Buckling Analysis of a New Type of Double-Steering Prestressed Plate Column

Abstract

A new type of dual-steering prestressed plate column is introduced. Compared to the previous prestressed strut column, this proposed column considers both bending and constraints at ends. The calculation results indicated that the support plate significantly improves the stable bearing capacity and buckling performance of the core steel column. When compared to the proposed column, the bearing capacity of the three-transverse prestressed beam column is 1.51 times smaller, the single-transverse prestressed beam column is 2.43 times lower, and the non-prestressed column is 4.51 times smaller. Moreover, this study examines the influences of effective length, buckling mode, stress nephogram detail, and prestress value. It explores the possibility of implementing this new type of dual-steering prestressed plate column in practical engineering. In addition, the variety and mechanical models of prestressed columns are expanded and refined.

1. Introduction

The compressive stability of slender steel columns is one of the main control factors in structural design [1]. Especially for steel columns with a large slenderness ratio (>150), the strength reduction is even as high as 70% due to stability problems, seriously reducing the strength utilization rate of steel columns in actual engineering [2,3]. The research results showed that the strength and stability of steel structures subjected to prestressing can be significantly improved if the prestressing method is reasonable [4,5]. The prestressed steel structure can save materials and funds, ensure safety, and reduce deformation [6,7]. The prestressed stayed column comprises three parts: the central steel column, which plays a vital role in bearing the upper load. The second part is the prestressed steel cable that can only bear tensile force and apply pre-pressure to the bearing column without increasing excessive weight. The third part is the transverse brace, which acts as a constraint on the central steel column under the action of prestress. Prestressed steel cables limit the deformation of the central steel column and prevent buckling of the central steel column [8,9,10]. However, the research on the structural types of prestressed stayed columns is relatively single, mainly focusing on single and three transverse prestressed beam columns [11,12]. The types and theories of new prestressed stayed columns with more reasonable force and higher strength utilization need further study.

The research on the buckling performance of prestressed stayed columns is mainly conducted from theoretical derivation, experimental verification, and numerical simulation. Many scholars have systematically investigated the theory of prestressed stayed columns [13,14,15,16]. Smith et al. [13] derived the theoretical buckling solution applicable to the single-transverse prestressed stayed column and applied it to numerical examples. They indicated that the theoretical buckling solution can predict the buckling behavior and bearing capacity of single and multiple transverse prestressed stayed columns.

Temple [15] proposed a method for calculating the buckling load of multi-transverse prestressed beam columns using the finite element method. The results were very close to the exact solution. Saito and Wadee [17] employed the Rayleigh–Ritz method to determine the geometric post-buckling behavior of prestressed stayed columns and verified it with numerical approaches. The research showed that the nonlinear behavior was closely related to the initial prestress. Wadee et al. [18] proposed a simplified theoretical model using discrete rigid connections, which considered the relaxation of prestressed steel cables and the destabilization effect of steel columns. Saito and Wadee [19] established a nonlinear model considering geometric imperfections utilizing the Rayleigh–Ritz method to compute the buckling behavior of prestressed beam columns. They demonstrated that prestressed columns were very sensitive to defects when bearing critical loads. However, the current theoretical research focuses on improving the bearing capacity of steel columns only by increasing the transverse braces.

Many experimental studies have also been conducted on the buckling behavior of prestressed stayed columns [10,20,21]. Osofero et al. [20] performed a systematic experimental study on the buckling performance, bearing capacity, and defect sensitivity of prestressed stayed columns. The experimental results verified the existence of symmetric and antisymmetric buckling modes in prestressed stayed columns. Hafez et al. [10] investigated the effect of the initial prestress on the single-transverse prestressed stayed column. The theoretical and experimental results indicated that the initial prestress greatly affected the buckling performance. De Araujo et al. [22] employed tests and finite element simulation methods to systematically examine the parameter laws of steel column height, diameter, and prestress size. Martins et al. [21] conducted experiments on the buckling behavior of prestressed stayed columns using ordinary and high-strength steel, respectively. They reported that high-strength steel has great advantages over prestressed stayed columns. The experimental and numerical results provide a reference for the design of three-dimensional prestressed stayed columns. At present, the research on the test of a prestressed stayed column is relatively few and only concentrates on the study of single transverse prestressed beam columns.

