Sensitivity Analysis of Initial Imperfection in Low Torsional Stiffness Cruciform Steel Columns
ZhengchaoLi1✉Phone+86 18360698127Emailxiyuan1988@126.com
ZhengLi2
LongJin3
QicaiLi3
1School of Civil EngineeringLianyungang Technical College222000LianyungangChina
2College of Material Science and EngineeringNortheast Forestry University150040HarbinChina
3School of Civil EngineeringSuzhou University of Science and Technology215011SuzhouChina
Zhengchao Li 1,*, Zheng Li 2, Long Jin 3 and Qicai Li 3
1 School of Civil Engineering, Lianyungang Technical College, Lianyungang 222000, China
2 College of Material Science and Engineering, Northeast Forestry University, Harbin 150040, China
3 School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
* Correspondence: xiyuan1988@126.com; Tel.: +86 18360698127
A
Abstract
Cruciform steel columns are widely employed in diverse structural applications due to their advantages, which include flexible layout configurations and reduced self-weight. However, these columns possess inherently low torsional stiffness, making them particularly vulnerable to the effects of initial imperfections, especially initial torsion, which can be especially detrimental. Current design standards, such as GB 50017, Eurocode 3, and AISC 360, do not take initial torsional imperfections into consideration. This omission can result in potential inaccuracies in structural evaluations and may compromise overall safety. In this study, nonlinear finite element modeling (FEM) was performed using ANSYS software with Solid45 elements. The analysis accounted for both geometric imperfections and residual stresses, and its accuracy was validated through eigenvalue and nonlinear buckling analyses, with error margins remaining below 5%. The main findings indicate that the width-to-thickness ratio (b/t ≈ 18.5) is a critical factor influencing the transition between elastic and plastic behavior, as well as contributing to a reduction in load-bearing capacity. Initial torsion within the range of 0–5° was found to have the most significant negative impact on the bearing capacity of the columns. Residual stresses were shown to further aggravate this effect, with notable coupling interactions. Additionally, when the slenderness ratio (
) relative to the b/t ratio (k < 5.18) is low, the detrimental influence of initial torsion is considerably amplified. This research helps to fill existing theoretical gaps, reinforces the foundational basis for stability analysis, and offers valuable insights that can inform future revisions of structural design codes.
Keywords:
Cruciform column
Torsional buckling
Buckling resistance
Axial compression
Finite element simulation
Introduction
Cruciform steel columns, which constitute a prominent category of specially shaped structural components, have been extensively utilized in multi-story steel frame systems, spatial lattice frameworks, and residential steel constructions 1. Their broad application is mainly due to several distinct advantages: they enable highly flexible spatial layouts by effectively reducing the constraints caused by protruding corners, thereby improving overall spatial utilization efficiency; their lightweight characteristic significantly enhances seismic performance; and they feature two primary axes with relatively high moments of inertia, which contribute to achieving uniform structural stability and simplifying the installation process2–4. Research results demonstrate that, in comparison with conventional I-shaped section columns, cruciform column (CCUB) can achieve a substantial decrease in column weight, ranging from 17% to 66%, thus exhibiting considerable potential for weight reduction 5. Moreover, with appropriate design considerations, these cruciform sections can retain satisfactory levels of axial compressive load-bearing capacity and structural stiffness 6,7.
Compared to closed cruciform-sections like circular or box-shaped profiles 8, the naturally open design of a cruciform section results in comparatively lower torsional stiffness 9. In real-world engineering applications, steel components are inevitably subjected to various unfavorable factors during processes such as fabrication, transportation, and installation 10. These include initial geometric imperfections (for example, initial flexural and initial torsion) as well as residual stresses. Among these, initial torsion imperfection is particularly detrimental, as it can significantly diminish the critical stress level and the overall stability-bearing capacity of the components. For cruciform sections, which already exhibit inherently poor torsional performance, the predominant modes of overall failure are often governed by either pure torsional buckling or flexural-torsion coupled buckling 11. Consequently, conducting an in-depth investigation into the effects of initial imperfections, especially initial torsion, on the stability performance of such sections is of paramount importance.
