Seismic Design and Ductility Evaluation of Thin-Walled Stiffened Steel Square Box Columns

Abstract

This paper investigates the seismic performance of thin-walled stiffened steel square box columns, modeling bridge piers subjected to unidirectional cyclic lateral loading with a constant axial load, focusing on local, global, and local-global interactive buckling phenomena. Initially, the finite element model was validated against existing experimental results. The study further explored the degradation in strength and ductility of both thin-walled and compact columns under cyclic loading. Thin-walled, stiffened steel square box columns exhibited buckling near the base, forming a half-sine wave shape. The research also addresses discrepancies from different material models used to analyze steel tubular bridge piers. Analysis using a modified two-surface plasticity model (2SM) yielded results closer to experimental data than a multi-linear kinematic hardening model, particularly for compact sections. The 2SM, which accounts for cycling within the yield plateau and strain hardening regime, demonstrated enhanced accuracy over the multi-linear kinematic hardening model. Additionally, a parametric study was conducted to assess the impact of key design parameters—such as width-to-thickness ratio (R_f), column slenderness ratio (λ), and magnitude of axial load (P/P_y)—on the performance of thin-walled stiffened steel square box columns. Design equations were then developed to predict the strength and ductility of bridge piers. These equations closely matched experimental results, achieving an accuracy of 95% for ultimate strength and 97% for ductility.

1. Introduction

Thin-walled steel box columns, commonly used as bridge piers, are vital to urban transportation infrastructure. Structural hollow sections offer several advantages over open sections, such as enhanced aesthetic appeal, excellent flexural resistance about all axes, and superior torsional rigidity. These features make them ideal for various applications, including columns in buildings, members in lattice girders, and bridge piers.

Typically produced through cold-forming techniques [1,2,3], thin-walled steel box columns are created by welding steel sheets longitudinally to form the final section profile without prior heating or subsequent heat treatment. This method not only improves structural performance but also provides an economical solution for withstanding instability due to severe environmental loads, such as seismic forces [4,5,6].

Research on the local [7,8,9,10,11,12,13] and global [14,15,16,17,18,19,20,21,22] buckling of cold-formed hollow sections is extensive. While some studies have explored the interaction between local and global buckling in welded box sections [23,24,25], research on cold-formed tubular members with slender cross-sections remains limited [26,27,28,29,30].

Steel bridge piers with box sections are generally designed as single cantilever columns or one- to three-story frames. Highway bridge systems frequently utilize thin-walled members with closed-box cross-sections due to their high strength and torsional rigidity. These structures differ markedly from building columns, often failing due to local buckling in thin-walled members, irregular distribution of story mass and stiffness, strong beams and weak columns, low rise (one to three stories), and the need for residual displacement evaluations. Consequently, thin-walled steel box columns in bridge piers are particularly susceptible to damage from local and overall interaction buckling during severe earthquakes [31,32,33,34].

For example, Figure 1 depicts the performance of thin-walled steel box section bridge piers during the Hyogoken-Nanbu earthquake near Kobe City, Japan, on 17 January 1995. The steel bridge piers with rectangular box sections experienced collapse, weld seam tear-outs, and severe local buckling damage, as shown in Figure 1 [32,33]. Local buckling was observed near the base of the column (Figure 1c).

Post-Kobe earthquake research has increasingly focused on steel tubular columns with various cross-sections subjected to constant axial loads and cyclic lateral loads in one or multiple directions. To enhance the seismic capacity of steel bridge piers with tubular box sections, researchers have investigated both experimentally and theoretically the stability and plastic ductility of steel tubular columns [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52]. The strength of these tubes is influenced by local buckling of the tube walls, which depends on the slenderness of the plate elements forming the tube. Local buckling patterns in rectangular or square bare steel tubes can include inward and outward buckles (see Figure 1c), which are considered in major steel design specifications.

Thin-walled steel tubular columns typically experience local buckling near the base at heights corresponding to the diameter and width of circular and square thin-walled columns, respectively [2,32,33,34,35,36]. Research indicates that the ability of these piers to withstand severe earthquakes depends heavily on their strength and ductility [2,15].To improve seismic performance, studies have explored incorporating stiffeners and diaphragms to enhance both strength and ductility [2,6,39,40].