Compared to theoretical and experimental methods, numerical approaches for prestressed stayed columns have high efficiency, low cost, and broad applicability [23,24,25]. Saito and Wadee [26] investigated the interactive buckling behavior using numerical analysis methods. They exhibited that interaction buckling behavior can significantly reduce the maximum bearing capacity of prestressed stayed columns. Pichal and Machacek [27] examined the effect of geometric and material nonlinear imperfections on the post-buckling behavior of prestressed stayed columns using numerical analysis methods. They indicated that the buckling behavior of prestressed columns depends on geometry, material properties, prestressed magnitude, and boundary conditions. Wang et al. [28] utilized numerical methods to analyze the post-buckling performance of prestressed stayed columns under different cross-brace lengths. The sensitivity analysis of the buckling load was conducted, and an optimization algorithm was applied to obtain the optimal brace length. The research results provide a reference for the design of prestressed columns. Lapira et al. [29] analytically derived the optimal prestress solution of multi-cross-braced prestressed columns and verified it with the finite element method. The research results can provide a reference for the prestress value of prestressed columns considering geometric nonlinearity. Currently, the research on numerical simulation of prestressed stayed columns primarily focuses on the buckling behavior, post-buckling behavior, and parameter optimization of single-transverse and multi-transverse prestressed beam columns. There are relatively few innovations in the structural forms of prestressed stayed columns.

In summary, numerous scholars have drawn many beneficial conclusions on prestressed stayed columns through large numbers of theories, simulations, and experiments. However, there are few relevant studies on improving the transverse form and increasing the end constraint to enhance the bearing performance of prestressed stayed columns. This study analyzes and compares various prestressed beam-column structures and proposes a new type of prestressed column to enrich and improve the theoretical basis and mechanical performance of prestressed stayed column structures. This research modifies the existing shape from a beam to two steering brace plates to turn the prestressed steel cables. The original prestressed steel cables are divided into two evenly: four prestressed steel cables are divided into four groups of eight, and the prestressed steel cables are turned twice around the compression column through the plate braces. The steel cable can transmit more effective elastic support through supporting plates, while the upper and lower supporting plates are the factors that can make use of the end of the pressure column. Compared to the previous single-point constraints, increasing end constraints can improve the ultimate bearing capacity of the bearing column. Figure 1 depicts the schematic diagrams of a traditional single-transverse prestressed beam column (Figure 1a), a three-transverse prestressed beam column (Figure 1b), and a new double-steering prestressed plate column (Figure 1c).

The primary contribution novelty of this study is the introduction of a novel double-steering prestressed plate column. Compared to the previous prestressed columns, the design considers both the bending constraint and the two-end constraint. These features significantly enhance the stable bearing capacity and buckling performance of the core steel column. This research investigates the buckling performance of dual steering prestressed brace columns and develops a finite element model. The model considers initial geometric defects and linear contact, analyzing the impact of prestress position distribution and steering plate size. Prestressed columns are widely used in large commercial centers, gymnasiums, terminal buildings, and other long-span structures. The innovative prestressed columns presented in this study offer considerable steel savings.

2. Calculation Model

The research object of this study is a double-steering prestressed plate column. A nonlinear buckling analysis method is selected due to its high nonlinearity of contact. In this section, the calculation model of the double-steering prestressed plate column is developed and calculated.

2.1. Assumptions

(1)

The steel column and the stayed brace are rigidly connected.

(2)

The steel column is simplified as isotropic beam elements, and the impact of initial geometrical defects is considered.

(3)

The prestressed steel cables are simplified as cable elements, which will generate large tensile strains but will not produce any form of compressive and bending strains.

(4)

The separable motion contact algorithm is used for line-to-surface contact.

(5)

The lower supporting plate is restrained by fixed constraints, and the upper supporting plate is constrained in a horizontal direction. Vertical loads are imposed on the column’s top along the Z axis.

(6)

Plane torsional buckling is ignored.

2.2. Initial Prestress

The initial prestress of steel cable is essential for improving the steel column’s bearing capacity. Figure 2 accurately describes the relationship between the initial prestress value and the critical buckling load of the prestressed stayed column [10,11].