The current principal steel structure design codes, including GB 50017 12, Eurocode 3 13, and AISC 360 14, primarily base their stability design theories for axial compression members on buckling models that account for initial imperfections such as initial flexural and residual stresses. These models typically determine the stability factor φ through either multiple curve charts for individual column profiles or a single comprehensive curve. It is crucial to recognize that for torsional buckling issues, which are inherently associated with certain cruciform geometries, different standards employ varied approaches. For example, GB 50017 uses a conversion slenderness ratio method, Eurocode 3 applies flexural-torsion buckling theory, and AISC 360 calculates elastic torsional buckling stresses. Nonetheless, these theoretical frameworks do not explicitly consider the effects of initial torsional imperfections or defects. This omission creates a theoretical deficiency that may lead to greater calculation inaccuracies and potentially unsafe design decisions 15. Consequently, in the analysis of torsional and combined flexural-torsional buckling behavior of cruciform columns, it is of great importance to systematically incorporate the effects of initial torsional defects, which is essential for advancing the theoretical basis and enhancing the precision of stability calculations in structural design. Recent experimental studies on cold-formed stainless steel cruciform columns 16 and high-strength steel welded cruciform columns 17 have also highlighted the significant influence of initial imperfections on buckling behavior, corroborating the need for code revisions.
Regarding the stability challenges associated with cruciform columns, especially torsional buckling, a substantial body of research has been carried out by numerous investigators. Chen G and Trahair N S 18,19 conducted in-depth analyses, utilizing both analytical and numerical methodologies, to critically evaluate the significant detrimental effects that initial twist angles and residual stresses exert on the ultimate torsional buckling capacity. They put forward predictive models capable of assessing the structural behavior both prior to and following the onset of yielding. Trahair N S 20 noted that, in essence, local buckling phenomena and torsional buckling in cruciform columns share fundamental similarities. He made efforts to adapt the design criteria for flexural instability to the specific context of torsional buckling. Nicos Makris 21 and Schurig M et al. 22 focused their attention on the torsional buckling behavior during the plastic deformation stage. They investigated the influence of initial imperfections and changes in the plastic modulus on the buckling performance. Naderian H R 23 carried out numerical analyses on FRP (Fiber Reinforced Polymer) cruciform columns and found that short, thin-walled columns are more prone to torsional buckling failure. Dobri´c et al. 16 studied the buckling of cold-formed stainless steel cruciform columns under axial compression through experiments and simulations. Six specimens made of austenitic channels with varying lengths and fastener setups were tested, alongside material and imperfection measurements. ABAQUS models, validated by experiments, assessed end fastener effects. Results showed short columns failed by local buckling, medium ones by combined local and torsional buckling, and long columns by torsional buckling, with fastener spacing impacting shorter columns more. The development and improvement of finite element technology has demonstrated its excellent mechanical performance prediction capabilities in both traditional construction 24–27 and emerging 3D printing fields 28,29. This characteristic indirectly confirms that finite element technology can provide scientific and effective theoretical guidance and technical support for related engineering and research work.
In engineering practice, initial torsional imperfections often arise from welding misalignments, fabrication deviations, and transportation-induced deformations. These imperfections are observed in on-site inspections of large steel frame systems and are rarely covered by current design codes. Therefore, the influence of these torsional imperfections requires further theoretical and numerical exploration. The derivations herein extend existing torsional buckling theory to incorporate initial torsional imperfections and residual stresses, providing a unified framework for stability assessment of cruciform columns. Based on the research background and identified issues, this investigation employs a nonlinear finite element simulation (FEM) approach using ANSYS software to evaluate the stability of welded axial compression members that possess inherent torsional imperfections. The study examines critical influencing factors such as the initial torsion angle, the plate b/t ratio, the slenderness ratio, and residual stress effects by constructing detailed models that accurately represent geometric imperfections and residual stress distribution patterns. A comprehensive set of simulations, including eigenvalue analysis and nonlinear buckling assessments, will be conducted to explore how these variables affect instability modes, buckling progression, and the ultimate load-carrying capacity of the members. The findings are intended to deepen the understanding of torsional imperfections impacts, contribute to the development of more robust stability design guidelines for cruciform columns, and support potential updates to relevant structural codes.
The theoretical formula for the direct analysis method
Fig. 1
Details of component dimensions: (a)Top view and (b) three-dimensional illustration.