In response to the damage observed in steel bridge piers during the Kobe earthquake, a critical consideration in the seismic design and retrofit of steel tubular columns is to increase ductility while maintaining nearly unchanged ultimate strength [2,31]. Therefore, understanding the cyclic inelastic behavior of thin-walled steel tubular columns is essential for developing rational performance-based seismic design methodologies and evaluating ductility.

Finite element analysis (FEA) of thin-walled steel tubular columns has utilized both isotropic and kinematic hardening material models for cyclic analysis. The kinematic hardening model generally aligns better with test data compared to the isotropic hardening model [47,49,51]. However, it may overestimate strengths under cyclic loading [50]. To improve numerical accuracy, further investigation into additional material models, such as those implemented through ABAQUS-UMAT subroutines in FE ABAQUS 2023 software [52], is necessary. Recent advancements, particularly in Japan, have introduced alternative material models that better replicate test results, such as the fiber-based model with a failure segment [47].

FE analysis of cyclic behavior involves solving a system of non-linear equations derived from applied nodal forces and internal equivalent nodal forces [53,54,55]. The constitutive material model of structural steel is described using incremental plastic theory, which includes strength rules, yield criteria, flow rules, and criteria for plastic loading and unloading [56,57,58]. Accurate finite element methods are crucial for predicting the cyclic behavior of these structures. The modified two-surface plasticity model (2SM), developed by Mamaghani et al. [6,58,59], which accounts for cyclic strain hardening and the reduction of the yield plateau and elastic range (Bauschinger effect) due to cyclic loading, was employed in the commercial software ABAQUS 2023 (ABAQUS-2SM).

This study aims to thoroughly investigate the stability behavior of cold-formed hollow steel box sections subjected to cyclic lateral loads combined with axial compression, focusing on local, global, and local-global interactive buckling phenomena. Thin-walled, stiffened steel square box columns are susceptible to damage from local buckling, global buckling, or a combination of both. Local buckling is influenced by the width-to-thickness ratio (R_f) of the flange plate, while flexural buckling is determined by the column’s slenderness ratio (λ) [2]. Extreme seismic loads can cause significant damage or collapse due to these buckling modes [32,33]. Large earthquakes generate substantial cyclic lateral loads, necessitating precise design parameters. This study employs ABAQUS-2SM to validate the inelastic cyclic behavior of thin-walled, stiffened steel square box columns [45]. A parametric analysis assessed key design parameters, including steel material modeling, diaphragms, stiffeners, the width-thickness ratio for the flange (R_f), and the column slenderness ratio (λ). The study developed interaction equations for these parameters, facilitating a safe and cost-effective design approach for thin-walled, stiffened steel square box columns used in bridge piers.

2. Characteristics of Thin-Walled Stiffened Steel Square Box Columns

Thin-walled stiffened steel square box columns modeling bridge piers are designed by monitoring specific ranges for various parameters: the width-to-thickness ratio for the flange (R_f), the slenderness parameter (λ), and the axial force parameter (P/P_y). Where P_y is the squash load of the column. The following equations describe these parameters [44]:

$$R_f={\displaystyle \frac{d}{t}}{\displaystyle \frac{1}{n\pi }}\sqrt{3\left(1-v^2\right)}\sqrt{{\displaystyle \frac{{\sigma }_y}{E}}}$$

(1)

$$\lambda ={\displaystyle \frac{2h}{r}}{\displaystyle \frac{1}{\pi }}\sqrt{{\displaystyle \frac{{\sigma }_y}{E}}}$$

(2)

where, d = column depth (square section size), h = column height, t = plate thickness, v = Poisson’s ratio, E = modulus of elasticity, σ_y = yield strength, n = the number of sub-panels separated by the stiffeners [44,47].

A series of experiments on thin-walled, stiffened steel square box columns has been conducted by [44,48]. Based on these experiments, the practical values for R_f for the thin-walled stiffened steel square box columns have been proposed in the range of 0.2 to 0.5, with the axial load not exceeding 20% of the yield load [39,47]. A schematic diagram of the thin-walled, stiffened steel square box column is illustrated in Figure 2a,c.

3. Finite Element Analysis

In this study, the results from two tested columns available in the literature—the TPRC9-9 column [43] and the B1 column [3]—were used to validate the finite element modeling with the 2SM model for material nonlinearity [58]. The modeling was conducted using ABAQUS [52] employing the 2SM for material nonlinearity [58], hereafter referred to as ABAQUS-2SM.

