Steel Columns under Compression with Different Sizes of Square Hollow Cross-Sections, Lengths, and End Constraints

Abstract

This work presents several results of the stability in steel columns subject to pure compression. A square hollow cross-section with different sizes was considered. This study presents all the analytical equations that need to be used to verify the stability of each column with different lengths and boundary conditions. A finite element program was also chosen to achieve the most critical loads (Euler and buckling resistance loads) in the calculation for each element under study, using linear and nonlinear geometric and material modeling. Steel material was used for the columns, where damage due to plasticity was included, through plastic behavior with isotropic hardening. Comparing the results allows us to conclude that the use of the finite element method is an alternative methodology to be used in other types or configurations of columns, where parameterized tests can be easily implemented and to contribute to the development of a wide-ranging database. The finite element method led to an accurate solution when compared with the analytical results with a maximum deviation of 14.7%. By increasing the column length and reducing the cross-section size, the design buckling resistance of the studied columns also decreases. These studies demonstrate that the length and size of the column cross-section can meaningfully increase the structural behavior of the columns.

1. Introduction

The present study focuses on two different methodologies (analytical and numerical) to be used in the analysis of the stability of columns with different sizes, lengths, and end constraints.

Some studies have been used to assess the strength and deformation of columns as well as the failure modes [1] in mixed materials such as concrete filled with steel. These authors refer to the excellent interaction between steel and concrete, such as general application [2,3]. Other authors tested different solutions of buckling in steel columns [4] where the influence of slenderness ratio, cross-section, and boundary conditions have a great influence on the behavior. Other studies focus on the analysis of columns with hollow cross-sections under pure compression through testing and finite element modeling using hybrid or simple materials [5,6]. The main purpose was to obtain the bearing capacity which these columns support at different conditions. From the test results, some authors concluded that the axial compression for tubular columns is influenced by the material strength, section type, and section hollow rate [7].

Design methodologies have been developed in the past decades and employed in design standards, where Eurocode 3 is an example with different procedures that can be used for beam-columns analysis. This standard has the most advanced and precise beam-column formulae to be used [8].

According to the applications, circular, rectangular, and square cross-sections are commonly used in column construction, in mid- and high-rise buildings, bridges and piles, metro station columns, wind turbines, submarine pipelines, and other structures [9]. Some of them are concrete filled with steel tubes due to their high flexibility, stiffness, energy absorption, and seismic and fire resistance [9]. More recently, some studies have focused on analytical formulation to assess the axial capacity and compressive load of recycled concrete-filled steel tubular cross-sections [10].

Nevertheless, square and rectangular cross-sections are the most frequent in construction [11], due to the simpler design in column-beam connections applications, their considerable bending resistance, and aesthetic concerns. Recently, high-strength steel materials have also been used for these column types to define the axial load-carrying capacity, namely, in conjunction with the concrete to measure the compressive strength using the current design standards [12]. Also, welded thin-walled box sections have been studied because of their broad application in the industry, as they deal with economic and safe-scope projects [13].

Different approaches with numerical methodologies and experimental tests have been used to carry out typical applications on column buckling and to predict the behavior of these structures under applied loads in real situations [14,15]. For example, buckling on composite columns has been a studied topic due to the requirements needed in the construction of large-span structures or even high-rise buildings and seismic areas [16]. Studies have been related to the application of numerical solutions as a good methodology in the study of failure modes that can be observed in columns [14]. Other studies experimentally evaluate the local buckling failure of columns when submitted to different loads as bending [15], where the circular hollow cross-sections are chosen due to being the most efficient and useful flexible structures. Other investigations focus on the presence of internal voids and opening cross-sections to study the effect on the ultimate load resistance and the web post-buckling as the predominant failure mode in the column [15,17,18].

In this study, the author uses the ANSYS ^® 2022 R2 program, with finite solid elements, to calculate the bearing capacity of different columns widely used in construction focusing on the study of the effective length, different sizes of square hollow cross-sections, and different end constraints. This study is a wide large application in the design of different real columns in continuation with the previous work from the same author, recently published but only for one condition type [19].