Zone one indicates that when the prestress is relatively small (0 ≤$T$ <${ T}_{min}$), the steel column’s critical bearing capacity is the minimum value of $P_{min}^C$, and the variation amplitude is small. Zone two demonstrates that when the prestress is moderate ($T_{min}$≤$T$<$T_{opt}$), the steel column’s critical bearing capacity significantly rises with the increase in the initial prestress value, and the maximum value can reach $P_{min}^C$. Zone three exhibits that when the prestress is too large ($T_{opt}$≤$T$), the steel column’s critical bearing capacity gradually decreases as the initial prestress value grows. When the prestressed value reaches $T_{max}$, the steel column’s critical bearing capacity is 0. The optimal prestress value is at the junction of zones two and three [30,31].

The relationship between initial prestress value and critical buckling load can be described by Equation (1):

$$\left\{\begin{array}{c}{P}_{min}^C, 0\le T<T_{min}\\ \frac{P_{man}^C-P_{min}^C}{T_{opt}{-T}_{min}}\times \left(T{-T}_{min}\right)+P_{min}^C,T_{min}\le T<T_{opt}\\ \frac{P_{man}^C}{T_{opt}{-T}_{man}}\times \left(T{-T}_{opt}\right)+P_{man}^C,T_{opt}\le T\le T_{max}\end{array}\right.$$

(1)

The optimum prestress expression of the prestressed stayed column is shown in Equation (2):

$$T_{opt}=\frac{f_p}{f_b}{P}^C$$

(2)

where $f_p$ and $f_b$ are the coefficients related to the prestressed steel cable and brace parameters; $P^C$ is the critical bearing capacity of the prestressed stayed column $T_{opt}$ is the optimum prestress value.

2.3. Geometric Defects

Due to the inevitable existence of certain geometric defects in prestressed stayed columns, the influence of geometric defects is considered in this study, and the application method of geometric defects is consistent with the literature [32,33,34]. Perpendicularity deviation of the main steel structure ${\epsilon }_0$ is limited by length, as shown in Equation (3):

$${\epsilon }_0=L/1000$$

(3)

where $L$ is the height of the prestressed column.

2.4. Line-Surface Contact Algorithm

This study allows the contact between the prestressed cable and the brace to be “hard contact”. The contact algorithm is depicted in the following equation [35]:

No contact

$$F_P=0 \mathrm{for} \Delta <0$$

(4)

Contact

$$\Delta =0 \mathrm{for} F_P>0$$

(5)

where $F_P$ is the contact pressure; $\mathsf{\Delta }$ is the gap.

The virtual work of the contact force can be calculated according to the Lagrange principle:

$$\delta \prod =\delta F_P\Delta +F_P\delta \Delta$$

(6)

The friction force $f_F$ can be expressed as

$$f_F={\mu }_{fric}{F}_P$$

(7)

$${\tau }_{sh}=\sqrt{{\tau }_x^2+{\tau }_y^2}$$

(8)

where $f_F$ is the friction force; ${\tau }_1$ and ${\tau }_2$ are shear stresses; ${\mu }_{fric}$ is the coefficient of friction; ${\tau }_{sh}$ is the equivalent shear stress.

2.5. Force Equation of Double-Steering Prestressed Plate Column

Let $Q_k$ and $M_k$ be the horizontal reaction and vertical bending moment at the $k$ node of the double-steering prestressed plate column, respectively; $\overline{{\alpha }_{kn}}$ and $\overline{{\beta }_{kn}}$ are the horizontal displacement spring coefficient and the vertical rotation spring coefficient at the $k$ node, respectively.

The steel column’s lateral displacement at the $k$ node is

$$V_k={\overline{{\alpha }_{kn}}Q}_k$$

(9)

The steel column’s angle at the $k$ node is

$${\theta }_k={-\overline{{\beta }_{kn}}Q}_k$$

(10)

The elastic buckling equilibrium relationship between the $k$ steel column’s joints is as follows:

$$EI{y}^{″}(k)+Ny(k)-R_0{x}_k+{\sum }_{i=1}^{k-1}(x_k-ih)Q_i+{\sum }_{i=1}^{k-1}{M}_i=0$$

(11)

where $y\left(k\right)$ is the horizontal displacement of $k$ node; $R_0$ is the horizontal reaction force of the bottom support.