As shown in Figure 2c, the column features eight longitudinal stiffeners for the TPRC9-9 column and twelve longitudinal stiffeners for the B1 column, each 6 mm thick and 60 mm wide, and two diaphragms, spaced 450 mm apart and 6 mm thick. The cross-sectional plate thickness is 9 mm for both the flange and the web.

Table 1. Geometric and material parameters for the specimens.

Column	h (mm)	d (mm)	E (MPa)	v	${\mathit{\sigma }}_{\mathit{y}}$ (MPa)	${\mathit{R}}_{\mathit{f}}$	$\mathit{\lambda }$	P/Py
TPRC9-9	2141	449	204,000	0.26	266.0	0.30	0.29	0.150
B1	3403	900	206,000	0.30	378.6	0.56	0.26	0.122

lists the material and geometric properties of the analyzed specimens.

3.1. Brief Description of the 2SM Model

The 2SM model can more effectively capture the cyclic behavior of structural steel compared to existing material models, as it incorporates the yield plateau and explicitly accounts for its reduction and eventual disappearance under cyclic loading [6,58]. A detailed description of the 2SM and its application to simulating the cyclic behavior of structural steel is provided by Shen et al. [58]. This model requires more constants than the multi-linear kinematic hardening (MKH) model, totaling 18 parameters: 5 material constants (σ_y, E, ν, ε_st^p, and E_st) and 13 model parameters [6,58]. Here, ε_st^p represents the plastic strain, and E_st denotes the initial hardening modulus at the end of the yield plateau. The 13 model parameters were calibrated using experimental data from a simple load test on a steel section [6,58].

The cyclic behavior of structural steel with a yield plateau influences the constitutive model based on the principles of time-independent plasticity theory. This model employs a Von Mises yield surface in conjunction with an associative flow rule [58]. The flow rule is described by the following equations:

$$f={\left[{\displaystyle \frac{3}{2}}\left(\zeta -\mathsf{\eta }\right):\left(\zeta -\mathsf{\eta }\right)\right]}^{{\displaystyle \frac{1}{2}}}-\kappa \le 0$$

(3)

$$k=R+\rho$$

(4)

$$d{\epsilon }^p=\xi {\displaystyle \frac{\partial f}{\partial \sigma }}={\displaystyle \frac{3}{2}}\xi {\displaystyle \frac{\zeta -\mathsf{\eta }}{{\rm K}}}$$

(5)

$$dp={\left[{\displaystyle \frac{2}{3}}d{\epsilon }^p:d{\epsilon }^p\right]}^{1/2}=\xi$$

(6)

In these equations, bold letters represent second-order tensors, ε^p represents the plastic strain tensor, σ represents the stress tensor, ζ is the stress deviator, ƞ is the back stress, ρ is the initial size of the yield surface, k is the size of the yield surface, R is the isotropic hardening/softening variable, ξ is the plastic multiplier, p is the accumulated plastic strain, and the double dot represents the inner product [6,58]. Although the stiffeners and diaphragms have different geometric properties, they share the same material properties as the column’s component plate. The 2SM model is also applied to these components since they are integral to the column and affect its buckling behavior.

3.2. Loading Description

Strength and ductility analysis of bridge piers typically involves evaluating their performance under various loading conditions, including axial, monotonic, and cyclic loads. Axial loading applies a force along the axis of the bridge pier, simulating dead loads, live loads, and other service loads within the bridge [54]. Monotonic lateral loading involves the gradual application of force or displacement in a single lateral direction without reversal.

Cyclic lateral loading, which represents the horizontal component of seismic excitation, can be either force-controlled or displacement-controlled. This loading involves increasing the lateral load or displacement by integer multiples of the initial yield load or displacement in alternating directions. Experimentally, cyclic loading is commonly applied using a push-pull hydraulic jack [39]. In this study, a unidirectional displacement-controlled cyclic loading protocol is used, as illustrated in Figure 2d. The solid line represents one-cycle loading (N = 1), while the dashed line denotes three-cycle loading (N = 3). During the loading history, a quasi-static cyclic lateral displacement (δ) is applied to the top of the column, accompanied by a constant axial force (P), see Figure 2a. The amplitude of the cyclic displacement is incrementally increased in multiples of the yield displacement (δ_y), which is defined by Equation (7).

$${\delta }_y={\displaystyle \frac{H_y{h}^3}{3EI}}$$

(7)

$$H_y=\left({\sigma }_y-{\displaystyle \frac{P}{A}}\right){\displaystyle \frac{s}{h}}$$

(8)

Equation (8) is used to calculate the lateral yield load (H_y), where A, h, E, I, and S represent the cross-sectional area, height, Young’s modulus, the moment of inertia of the cross-section, and the elastic section modulus of the column, respectively [6,44].