In real-world situations and due to manufacturing tolerances and handling processes, the columns are rarely straight. These phenomena, designated as initial imperfections, can affect the critical buckling load, reducing it from the idealized values calculated mathematically, assuming a perfect geometry. This is a gap in some numerical investigations if this behavior is not considered. Therefore, the great significance of the present investigation is also to include the effect of initial imperfections of columns, including simultaneous nonlinearity to the plasticity damage of the material, and studying all these effects on the buckling resistance impact.

This work presents the mathematical background to study this buckling phenomenon to allow the comparison with all numerical results. In the function of the standards and using simplified equations, newly developed computational models will be available and allow us to compare the solutions.

The finite element method with shell elements will be used as an alternative numerical methodology to calculate the buckling resistance in different types of columns. Using shell finite elements, it is possible to explore the elastic and inelastic problem, also considering geometric initial imperfections. A linear and nonlinear material analysis with the influence of imperfections was conducted to study the structural stability in each column and identify the critical loads (Euler and the design buckling resistance loads).

2. Stability of Columns, Mathematical Formulation for Determining the Critical Loads

2.1. Euler Load

Columns submitted to an axial load have a limit on their resistance, identical to the material’s yield limit $f_y$ multiplied by the cross-section area $A$ [19,20], expressed by plastic load $N_{pl}$, in Equation (1).

$$N_{pl}=A f_y$$

(1)

Short columns fail with a critical buckling stress that can be larger than the material yield. In this zone, the material no longer performs by elasticity [20,21] and the buckling of this region is called inelastic [20,21].

The long columns will tend to fail with the application of lower loads by elastic buckling, reaching the critical buckling stress [20,21], and it is called elastic buckling [21,22] or Euler critical load $N_{cr}$. This critical load gives the value of the slenderness of a member in compression [22] (Equation (2)).

For the applied boundary conditions, an effective length $L_e=k L$ is needed to include a function of the column length L. For fixed-fixed end column k = 0.5, in fixed-pinned end column k = 0.7, pinned-pinned k = 1, and in fixed-free ends k = 2. The effective length is explained as the column length involving two zero-moment locations.

$$N_{cr}={\displaystyle \frac{{\pi }^2}{L_e^2}}E I$$

(2)

In Equation (2), E is the modulus of elasticity, and I the minimum moment of inertia. Figure 1 shows all the boundary conditions applied to the columns and the function of the effective length.

The Euler load $N_{cr}$ is the lowest or ideal quantity obtained when an ideal column is overloaded. For the elastic buckling problem, the material is linearly elastic and homogeneous, the columns are initially perfectly straight without imperfections or residual stresses, the applied load is centrally in the cross-section, and its restraints are such that only plane buckling in one direction is relevant.

In review, buckling loads are sensible to material properties, column dimension, cross-section, boundary conditions, and primary eccentricity.

The properties of steel S235 assumed in this work are the following [23]: Young Modulus is 210 GPa, yield strength equal to 235 MPa, and the ultimate tensile stress is 360 MPa. Also, in the finite element analysis, it is important to give appropriate material properties. It was considered that the stress–strain is elastoplastic, and it follows the parameter of the isotropic hardening plasticity.

2.2. Buckling Resistance

Elastic buckling theory linked with empirical parameters is converted into recommendations [21], explained in Eurocode 3, part 1-1 [23].

According to Eurocode 3, part 1-1 [23], the buckling resistance of members in pure compression is calculated using the following simplified equations for different classes of steel cross-sections designed: class 1 (plastic section), class 2 (compact section), class 3 (semi-compact section), and class 4 (slender section). The classification of a cross-section depends on the width-to-thickness ratio and the yield strength of the material, which allows us to know how the resistance and rotation ability of a column cross-section can be affected by plasticity or local buckling phenomena [19,22].