2.6. Calculation Model

The finite element software ANSYS 15.0 is utilized to calculate and analyze the proposed new type of prestressed column. This study compares non-prestressed bearing columns, a traditional single-transverse prestressed beam column, a traditional three-transverse prestressed beam column, and the new double-steering prestressed plate column. The model’s basic mechanical and geometric parameters are as follows.

The elastic modulus $E$ of steel is considered 210 GPa, the height of core steel columns is 3 m, the Poisson’s ratio $\nu$ is 0.3, and the column’s section diameter is 50 mm. For a single-transverse prestressed beam column, the length $H_1$ of the brace is set to 150 mm, the cross-sectional diameter $D_1$ is placed to 50 mm, and the prestressed reinforcement is Φ 8. For a three-transverse prestressed beam column, the length $H_1$ of brace one is considered to be 150 mm, the length $H_2$ of brace two is 110 mm, and its cross-sectional diameter $D_1$ is 50 mm; the prestressed reinforcement is Φ 8. For the double-steering prestressed plate column, the thickness of plates ($h$) is set to 10 mm, and the hole ($r$) radius is 5 mm. The diameter $d_1$ of supporting plates is 330 mm, and the distance $s_1$ from the center of the small hole to the central column is 150 mm. The diameter $d_2$ of steering plates is 220 mm, and the distance $s_2$ from the center of the small hole to the central column is 100 mm. The prestressed reinforcement is Φ 4.

The geometric models of traditional prestressed beam columns and dimensional parameters are illustrated in Figure 3, and the section parameters are listed in

Table 1. Prestressed beam columns section parameters.

	h (mm)	H (mm)	D (mm)
Single-transverse prestressed beam column	1500	150	50
Three-transverse prestressed beam column	600	100	50
	1500	150
	2400	100

.

Figure 4 depicts the mesh convergence results for the supporting plate and steering plate. It indicates that the maximum principal stress converges when the number of elements reaches 50,000. The total number of mesh units used in this study is 53,192, satisfying the required calculation accuracy.

Figure 5a depicts the geometric model and dimensional parameters of a double-steering prestressed plate column, whereas Figure 5c illustrates the angle diagram, and Figure 5b displays the mesh model.

Table 2. The double-steering prestressed plate column section parameters.

	h (mm)	α1 (°)	α2 (°)	α3 (°)	d (mm)	s (mm)	r (mm)	D (mm)
End supporting plates	0	32.2	12.8	90	330	150	5	50
	3000	32.2	12.8	90	330	150	5
Middle supporting plates	1500	32.2	12.8	90	330	150	5
Steering plates	600	24.3	20.7	90	220	100	5
	2400	24.3	20.7	90	220	100	5

lists the section parameters. The beam 188 elements are selected for the core steel column and brace. The link ten elements are used for the cable, which can accurately express the mechanical performance model of the material that is only subjected to tensile stress and not compressed. In addition, the input of its mechanical performance can accurately simplify the cable calculation, and the correct results can be determined. This study adopts a sliding form of the connection part. Therefore, Solid185 units are adopted by the plates, which can accurately achieve sliding contact between the lock and steering plates. Figure 6 shows the element details, including linear and quadratic element, load, and boundary conditions of the new prestressed column.

Load conditions: The buckling analysis consists of two load steps: In the first loading step, static analysis under defect conditions is considered. In this study, the initial strain of the Link 10 element of the prestressed cable must be determined iteratively for the prestressed cable to be prestressed in the static analysis result. The second load step is buckling analysis.

3. Results and Discussion

3.1. Buckling Analysis

3.1.1. Buckling Calculation of Different Types of Bearing Columns

This section compares the numerical and analytical solutions of non-prestressed bearing columns. Based on Euler’s formula for critical buckling load in material mechanics:

$$F_{cr}={\pi }^{2}EI/{\left(\mu h\right)}^2$$

(12)

where $E$ is the elastic modulus; $I$ is the moment of inertia; $\mu$ is the length coefficient; $h$ is the column length.