3.3. FE Meshing

In thin-walled, stiffened steel square box columns, excessive local deformation can inhibit stress redistribution. Test results indicate that local buckling typically occurs near the column base over a region approximately equal to the cross-sectional size (d) [3,44,46]. To address this, a two-node beam element (B31) is used for the upper segment of the column, with a height of h − 3d, while reduced integration four-node shell elements (S4R), which effectively capture localized deformation, are applied to the lower segment, with a height of 3d, as shown in Figure 2b. The interface between the shell (S4R) and beam (B31) elements is modeled using a multi-point constraint (MPC). The elements used are available in the Abaqus/Standard library [52].

For computational efficiency and accuracy, the bottom and middle portions of the lower segment of the column (equal to the side width, d) are, respectively, divided into 30 and 20 S4R elements, while the remaining height (d) is divided into 15 elements, with an additional 20 elements used in the width direction for the lower segment, see Figure 2b. The upper segment of the column, with a height of h − 3d, is divided into beam elements (B31), each sized at 90 mm. Finally, each longitudinal stiffener and subpanel between the stiffeners, respectively, have 3 and 6 columns of S4R. These mesh sizes were optimized through trial and error to ensure efficient and accurate results. A default of five integration points through the thickness of the S4R element is used. The analysis employs a displacement convergence criterion with a tolerance of 10⁻⁵. Initial geometric imperfections and residual stresses are not considered, as these were not quantified in the tested columns [3,43]. Additionally, these factors are believed to have a negligible influence on the overall cyclic behavior after the first half-cycle [6,31]. Further details on elastoplastic large displacement analyses are provided in [6].

4. Numerical Results

4.1. FE Model Validation for Column TPRC9-9

Finite element (FE) analysis was performed using both the multi-linear kinematic hardening (MKH) model and the 2SM model, and the results were compared with experimental data for the column TPRC9-9 by Fukumoto et al. [43]. Figure 3a,b compare the test and analysis results for the normalized lateral load versus lateral displacement hysteresis loops and envelope curve, respectively, for the column TPRC9-9. Deformation visualizations are shown in Figure 4a,b, where the buckled shape near the base displays a half-sine wave formation. The ABAQUS-2SM more accurately predicted the strength values compared to the ABAQUS-MKH in the post-buckling range. For thin-walled, stiffened square box columns, the maximum strength predictions were similar for both MKH and 2SM models, but the accuracy diverged beyond this point (in the post-buckling regime). This discrepancy is attributed to the 2SM model’s superior ability to predict the cyclic elastoplastic behavior of structural steel [58] used in the analysis.

4.2. Model Validation for Column B1

Column B1, as reported in the literature [3], was also used to validate the 2SM model against experimental results. The comparison of results from the ABAQUS-2SM model and the multi-linear kinematic hardening (ABAQUS-MKH) model is shown in Figure 5a,b. The ABAQUS-MKH model predicted higher strength values than both the experimental data and the ABAQUS-2SM model. However, the ABAQUS-2SM model provided strength values closer to the experimental results compared to the ABAQUS-MKH model. Figure 6a,b show the comparison of buckling shapes for the two material models. The lower portion of the stiffener exhibited a buckling shape resembling a half-sine wave, while the plates buckled alternately inwards and outwards.

4.3. Comparison of Material Model Performance

To evaluate the performance of the 2SM model, further analysis was conducted using a column with compact section properties. The material and geometric properties of this compact section were similar to those of column B1, as detailed in Table 1, except that the plate thickness of the compact section is 25 mm instead of 9 mm. Figure 7a,b show that the MKH model predicted higher strength values for the compact section compared to the 2SM model. This observation suggests that the 2SM model yields more reliable results for FE analysis of compact sections. For the compact column under cyclic lateral loading, material nonlinearity predominates over geometric nonlinearity. Since the 2SM model accurately represents the material characteristics under cyclic loading, it provides a more reliable analysis.