To calculate the design buckling resistance $N_{b, Rd}$, we need to apply Equation (3), depending on the reduction factor for the buckling mode χ (Equation (4)), and the partial member resistance factor to instability ${\gamma }_{M1}$ equivalent to 1, for buildings.

$$N_{b, Rd}={\displaystyle \frac{\chi A f_y}{{\gamma }_{M1}}}$$

(3)

$$\chi ={\displaystyle \frac{1}{\Phi \sqrt{{\Phi }^2-{\overline{\mathsf{\lambda }}}^2}}} but \chi \le 1$$

(4)

where

$$\Phi =0.5\left[1+\alpha \left( \overline{\lambda }-0.2\right)+{\overline{\mathsf{\lambda }}}^2\right]$$

(5)

α is an imperfection factor, relating to the correct buckling curve in function to the cross-section [23], the fabrication process, the direction buckling appears, and the yield strength. The non-dimensional slenderness $\overline{\lambda }$ is obtained according to Equation (6) for classes 1, 2, or 3.

$$\overline{\lambda }=\sqrt{{\displaystyle \frac{A f_y}{N_{cr}}}}$$

(6)

For tubular cross-sections and material S235, dependent on hot or cold finish, the constant quantity is equal to 0.21 or 0.49, correspondingly.

For slenderness $\overline{\lambda }\le 0.2$ or for ${\displaystyle \frac{N_{Ed}}{N_{cr}}}\le 0.04$, the buckling effects can be disregarded [23].

3. Analytical Solution of Different SHS Columns, Cross-Section Sizes, Lengths, and End Constraints

Results of the Critical Loads

In this study, columns under pure compression were tested on pinned-pinned, fixed-free, fixed-fixed, and fixed-pinned ends.

Table 1. Analytical solution in different cross-sections, lengths, and constraints.

			Fixed-Fixed		Fixed-Pinned		Pinned-Pinned		Fixed-Free
	L	Npl	Ncr	Nb,Rd	Ncr	Nb,Rd	Ncr	Nb,Rd	Ncr	Nb,Rd
	mm	kN	kN		kN		kN		kN
B50 t = 1.5 mm B = 50 mm	500	68.4	3786.9	69.4	1932.1	68.6	946.7	67.3	236.7	62.4
	1000		946.7	67.3	483.0	65.6	236.7	62.4	59.2	41.9
	1500		420.8	65.1	214.7	61.8	105.2	54.2	26.3	22.5
B60 t = 1.5 mm B = 60 mm	500	82.5	6643.4	84.1	3389.5	83.3	1660.9	82.1	415.2	77.6
	1000		1660.9	82.1	847.4	80.4	415.2	77.6	103.8	61.0
	1500		738.2	80.0	376.6	77.0	184.5	71.1	46.1	37.1
B80 t = 1.5 mm B = 80 mm	500	110.7	16,047.5	113.5	8187.5	112.7	4011.9	111.5	1003.0	107.4
	1000		4011.9	111.5	2046.9	109.9	1003.0	107.4	250.7	95.6
	1500		1783.1	109.5	909.7	106.9	445.8	102.4	111.4	73.9

shows the analytical calculations of the stability for different column lengths and cross-sections of SHS. By increasing the length of the column from 500 to 1500 mm, the critical load and the design buckling resistance of the SHS column always decrease. By increasing the cross-section size of the column (50, 60, and 80 mm), the critical loads of the SHS column always increase.

Columns with fixed-fixed ends have higher resistance when compared to the other end constraints with a maximum value of the critical load equal to 16,047.5 kN and for the design buckling resistance equal to 113.5 kN, for cross-section B80 with length equal to 500 mm. The design buckling resistance always presents lower values concerning the critical load of the column.

This study shows that column length, cross-sectional size, and applied boundary conditions affect the structural behavior of SHS columns.

4. Numerical Solution of Elastic Deformation, Plastic and Design Buckling Resistance in Columns with Initial Imperfections and Nonlinearity Material

4.1. SHS Columns, Cross-Section Sizes, Lengths, and End Constraints

All SHS cross-sections in pure compression (Table 1), were tested using the finite element method. The most important purpose was to examine the shape cross-section of the column and calculate the critical stability load (Figure 2a).