Table 3. Non-prestressed bearing column calculation parameters.

D (mm)	I (mm4)	h (mm)	E (MPa)	μ
50	6.14e5	3000	2.1e5	0.7

lists the calculation parameters.

Table 4. Comparison between the theoretical and numerical analysis results.

	Numerical Solution	Theoretical Solution	Error
Non-prestressed bearing column	144 kN	144.2 kN	0.14%

reveals the obtained results using theoretical calculation and finite element simulation.

The calculation shows that the numerical solution is 144 kN, and the theoretical solution is 144.2 kN. The difference is only 0.2 kN, and the error rate is only 0.14%.

Figure 7 depicts the buckling load calculation results of the traditional single-transverse prestressed beam column, three-transverse prestressed beam column, and the new double-steering prestressed plate column under different prestress.

As stated previously, the elastic yield load F_cr0 of a non-prestressed column subjected to the same constraint is 144 kN. By comparing the buckling loads in various cases, the variation of bearing capacity is as follows.

(1)

When the pretension of the prestressed cable is 0–5 kN, the yield loads F_cr of the three prestressed stayed column models have little change. When the pretension of prestressed cable is 5–15 kN, the yield loads F_cr of the three prestressed column models rise with the increase in prestressed stress. When the pretension force exceeds 60 kN, the yield loads F_cr of the three models gradually decrease as the prestress grows. This phenomenon is due to the role of the pretension force to ensure that the prestressed cables do not relax during the buckling of the central column, which can provide elastic support. When the prestress value is too high, the tensioning force will cause the initial compressive stress of the central column to be too large, reducing its ability to withstand external loads.

(2)

Compared to the simulation results of a single-transverse column, the buckling load of the three-transverse column is significantly increased due to adding two additional cross braces, increasing two elastic supports, and reducing the slenderness ratio.

(3)

Compared to simulation results of prestressed beam columns with double-steering prestressed plate columns proposed in this study, the buckling load F_cr of the new model is significantly increased because using steering plates raises the single point constraint on joints and increases the constraint ratio.

3.1.2. Effect of Effective Length

Figure 7 demonstrates that the prestressed value of the three prestressed stayed column models is 15 kN when the F_cr simulated reaches the maximum value. In this case, the F_cr values of the three simulated prestressed stayed columns are compared to the critical elastic buckling load F_cr0 of the non-prestressed column (which has the same parameter as the central column). The effective length coefficient values are illustrated in

Table 5. Buckling load and effective length coefficient.

	Non-Prestressed Column	Single-Transverse Prestressed Beam Column	Three-Transverse Prestressed Beam Column	Double-Steering Prestressed Plate Column
Buckling load Pcr (kN)	144	267	430	650
Effective length coefficient	1	0.734	0.579	0.47

($\mu =\sqrt{P_{cr0}/P_{cr}}$).

From the buckling load values in Table 5, it can be concluded that prestressed columns improve the buckling performance of compression columns. Comparing the values of the three kinds of prestressed stayed columns indicates that the F_cr values of the double-steering prestressed plate column proposed in this study are all higher than those of the prestressed beam columns, and the length coefficients of the bearing columns are further reduced.

Compared to the 144 kN buckling load of the non-prestressed column, the maximum buckling load of a single-transverse prestressed beam column is 267 kN, which is 1.85 times higher, and three-transverse prestressed beam column is 430 kN, which is 2.99 times greater, while the new column proposed in this study is 650 kN, which is 4.5 times higher. The results indicated that the steering plate brace plays a significant role in improving the buckling load value.

The prestress value significantly affects the buckling load value, and a specific law exists. When the prestress value initially rises, the buckling load value rapidly increases until the effect becomes optimal. As the prestress value continues to grow, the buckling load reduces slowly, but the rate of decrease is much lower than the increase. In the design of the prestressed value, in addition to factors such as the prestressed loss and installation deviation, it is necessary to consider a larger value of the prestress design value to avoid significant buckling load value errors caused by a smaller prestress design value.

3.2. Buckling Mode Analysis

3.2.1. Buckling Mode Coefficient and Load

The first six orders of modes are considered for analysis. The buckling loads are calculated relying on the buckling coefficient listed in

Table 6. Buckling mode orders and loads.