5. Parametric Study

Using the ABAQUS-2SM, a parametric study was conducted on thin-walled, stiffened steel square box columns to investigate their overall interaction and the impact of key design parameters on their cyclic behavior. The parameters examined include the width-to-thickness ratio (R_f), the column slenderness ratio (λ), and the magnitude of axial load (P/P_y). Based on previous studies, the practical ranges for these parameters are: 0.2 ≤ R_f ≤ 0.5, 0.2 ≤ λ ≤ 0.5, and P/P_y ≤ 0.2 [39,47].

In this study, the size of diaphragms and stiffeners were kept constant at a thickness of 9 mm, while the web and flange thicknesses were calculated for forty columns by varying R_f and λ values. Details of the column properties used in the parametric study are listed in

Table 2. Proposed columns for the parametric study.

Column Name	Axial Load	Rf	λ	tf (mm)	Height (mm)
T-20-29-0.00	0.00	0.20	0.29	14.39	2269
T-30-29-0.00	0.00	0.30	0.29	9.00
T-40-29-0.00	0.00	0.40	0.29	7.20
T-50-29-0.00	0.00	0.50	0.29	6.00
T-20-29-0.05	0.05	0.20	0.29	14.39
T-30-29-0.05	0.05	0.30	0.29	9.00
T-40-29-0.05	0.05	0.40	0.29	7.20
T-50-29-0.05	0.05	0.50	0.29	6.00
T-20-29-0.10	0.10	0.20	0.29	14.39
T-30-29-0.10	0.10	0.30	0.29	9.00
T-40-29-0.10	0.10	0.40	0.29	7.20
T-50-29-0.10	0.10	0.50	0.29	6.00
T-20-29-0.15	0.15	0.20	0.29	14.39
T-30-29-0.15	0.15	0.30	0.29	9.00
T-40-29-0.15	0.15	0.40	0.29	7.20
T-50-29-0.15	0.15	0.50	0.29	6.00
T-20-29-0.20	0.20	0.20	0.29	14.39
T-30-29-0.20	0.20	0.30	0.29	9.00
T-40-29-0.20	0.20	0.40	0.29	7.20
T-50-29-0.20	0.20	0.50	0.29	6.00
T-30-29-0.00	0.00	0.30	0.20	9.00	1565
T-30-29-0.00	0.00	0.30	0.29	9.00	2269
T-30-29-0.00	0.00	0.30	0.50	9.00	3912
T-30-29-0.05	0.05	0.30	0.20	9.00	1565
T-30-29-0.05	0.05	0.30	0.29	9.00	2269
T-30-29-0.05	0.05	0.30	0.40	9.00	3130
T-30-29-0.05	0.05	0.30	0.50	9.00	3912
T-30-29-0.10	0.10	0.30	0.20	9.00	1565
T-30-29-0.10	0.10	0.30	0.29	9.00	2269
T-30-29-0.10	0.10	0.30	0.40	9.00	3130
T-30-29-0.10	0.10	0.30	0.50	9.00	3912
T-30-29-0.15	0.15	0.30	0.20	9.00	1565
T-30-29-0.15	0.15	0.30	0.29	9.00	2269
T-30-29-0.15	0.15	0.30	0.40	9.00	3130
T-30-29-0.15	0.15	0.30	0.50	9.00	3912
T-30-29-0.20	0.20	0.30	0.20	9.00	1565
T-30-29-0.20	0.20	0.30	0.29	9.00	2269
T-30-29-0.20	0.20	0.30	0.40	9.00	3130
T-30-29-0.20	0.20	0.30	0.50	9.00	3912

. The finite element analysis (FEA) results are summarized in

Table 3. Parametric study: strength and ductility evaluation.