According to the procedural steps outlined in the direct analysis method 18, the analytical process leads to the derivation of three key mathematical expressions. Firstly, it involves the establishment of the equilibrium differential equation that characterizes the overall torsional instability while explicitly incorporating the influence of initial geometric or structural defects. Secondly, an internal force expression is formulated to comprehensively reflect how these initial defects manifest their effects throughout the entire structural component. Thirdly, the maximum internal force expression is derived to accurately represent the actual impact of initial defects under critical loading conditions. Collectively, these three derived expressions provide a comprehensive framework for evaluating both the torsional instability behavior and the ultimate internal force characteristics of the structural component when initial defects are presented.
The ideal elastic torsional buckling load can be derived from the torque balance equation, and its expression is shown in Eq. (1).
1
Where 
means the cross-sectional area, b and t represent the width and thickness of the steel column, respectively, as shown Fig. 1. E represents the Young’s modulus of elasticity.
,
,
is the warping moment of inertia. G denotes the shear modulus of elasticity and G = 79000N/mm2. J represents the torsional constant that remains consistent across the cross-section, indicating a uniform distribution of stiffness against torsional forces.
The torsional moment of inertia (
) of cross-sectional members is typically very small, and for practical purposes in engineering calculations, it is commonly simplified as zero. Based on this simplification, the derived expression for the ideal elastic torsional buckling load is presented in Eq. (2).
2
Accordingly, the torsional buckling stress can be calculated using Eq. (3).
3
λ Derivations based on theoretical models incorporating initial torsional imperfections
Fig. 2
Initial torsional deformation diagram.
Assuming that the initial torsional angle
.
of the member varies along its length, the cross-section is subjected to an axial compressive force. When the applied load reaches a critical level, the member will undergo rotation about its shear center. The rotation angle
is also considered as a function of the variation along the length of member, as illustrated in Fig. 2. At a distance z from point o, the torsional angle is denoted as
, and at a position
, it is represented as
. Within the infinitesimal segment
, point D and E move to a new position
and
due to torsion of the member, forming an angle
with the vertical axis, and the horizontal distance from this point to the shear center of the cross-section is given by
Since the value of
is relatively small, the equilibrium relationship expressed in Eq. (4) can be derived from the deformation relationship illustrated in Fig. 2.
4
As shown in Fig. 2, and by applying the previously derived formula, Eq. (5) can be further derived.
5
Furthermore, the non-uniform distribution of torque across the entire cross-sectional area of the component under torsional loading can be mathematically represented by Eq. (6).
6
Under the axial load P, the torsional equilibrium condition involves the sum of the free torque (
), originating from Saint-Venant torsion 30, and the warping torque (
), resulting from section warping restraint. The calculation process is represented by Equations. (7) and (8):
7
8
Where
.
The component will reach its maximum rotation at position (x) in the column, while the rotations at both ends are zero. Here,
and
represent the maximum initial torsion angle and the maximum torsion angle, respectively. Substituting these parameters into Eq. (8) yields Eq. (9).
9
Equation (10) is derived by considering the influence exerted through the action of shear forces, which introduces additional stress components and modifies the overall equilibrium conditions within the system.
10
The column is susceptible to buckling when the upper segment is subjected to axial compressive forces. This buckling behavior is generally characterized by the compressive stress in the outermost edge fibers reaching the material's yield strength. Such a condition signifies that the stress state at the extreme edge fibers has met the yield criterion, potentially initiating plastic deformation within the material and ultimately resulting in global buckling failure of the column. Based on the von Mises yield criterion 31, which accounts for the combined effect of normal and shear stresses, Eq. (11) can be derived to quantitatively describe this critical stress state and predict the onset of yielding under axial compression conditions.
11
The maximum initial torsion angle is assumed to be
. This value is substituted into Eq. (11) to derive Eq. (12), thereby enhancing the precision of the measurement and analysis of the torsion angle.
12
where the normalized slenderness ratio is
.