The finite element chosen to apply in all models was SHELL281. This element has eight nodes and six degrees of freedom (DOF) per node (three translations: Ux, Uy, Uz; and three rotations: ROTx, ROTy, ROTz) with full integration, and it is appropriate for investigating thin or moderately thick shell elements, for linear, high rotation, and/or large nonlinear deformations [19,24]. Shell finite elements were used in the steel hollow SHS cross-section columns and supported with an analytical solution. Due to the kind of parameters in this study, a mesh study was applied initially. Convergence studies were given to reach the appropriate mesh size, where, after several trials, the minimum mesh size considered was 10 mm (Figure 2b). Figure 2c represents a local side of the updated geometry to simulate the imperfections.

The studied boundary conditions were fixed-fixed, where all six DOF are fixed on the bottom and top columns; fixed-pinned, with the bottom column fixed with all six DOF but the top column with only three transversal restrictions (UX, UY, and UZ); pinned-pinned column, where both top and bottom column had only restraints with transversal displacements (Ux, Uy, Uz); and fixed-free, with six DOF on the bottom column and free of restrictions on the top column.

4.2. Elastic Deformation in Columns to Obtain Euler Critical Load

The numerical procedure is divided into several steps. First, an eigenbuckling analysis was performed to assess a scaled shape of local buckles, for each studied column. This step allows us to obtain the Euler load, where the prestress on the linear elastic buckling behavior of columns is employed with ANSYS ^® 2022 R2 program [24]. Prestress effects confirm the computation of the stress stiffness matrix. This is completed by first completing a structural analysis on an ideally loaded structure, and then evaluating the stress field to continue a modal analysis that results in the buckling mode, like the example given in Figure 3. The eigenvalue solver applies a unit load on the column to establish the needed buckling load.

4.3. Plastic Resistance in Columns without Imperfections and with Nonlinear Material Properties

To obtain the plastic load, a nonlinear elastic analysis was used in the ANSYS ^® 2022 R2 program [24] with the nonlinear steel material behavior of an ideal column. The equations are nonlinear and solved iteratively using the Newton–Raphson method, where the applied load is divided into small increments to have a better numerical performance. The maximum applied incrementally load gives the plastic collapse in the column without any imperfections, based on the achieved elastic–perfectly plastic material behavior. In this process, a convergence criterion was based on displacement with a tolerance of 9%, and the solution is obtained when the difference between the displacements is less than or equal to this tolerance. Figure 4 represents the equivalent stresses obtained for the applied maximum incremental load that achieves the plasticity failure of the material.

The columns reach the same value maximum yield stress value in the length (235 MPa) for any SHS column, regardless of the applied boundary conditions. Only slightly higher values are located near the load application, which is insignificant, and lower values in the restrictions. The range of these values is automatically obtained in the finite element program in all columns. The corresponding plastic load in each column is 67.4, 61.6, 61.9, and 65.8 kN for fixed-fixed, fixed-pinned, pinned-pinned and fixed-free columns, close to the analytical value equal to 68.4 kN (Table 1).

4.4. Design Buckling Resistance in Columns with Initial Imperfections and with Nonlinear Material Properties

To obtain the design buckling resistance, a nonlinear inelastic buckling analysis in ANSYS ^® 2022 R2 program [24] was applied, including geometric imperfections and nonlinear properties of the material (Figure 2c). The shape of the global imperfections may be obtained from the column elastic buckling [23]. The longitudinal curvature, which is normally present in columns, is an imperfection due to manufacturing processes, handling, and transporting of the components. According to the authors, initial imperfections can be described by a harmonic function, among other solutions [19,25,26], where L/1000 is the maximum measured amplitude imperfection.

$$u(x)=\left({\displaystyle \frac{L}{1000}}\right)sin\left({\displaystyle \frac{\pi x}{L}}\right)$$

(7)

To update the geometry of the straight column and insert the initial imperfection, the UPGEOM command from ANSYS ^® 2022 R2 program [24] was used. The process is automatically executed according to the following command: </prep7> UPGEOM, maximum deflection, Load step, Sub step, File name >. This command adds displacements from the previous analysis (from the elastic buckling analysis eigenvectors) and updates the model geometry to the deformed configuration [24] (Figure 2c). With this command, it is possible to establish the amplitude of the displacement for each nodal point. To implement it, it is necessary to calculate the FACT factor, achieved with the lateral displacement $u\left(x\right)$ of linear elastic buckling, which is multiplied by the displacements being included into the coordinates.