	First Order (kN)	Second Order (kN)	Third Order (kN)	Fourth Order (kN)	Fifth Order (kN)	Sixth Order (kN)
Non-prestressed column	144	144	187.2	187.2	388.8	388.8
Single-transverse prestressed beam column	267	267	507.3	507.3	881.1	881.1
Three-transverse prestressed beam column	430	430	731	731	1075	1075
Double-steering prestressed plate column	650	650	1040	1040	1690	1690

.

Table 6 indicates that since the research objects are spatially symmetric structures, each buckling load coefficient of the four models has the same coefficient as the other, and the buckling modes under identical buckling loads only have different buckling directions under the spatial coordinate axis. Therefore, only the first, third, and fifth-order modes must be compared.

3.2.2. Buckling Mode Cloud Diagram

Figure 8, Figure 9 and Figure 10 illustrate the stress cloud diagram of the first, third, and fifth-order modes.

The three-order modal cloud diagrams exhibit that the prestressed cables transmit force to the transverse brace for prestressed beam columns and lateral support is applied in the deformation direction of the central column, thereby slowing the central column deformation and increasing its bearing capacity.

The prestressed beam columns are prestressed steel bar that exerts lateral support along the deformation direction of the central column through transverse bracing, reducing the central column deformation, thereby enhancing the ultimate bearing capacity. For the proposed column, due to increased end constraints and the existence of two steering plates, the directions of the prestressed cables are altered, which better inhibits the buckling deformation of the central column and even changes the buckling modes of the central column.

3.2.3. Detail Analysis

Detailed analysis is performed on the double-steering prestressed plate column, and the corresponding stress cloud is revealed in Figure 11, Figure 12 and Figure 13.

The stress cloud diagram demonstrates that in the first mode, the deformation form of the non-prestressed column is a single-wave symmetric instability deformation, with lateral displacement at the upper, middle, and lower parts of the column. In contrast, the upper and lower steering plates and intermediate supporting plates of the double-steering prestressed plate column both transmit considerable stress to the central column, thereby changing the buckling mode of the central column. In the third-order mode, the non-prestressed column deformation is a dual wave antisymmetric instability deformation, with lateral displacement at the upper and lower parts of the column, no lateral displacement at the midpoint of the column, and only rotation angles. Under this deformation, only the upper and lower steering plates transfer stress to the central column, thus changing the buckling mode of the central column for the prestressed column. In the fifth-order mode, the deformation form of the non-prestressed column is three-wave symmetric instability, with lateral displacement at the upper, middle, and lower parts. For the double-steering prestressed plate column, the upper and lower steering plates and intermediate supporting plates all transmit significant stress to the central column, thus changing the buckling mode of the central column.

4. Conclusions

This study proposes a scheme with a double-steering prestressed plate column and compares it with traditional compression columns. The comparative analysis includes theoretical analysis and buckling performance. The results indicated that the improved prestressed stayed column proposed in this research is reasonable. In addition, the influencing factors of the proposed prestressed column are analyzed. The following are the study’s findings:

(1)

The study introduces a new type of prestressed stayed column with double-steering. The research on the proposed column addresses the gap in the literature for increasing end constraints to enhance the bearing performance of prestressed stayed columns.

(2)

The results obtained by the numerical analysis method are identical to the actual calculations, making it suitable for general engineering practice. The prestressed stayed column with double-steering plates is 2.17–4.51 times the characteristic buckling load of different traditional bearing columns. The proposed column has improved material utilization and restricted deformation.

(3)

When the pretension of the proposed prestressed cable is 5–15 kN, the yield load F_cr of the steel column increases significantly with the increase in the prestressed cable. The influence law is consistent with the traditional prestressed beam column, while the steel column’s bearing capacity is significantly higher than prestressed beam columns.

(4)

The possibility of implementing a new type of dual-steering prestressed plate column in practical engineering is explored in this study. In addition, the types and mechanical models of prestressed columns are enriched and developed.

(5)

Although the prestressed column proposed in this study significantly enhances the stable bearing capacity and buckling performance, the ductility and seismic performance of the structure must be further assessed in future work.