λ Derivation of a theoretical model based on torsional post-buckling strength considerations
Cruciform cross-section plates fall within the category of unreinforced plate members, which are defined as plate elements supported solely along one side of the non-loaded edge while the opposite side remains unsupported or free. Experimental research in this area has conclusively shown that following the onset of buckling, unreinforced plate members cease to develop lateral membrane stresses. Nevertheless, as long as both loaded edges maintain their straightness, the load-carrying capacity of these members can continue to increase. This additional load-bearing capacity is predominantly sustained by the regions situated adjacent to the supported edges, with the stress at these edges progressively rising until it ultimately reaches the yield stress of the material. This behavior suggests that, provided the buckling of the flanges does not result in a displacement of the resultant stress force point, the post-buckling strength inherent to symmetrical sections can be effectively utilized 32. Based on this theoretical foundation, the current study adopts the load value associated with post-buckling strength as the ultimate bearing capacity of cruciform cross-section members when they experience torsional instability under axial compression.
Fig. 3
Axial compression deformation and cross-section torsion deformation diagrams: (a) Elevation and (b) Section.
As illustrated in Fig. 3, when subjected to an axial compressive force P, the structural member undergoes an axial displacement denoted as w. This total displacement is composed of two distinct components: the first component, wf, arises from axial elastic strain within the material, while the second component, ws, results from torsional deformation. Under these loading conditions, the resulting elastic compressive stress can be mathematically represented by Eq. (13).
13
The yielding behavior of material under these conditions can be theoretically determined by applying the von Mises yield criterion, which takes into full account the shear deformation effects of the material. This analysis becomes particularly relevant when the applied axial pressure P reaches a level sufficient to induce an initial yielding state in the edge fibers of the structural member. The corresponding calculation formula is presented in Eq. (14).
14
The maximum load resulting from compressive stress application is denoted as
. This value serves as the basis for deriving the torsional equilibrium equation, which is expressed as Eq. (15).
15
The ultimate stress can be calculated by Eq. (16) when the load on the structure reaches its ultimate load carrying capacity.
16
The derivation of Eq. (17) is accomplished by substituting Eq. (16) into Eq. (14).
17
Where
.
λ Derivation of a theoretical model incorporating residual stresses
Fig. 4
Distribution of residual stresses: (a) The proposed distribution model, (b) The distribution model used in practice and (c)the distribution range of residual stresses in tension and compression.
The residual stress distribution pattern of steel columns with a cruciform cross-section refers to the typical residual stress distribution observed in welded plates, as proposed by the European Committee for Steel Construction (ECCS) 33. The ECCS model was selected due to its widespread validation for welded European sections, which are structurally similar to the cruciform sections analyzed herein. It provides a representative and conservative estimation of residual stresses in welded I-sections and related shapes, making it a suitable choice for this parametric study. In this model, the specific values of residual compressive stress are derived from a theoretical relationship equation that correlates the residual compressive stresses in welded cruciform cross-sections with the b/t ratio of the extensible plate member. This theoretical relationship is established in the referenced literature 34,35. The distribution of these residual stresses is illustrated in Fig. 4, and the corresponding calculation formula is provided in Eq. (18).
18
The residual tensile stress reaches its ultimate value
, according to the intrinsic relationship of self-equilibrium in the residual stress system
, the distribution length of residual tensile stress and compressive stress within the material can be derived:
.
The torque equilibrium equations are adjusted in accordance with Equation s. (19) and (20), incorporating the specific characteristics of the residual stress distribution within the material or component.
19
20
Equation (21) is referred to in order to evaluate the effect of initial torsional defects and residual stresses on the ultimate stress.
21
A systematic evaluation of the effects of initial torsional defects and residual stresses on the ultimate stresses is possible by transforming Eq. (16) into Equation s. (22) and (23), which provides a more accurate theoretical basis for analyzing the safety of the structure, as shown in Eq. (24).
22
Where
.
23
24
Where
,
and
.
λ Derivation of theoretical models considering elastoplasticity
Under the combined influence of axial compressive stress and residual stress, a cruciform column experiences torsional deformation, which induces a redistribution of stresses across its cross-section. Throughout this deformation process, geometric nonlinearity gives rise to disturbance torque, thereby further diminishing the member's stability-bearing capacity. In severe instances, such deformation can lead to torsional buckling or flexural-torsion coupled buckling of the member. Once the material transitions into the plastic deformation phase, this disturbance torque must be counteracted by the non-elastic torsional stiffness inherent to the cross-section. The equivalent shear modulus of the steel should be ascertained in accordance with the methodology outlined in Reference 20, and the theoretical evaluation ought to be performed employing the von Mises yield criterion.
