5 × 070)	(99.5)
4	203CS64 × 1.7	11,537,300
	(8CS2.5 × 065)	(102.1)
5	152CS64 × 2.7	14,452,700
	(6CS2.5 × 105)	(127.9)
6	152CS64 × 2.2	11,492,100
	(6CS2.5 × 085)	(101.7)
7	152CS64 × 1.8	8,904,400
	(6CS2.5 × 070)	(78.8)
8	152CS64 × 1.6	7,797,000
	(6CS2.5 × 065)	(69.0)

.

3.4. Comparison

The AISI chart, the AISI DSM method, CUFSM, and ABAQUS nominal strengths for non-linear buckling are compared; the evaluation is shown in

Table 5. Comparison between the AISI chart, AISI DSM, CUFSM, and ABAQUS.

S/No	Section	AISI CHART	DSM		ABAQUS		CUFSM
		Mn kN-m (kip-in)	Mn kN-m (kip-in)	Percentage Difference with AISI CHART (%)	Mn kN-m (kip-in)	Percentage Difference with AISI CHART (%)	Mn kN-m (kip-in)	Percentage Difference with AISI CHART (%)
1	203CS64 × 2.7	19.89	18.32	−8.57	19.41	−2.47	21.76	+9.44
	(8CS2.5 × 105)	(176.00)	(162.10)		(171.80)		(192.61)
2	203CS64 × 2.2	14.80	14.90	+0.68	14.55	−1.72	16.11	+8.80
	(8CS2.5 × 085)	(131.00)	(131.90)		(128.73)		(142.53)
3	203CS64 × 1.8	11.27	11.83	+4.97	11.24	−0.27	12.34	+9.54
	(8CS2.5 × 070)	(99.70)	(104.70)		(99.50)		(109.21)
4	203CS64 × 1.7	10.12	10.63	+5.04	11.54	+14.03	11.04	+9.04
	(8CS2.5 × 065)	(89.60)	(94.10)		(102.10)		(97.7)
5	152CS64 × 2.7	14.13	12.43	−13.68	14.45	+2.26	14.88	+5.35
	(6CS2.5 × 105)	(125.00)	(110.00)		(127.90)		(131.69)
6	152CS64 × 2.2	10.66	10.14	−5.13	11.49	+7.79	11.61	+8.94
	(6CS2.5 × 085)	(94.30)	(89.70)		(101.70)		(102.73)
7	152CS64 × 1.8	8.16	8.40	+2.89	8.90	+9.07	8.91	+9.22
	(6CS2.5 × 070)	(72.20)	(74.30)		(78.80)		(78.86)
8	152CS64 × 1.6	7.36	7.76	+5.43	7.80	+5.98	8.13	+10.49
	(6CS2.5 × 065)	(65.10)	(68.70)		(69.00)		(71.93)

.

The results of Table 5 are graphically shown in Figure 12. The comments regarding the results shown in Table 5 are given below:

• The DSM results are in relatively good agreement with the AISI chart, with percentage differences ranging from −8.57% to +5.43%. The observed differences are likely due to the basic approximations in the DSM for predicting the strength of cold-formed steel sections.
• ABAQUS provides a more detailed and potentially more accurate prediction of moment capacities. It is particularly useful for detailed analyses in which capturing complex behaviors is essential. In this study, the AISI chart values are closer to the ABAQUS results, with differences mainly within ±10%. The observed range of −2.47% to +14.03% shows that AISI charts can accurately predict the complex stress–strain behavior of steel sections.
• CUFSM tends to predict higher moment capacities, with percentage differences ranging from +5.35% to +10.49%. This consistent overestimation suggests that AISI might have employed conservative assumptions for design purposes.

Figure 12. Comparison of nominal flexural capacities predicted by AISI charts, DSM, ABAQUS, and CUFSM.

Figure 12. Comparison of nominal flexural capacities predicted by AISI charts, DSM, ABAQUS, and CUFSM.

4. Conclusions

In this paper, a comparative study is conducted on eight selected CFS channel members. To control torsion, loads are applied at the shear center of each respective section, and to control torsional buckling, bracing is provided at a regular interval of L/3. The four-point bending method is adopted.

Manual calculations are performed on the basis of the AISI DSM method to capture the non-linear buckling behavior. CUFSM is used to perform linear analysis, then these results are used to perform a non-linear analysis via DSM to identify the failure strengths. In ABAQUS, models are developed to perform a non-linear FEM analysis.

It is observed that the governing failure modes for all the eight specimens were local, distortional, and a combination of these modes. The results acquired by all three methods mentioned are compared with the AISI charts, and the following conclusions can be drawn:

• The AISI charts can be used by designers with confidence for practical designs because of their conservative nature, in most cases. However, to achieve economy and for speedy design, the CUFSM software may be employed. For detailed designing, when sufficient time is available, ABAQUS can provide economical, safe, and reliable solutions.
• DSM is a scientifically sound method that aligns closely with the AISI chart, providing reliable predictions, with only minor discrepancies.
• The ABAQUS finite element analysis results show a small to moderate deviation from the AISI charts, with percentage differences ranging from −2.47% to +14.03%.
• CUFSM consistently predicts higher moment capacities compared to those in the AISI charts, with positive percentage differences ranging from +5.35% to +10.49%.
• The difference between the DSM and ABAQUS flexural strength results with respect to the AISI charts varies on both the plus and minus sides.
• In the selected sections, as the thickness decreases, the results from all the four approaches come closer to each other. This means that the thin wall behavior is captured well by all of these methods.
• It is to be noted that for the selected sections and their specific unbraced lengths, AISI charts predict higher strengths for three out of eight sections when compared with the ABAQUS results, with a maximum difference of 2.47%.
• The DSM values vary from those in the AISI charts on both the plus and minus sides, with a maximum difference of 13.68% and the charts being on the unsafe side.