Column	P/Py	tf (mm)	Rf	λ	(1 + P/Py). Rfλ	Rfλ	Hm/Hy	δm/δy	H0.95/Hy	δ0.95/δy
T-20-29-0.00	0.00	14.39	0.20	0.29	0.058	0.058	3.00	8.00	2.85	9.00
T-30-29-0.00	0.00	9.00	0.30	0.29	0.087	0.087	2.00	8.00	1.90	8.00
T-40-29-0.00	0.00	7.20	0.40	0.29	0.116	0.116	1.48	5.00	1.41	6.00
T-50-29-0.00	0.00	6.00	0.50	0.29	0.145	0.145	1.23	4.00	1.17	4.00
T-20-29-0.05	0.05	14.39	0.20	0.29	0.061	0.058	2.89	7.00	2.75	8.97
T-30-29-0.05	0.05	9.00	0.30	0.29	0.091	0.087	1.84	7.00	1.75	7.94
T-40-29-0.05	0.05	7.20	0.40	0.29	0.122	0.116	1.40	4.00	1.33	6.00
T-50-29-0.05	0.05	6.00	0.50	0.29	0.152	0.145	1.16	3.00	1.10	4.00
T-20-29-0.10	0.10	14.39	0.20	0.29	0.064	0.058	2.89	7.00	2.75	8.44
T-30-29-0.10	0.10	9.00	0.30	0.29	0.096	0.087	1.81	6.00	1.72	7.34
T-40-29-0.10	0.10	7.20	0.40	0.29	0.128	0.116	1.37	4.00	1.30	5.61
T-50-29-0.10	0.10	6.00	0.50	0.29	0.160	0.145	1.10	3.00	1.04	3.82
T-20-29-0.15	0.15	14.39	0.20	0.29	0.067	0.058	2.770	6.00	2.632	7.775
T-30-29-0.15	0.15	9.00	0.30	0.29	0.100	0.087	1.756	6.00	1.668	6.870
T-40-29-0.15	0.15	7.20	0.40	0.29	0.133	0.116	1.329	4.00	1.263	5.444
T-50-29-0.15	0.15	6.00	0.50	0.29	0.167	0.145	1.067	3.00	1.014	3.800
T-20-29-0.20	0.20	14.39	0.20	0.29	0.070	0.058	2.570	6.00	2.442	7.500
T-30-29-0.20	0.20	9.00	0.30	0.29	0.104	0.087	1.620	6.00	1.539	6.600
T-40-29-0.20	0.20	7.20	0.40	0.29	0.139	0.116	1.270	3.00	1.207	5.000
T-50-29-0.20	0.20	6.00	0.50	0.29	0.174	0.145	0.960	2.00	0.912	3.500
T-30-29-0.00	0.00	9.00	0.30	0.20	0.060	0.060	2.500	3.00	2.375	4.260
T-30-29-0.00	0.00	9.00	0.30	0.29	0.087	0.087	2.000	8.00	1.900	8.000
T-30-29-0.00	0.00	9.00	0.30	0.40	0.120	0.120	1.175	6.00	1.116	8.148
T-30-29-0.00	0.00	9.00	0.30	0.50	0.150	0.150	0.920	6.00	0.874	8.800
T-30-29-0.05	0.05	9.00	0.30	0.20	0.063	0.060	2.450	3.00	2.328	3.940
T-30-29-0.05	0.05	9.00	0.30	0.29	0.091	0.087	1.844	7.00	1.752	7.944
T-30-29-0.05	0.05	9.00	0.30	0.40	0.126	0.120	1.130	5.00	1.074	5.800
T-30-29-0.05	0.05	9.00	0.30	0.50	0.158	0.150	0.900	6.00	0.855	8.000
T-30-29-0.10	0.10	9.00	0.30	0.20	0.066	0.060	2.400	3.00	2.280	3.420
T-30-29-0.10	0.10	9.00	0.30	0.29	0.096	0.087	1.805	6.60	1.715	7.342
T-30-29-0.10	0.10	9.00	0.30	0.40	0.132	0.120	1.090	4.00	1.036	6.300
T-30-29-0.10	0.10	9.00	0.30	0.50	0.165	0.150	0.870	5.00	0.827	8.000
T-30-29-0.15	0.15	9.00	0.30	0.20	0.069	0.060	2.350	2.00	2.233	3.280
T-30-29-0.15	0.15	9.00	0.30	0.29	0.100	0.087	1.756	6.00	1.668	6.870
T-30-29-0.15	0.15	9.00	0.30	0.40	0.138	0.120	1.080	4.00	1.026	5.630
T-30-29-0.15	0.15	9.00	0.30	0.50	0.173	0.150	0.830	5.00	0.789	7.160
T-30-29-0.20	0.20	9.00	0.30	0.20	0.072	0.060	2.260	2.00	2.147	3.150
T-30-29-0.20	0.20	9.00	0.30	0.29	0.104	0.087	1.620	6.00	1.539	6.600
T-30-29-0.20	0.20	9.00	0.30	0.40	0.144	0.120	1.040	4.00	0.988	5.267
T-30-29-0.20	0.20	9.00	0.30	0.50	0.180	0.150	0.800	4.00	0.760	6.000

.