$$FACT=\left({\displaystyle \frac{L}{1000}}\right)/u(x)$$

(8)

For this model, an increased incrementally compressive load is applied until the beginning of buckling in the column (Figure 5). The nonlinear buckling solution requires the use of Newton–Raphson to calculate a system of equations iteratively, concerning the nonlinear material and geometry [27]. Therefore, the solution requires running of the model with incremental load until the column fails by convergence, and this is the point of buckling strength.

The results show the calculated design buckling resistance, as the last incremental load applied, obtained in a column with imperfections, which represents the deformation tendency.

5. Numerical Validation with Analytical Method

Critical buckling and resistance loads were established by applying the ANSYS ^® 2022 R2 program [24]. Figure 6 and Figure 7 represent the comparison of all results between the numerical and the analytical methods, each one correspondent to the end column applied boundary condition. The results were obtained with the analytical variation of $N_{cr}$, $N_{b, Rd}$ and the numerical solution $N_{cr\_A}$, $N_{b, Rd\_A}$ with the ANSYS ^® 2022 R2 program [24].

In Figure 6, the curve outlined by blue dots represents the limiting condition for elastic buckling results. The curve outlined by the orange dots represents the buckling resistance results. In both situations, there is an apparent convergence between numerical and analytical solutions. The results are presented as a ratio between the Euler elastic buckling load or the design buckling resistance and the yield load. The trend is always the same; the highest values correspond to the largest cross-section B80 until to the smallest B50. Furthermore, columns with shorter lengths achieve greater loads. Regarding the effect of boundary conditions, fixed-fixed columns have the highest critical load value and the lowest slenderness ratio. In fixed-pinned columns, the critical load level is lower, increasing the slenderness index. Pinned-pinned columns continue to record low values of critical load and increased slenderness, with fixed-free columns being those that present the lowest loads and highest slenderness indexes.

Figure 7 shows the same results, but now only compares the solution of the design buckling resistance.

It is noticeable that the numerical values present some differences relative to the analytical calculation. Generally, the results agree between analytical and numerical procedures, but some calculations are more conservative when using numerical tests. For numerical results, it was necessary to include the boundary conditions that should have the same real effect.

The analytical method is based on some constant empirical factors, such as the imperfection factor, which is related to the buckling curve and cross-section. In the numerical solution, the imperfections were allocated to all geometry, with the previous values of the buckling displacements not being a constant value.

The design buckling resistance is not the same across the slenderness range. These values represent a failure for very slender or high columns with all different boundary conditions and slenderness greater than 0.2. Plastic damage is the critical load supported in low slender or thick columns, with slenderness lesser than 0.2. The exception occurs in fixed-free columns, where the damage occurs only due to buckling, without a plastic effect.

According to the previous values, a relative error was obtained to compare the numerical with analytical values (

Table 2. Comparison of results in all different SHS columns.

		Fixed-Fixed				Fixed-Pinned				Pinned-Pinned				Fixed-Free
	L mm	Ncr_A	Nb,Rd_A	Ecr	Eb,Rd	Ncr_A	Nb,Rd_A	Ecr	Eb,Rd	Ncr_A	Nb,Rd_A	Ecr	Eb,Rd	Ncr_A	Nb,Rd_A	Ecr	Eb,Rd
		kN		%		kN		%		kN		%		kN		%
B50	500	3774.8	68.0	0.3	2.0	1938.4	68.0	0.3	0.8	949.7	67.2	0.3	0.2	235.3	63.1	0.6	1.2
	1000	941.7	65.8	0.5	2.3	486.1	58.5	0.6	10.8	235.9	56.1	0.3	10.1	61.7	48	4.2	14.5
	1500	423.2	55.5	0.6	14.7	211.4	63.0	1.5	2.0	98.5	51.8	6.4	4.4	28.1	24.4	6.8	8.6
B60	500	6646.0	82.1	0.1	2.4	3402.0	82.1	0.4	1.4	1659.4	78.7	0.1	4.1	415.3	80.2	0.0	3.3
	1000	1661.3	74.2	0.0	9.6	846.3	77.8	0.1	3.3	418.1	70.3	0.7	9.3	105.9	67.2	2.0	10.2
	1500	737.4	70.4	0.1	12.0	375.2	82.1	0.4	6.5	184.2	78.1	0.2	9.9	48.0	40.1	4.0	8.0
B80	500	16,025.3	109.4	0.1	3.6	8179.1	110.1	0.1	2.3	4012.3	106.9	0.0	4.1	1016.3	92.4	1.3	13.9
	1000	4015.0	110.1	0.1	1.3	2044.1	101.7	0.1	7.5	1001.6	103.7	0.1	3.4	250.2	100.8	0.2	5.4
	1500	1783.4	102.4	0.0	6.5	909.8	99.0	0.0	7.4	446.0	101.1	0.0	1.2	112.7	83.2	1.2	12.5

).

Equation (9) calculates the relative error according to the Euler critical load $E_{cr}$, and Equation (10) calculates the relative error according to the design buckling resistance $E_{b,Rd}$.

$$E_{cr}=\left({\displaystyle \frac{N_{cr}-N_{cr\_A}}{N_{cr}}}\right)\times 100\%$$

(9)

$$E_{b,Rd}=\left({\displaystyle \frac{N_{b,Rd}-N_{b,Rd\_A}}{N_{b,Rd}}}\right)\times 100\%$$

(10)

Table 2 gives, in detail, the calculated relative error using Equations (9) and (10).

The solution proved that the finite element method resulted in a close calculation of the analytical values for the elastic buckling of the SHS columns. The variation concerning the finite element method, and the analytical solution is between 0 and 6.8%. For the design buckling resistance, the deviation between the finite element method and the analytical results is between 0 and 14.7%. The higher relative errors are obtained for lesser cross-section sizes in the study.

In the following work, other numerical solutions to be implemented may provide closer results if the tolerance imposed on the convergence criteria is different. Parameters related to the nonlinear analysis performed on these numerical results used the large displacement method, but the arc length method can also be implemented to compare other numerical solutions.

The present results increase the solution already started by the author [19] but now cover more columns under analysis and with different variables under test.

6. Conclusions

In this study, three-dimensional linear and nonlinear finite element modeling of SHS columns under pure compression was carried out. Steel material was used, including plasticity damage, which includes the isotropic hardening plasticity behavior. The results from the numerical analysis of the present study were validated by the results from the mathematical formulation. A comparative study was performed, using finite element analysis to estimate the critical load and design buckling resistance with analytical calculations following Eurocode 3.

Focusing on the results and discussion presented in this study, the next conclusions are described below:

-

For the elastic buckling or Euler load determination, the finite element method resulted in a close calculation from the mathematical formulation, with a variation between 0 and 6.8%.

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For the design buckling resistance and the critical load determination, the deviation between the finite element method and the mathematical formulation is between 0 and 14.7%. The higher relative errors are obtained for lesser cross-section sizes in the study.

-

According to the needs for the design of the columns, by increasing the length and reducing the cross-sectional size, the design buckling resistance also decreases. These demonstrate that the length and size of the column cross-section can meaningfully increase the structural behavior of SHS columns. Also, the material properties stiffness is incorporated as a great effect in its resistance.

-

Three-dimensional linear and nonlinear finite element modeling was demonstrated to be the best instrument for researching the performance and behavior of steel columns under pure compression, incorporating the correct material properties and boundary conditions.

-

Finally, the great meaning of the present investigation was to include the effect of initial column imperfections, including simultaneous nonlinearity with material plasticity damage and the study of all these combined effects on the impact of buckling resistance.

-

The tested numerical method will be possible to implement in other future column analyses where other parameters could be different.

Additional and further studies are suggested to improve the evaluation and behavior of the numerical development presented but extended to other column cross-section configurations, including the variable column wall thickness.