5.1. Effect of Width-to-Thickness Ratio Parameter (R_f)

The influence of the R_f parameter on the hysteretic behavior of the analyzed columns was examined. As R_f increased, the thickness of the column decreased. Although the width of the column also affects this parameter, it was kept constant for this study. With λ = 0.29, both the ultimate strength and maximum displacement decreased by 59% and 50%, respectively, as R_f increased from 0.2 to 0.5, as illustrated in Figure 8a,b.

5.2. Effect of Slenderness Ratio Parameter (λ)

The height of the column is directly proportional to the λ parameter. While the plate thickness of the column remained constant as the λ parameter varied from 0.2 to 0.5, with R_f and P/P_y held constant, the height changed from 1563 mm to 3907 mm due to this variation. As shown in Figure 9a,b, both ultimate strength and maximum displacement decreased by 54% and 25%, respectively, when the λ parameter increased from 0.2 to 0.5.

5.3. Effect of P/Py

Envelope curves of the proposed columns were examined under varying axial loads (0 ≤ P/P_y ≤ 0.2). As shown in Figure 10, the ratio H_m/H_y decreased with increasing axial load. Although ultimate strength and maximum displacement were only slightly affected as the axial load increased from 0 to 0.2P_y, lateral strength and lateral deformation were significantly reduced at higher axial loads (e.g., for P/P_y ≥ 0.30).

5.4. Post-Buckling Behavior of the Columns

The post-buckling behavior of the columns was analyzed, with Figure 11 illustrating the variation in maximum strength and deformation for each column. The analysis revealed that columns with lower values of R_f and P/P_y exhibited higher maximum lateral strength and lateral deformation. To accurately capture failure and assess critical strength and ductility, a failure criterion corresponding to 95% of the maximum strength was used to define the post-buckling behavior [39,44,46]. This threshold represents the failure point beyond which the column is considered damaged and unable to support additional load, requiring retrofitting or replacement. Post-buckling strength and deformation values decreased as R_f and P/P_y increased for all columns.

6. Proposed Design Equations

The ultimate strengths and ductility of 40 analyzed columns were determined through finite element (FE) analysis using the 2SM for material nonlinearity in ABAQUS. Figure 12 plots H_m/H_y against the integrated parameters (1 + P/P_y)⋅R_f⋅λ, considering the interaction between R_f, λ, and P/P_y on column strength. The best-fitting equation relating to the computed ultimate strength is given by:

$${\displaystyle \frac{H_m}{H_y}}={\displaystyle \frac{0.1402}{{\left[\left(1+\frac{P}{P_y}\right)R_f\lambda \right]}^{1.084}}}$$

(9)

As shown in Figure 12, the ultimate strength improved as the integrated parameters (1 + P/P_y)⋅R_f⋅λ, decreased. For this study, failure of the thin-walled stiffened steel box columns was considered to occur at a displacement equal to either δ_m or δ_0.95. Here, δ_m represents the maximum displacement corresponding to H_m/H_y, while δ_0.95 is the displacement at which post-peak strength drops to 95% of H_m/H_y. These normalized parameters are key to evaluating the ductile behavior of the columns. The δ_0.95/δ_y parameter was used to analyze ductility, as column strength significantly deteriorates after this point due to local buckling.

As shown in Figure 12, the ultimate strength improved as the integrated parameters (1 + P/P_y)⋅R_f⋅λ, decreased. For this study, failure of the thin-walled stiffened steel box columns was considered to occur at a displacement equal to either δ_m or δ_0.95. Here, δ_m represents the maximum displacement corresponding to H_m/H_y, while δ_0.95 is the displacement at which post-peak strength drops to 95% of H_m/H_y. These normalized parameters are key to evaluating the ductile behavior of the columns. The δ_0.95/δ_y parameter was used to analyze ductility, as column strength significantly deteriorates after this point due to local buckling.

Figure 13a,b illustrate a reduction in both maximum deformation and post-buckling deformation as (1 + P/P_y)⋅R_f⋅λ and R_f⋅λ increased. Additionally, ductility equations were proposed:

$${\displaystyle \frac{{\delta }_m}{{\delta }_y}}={\displaystyle \frac{0.5654}{{\left(R_f\lambda \right)}^{0.938}}}$$

(10)

$${\displaystyle \frac{{\delta }_{0.95}}{{\delta }_y}}={\displaystyle \frac{1.0877}{{\left[\left(1+\frac{P}{P_y}\right)R_f\lambda \right]}^{0.783}}}$$

(11)

Equations (10) and (11) were derived from the design parameters to determine maximum deformation and post-buckling deformation, respectively, within the ranges 0.2 ≤ R_f ≤ 0.50, 0.2 ≤ λ ≤ 0.50, and P/P_y ≤ 0.2. These limits for parameters are in the practical range for bridge design [44,46].

Application of the Proposed Design Equations

To verify the proposed design equations, two columns were designed and analyzed using ABAQUS-2SM: Column-1 with P/P_y = 0.15, R_f = 0.35, and λ = 0.25; and Column-2 with P/P_y = 0.18, R_f = 0.30, and λ = 0.30. The verification results are shown in

Table 4. Results for design verification.

Column		P/Py	Rf	λ	t (mm)	H (mm)	Hm/Hy	δm/δy	δ0.95/δy
1	Proposed Equation	0.15	0.35	0.25	8.220	1956	1.690	5.550	6.567
	FE Analysis						1.745	5.500	7.000
	Accuracy (%)						97.00	100.00	94.00
2	Proposed Equation	0.18	0.3	0.3	9.590	2348	1.600	5.411	6.296
	FE Analysis						1.750	5.200	6.100
	Accuracy (%)						92.00	96.10	97.00

. Figure 14a,b present the FE analysis results for Columns 1 and 2, respectively. The strength was predicted with 92% accuracy, while maximum deflection and post-buckling deflections were predicted with 94% accuracy. This study thus establishes practical equations for engineers to use in designing and constructing thin-walled, stiffened square box columns for bridge piers.

7. Conclusions

This study examined the cyclic behavior of prismatic thin-walled, stiffened steel square box columns, which model bridge piers, using finite element (FE) analysis with the commercial software ABAQUS, focusing on local, global, and local-global interactive buckling phenomena. The analysis employed the 2SM model to capture material nonlinearity under cyclic loading. The 2SM model offers a more accurate representation of the cyclic elastoplastic behavior of structural steel compared to existing material models in ABAQUS. It effectively incorporates cyclic effects within the yield plateau, accounting for their reduction and eventual disappearance, as well as the cyclic strain hardening of the material. Key design parameters investigated included the width-to-thickness ratio (R_f), the slenderness ratio (λ), and the axial load ratio (P/P_y). The study led to the following conclusions:

FE Analysis Accuracy: The FE analysis using ABAQUS and the 2SM model (implemented via a UMAT subroutine) accurately captured column behavior under cyclic loading, closely matching experimental results. The thin-walled columns showed minimal variation when the FE model included a multi-linear kinematic hardening model. Local buckling was observed in thin-walled steel tubes, with geometric attributes influencing cyclic behavior more than material non-linearities, especially for thin-walled sections susceptible to local buckling. Significant differences in strength and ductility were found in compact columns when using different material models, as the material nonlinearity governs the column behavior in the absence of local buckling.

Effect of Axial Loads: Axial loads up to 20% of the yield load (P_y) had a negligible effect on the ultimate strength of the columns. However, higher axial loads (exceeding 30% of P_y) significantly lowered buckling, post-buckling, and ductility of the impacted columns.

Parametric Study Results: The ultimate lateral strength of thin-walled, stiffened steel square box columns was achieved when lateral displacement reached 5δ_y. Beyond this point, strength decreased due to the local buckling of the component plates. Increasing both R_f and λ resulted in lower ultimate lateral strength and ductility.

Design Equations: New design equations were introduced to predict ultimate lateral strength, maximum lateral displacement, and post-buckling displacement. These equations offer a practical approach for designing thin-walled, stiffened steel square box columns for bridge piers.

Validation of Design Equations: To validate the proposed design equations, two columns were designed, and their ultimate lateral strength and displacement values were confirmed through FE analysis. The design equations achieved 95% accuracy in predicting ultimate lateral strength and 97% accuracy in predicting ultimate lateral displacements. The discrepancies between the FE predictions and experimental results were 5% for strength and 3% fortool ductility.