Because of the alteration in shear modulus, the original Eq. (21) must be modified accordingly, resulting in Equations (25)-(28) to account for the effects of shear deformation.
25
26
Where
.
27
28
Where
.
Numerical model
λ Material and Element
Fig. 5
Division of mesh for columns.
The FE model development was conducted using ANSYS 36. Residual stresses were introduced following the distributions observed in welded sections 34,37. Material behavior was modeled using a three-stage stress–strain relation for stainless steels 38. Geometric imperfections were introduced based on eigenmode shapes scaled to experimental levels 39. These modeling strategies follow established practices in structural stability research. During the meshing process, a preference for hexahedral elements is emphasized by controlling the element shape to ensure optimal accuracy. The mapped meshing approach is applied to achieve a uniform and regular mesh distribution, as illustrated in Fig. 5. A mesh sensitivity analysis was performed using element sizes of 10 mm, 15 mm, and 20 mm. The 15 mm mesh size was selected based on convergence of buckling load results within 2% deviation, ensuring computational efficiency without sacrificing accuracy. For a detailed comparison, see the supplementary materials.
Fig. 6
Stress–strain curve of steel.
The material model utilizes a bilinear isotropic hardening approach, which simplifies the post-yield behavior by defining two distinct stages: the initial elastic region and the subsequent hardening phase. The steel material conforms to the Von Mises yield criterion, a widely accepted criterion for ductile metals 40, indicating that yielding occurs when the second deviatoric stress invariant reaches a critical value corresponding to the yield stress. The steel employed in this study is Q235 grade low-carbon steel (equivalent to ASTM A36 or EN S235), known for its good weldability and mechanical properties. Its stress-strain relationship is depicted in Fig. 6, illustrating the transition from the elastic regime, characterized by linear behavior, to the plastic hardening stage, where the material undergoes permanent deformation. The bilinear model employs a yield stress of 235 MPa, an ultimate tensile strength of 375 MPa, a post-yield tangent modulus of 2.0 GPa, and a failure strain of 0.20. These parameters are representative of the typical mechanical behavior observed in low-carbon steels 38. Although this model exhibits certain limitations when applied to reinforced concrete structures 41, the findings derived from its application in steel structure research 39,42,43 indicate that its performance in characterizing steel stress remains reasonably reliable and acceptable. This model is widely adopted in structural steel simulations as it effectively captures the essential elastic-plastic behavior, including the Bauschinger effect, providing a good balance between computational efficiency and accuracy for post-yield buckling analysis.
λ Boundary condition and loading condition
The boundary conditions of the model are defined by fixing the bottom of column, while horizontal constraints in the x and y directions are applied to the column top to prevent any lateral movement 22. Prior to applying the load, the longitudinal (z-axis direction) degrees of freedom of all nodes at the column top section are coupled to create a unified coupling master node. With this setup, a unit load is applied to the column top coupling master node in the negative z-axis direction, followed by a static analysis to establish the initial stress state of the structure. After obtaining the initial stress state, an eigenvalue analysis method is employed to perform a buckling characteristic analysis. This analysis calculates the deformation response of the first-order mode and its corresponding eigenvalue, thereby evaluating the stability and deformation behavior of the structure under this specific buckling mode. The boundary conditions and loading of the model are shown in Fig. 7.
Fig. 7
Boundary condition and loading condition.
λ Initial torsional imperfection
Fig. 8
Deformation pattern of the first-order torsional buckling mode.
The modal shape of the structure at the onset of first-order instability was adopted as the fundamental representation of the defects, as shown in Fig. 8. The corrected nodal coordinates were derived by scaling the original coordinates of each node in the first-order modal state with a specific coefficient. This method is fundamentally grounded in the deformation characteristics inherent to the first-order mode (specifically, the torsional unstable mode). An initial torsion angle
is deliberately introduced, which is then translated into corresponding coordinate displacements
of the structure along the x-axis and y-axis directions. Through this transformation process, the structure undergoes displacement in both the x and y directions within the original coordinate system, thereby achieving the final update of the node coordinates.
λ Consideration of residual stress
Fig. 9
Results of applied residual stress.
The influence of residual stresses was simulated by incorporating initial stresses at the integration points within the finite elements 44. The modeling process for residual stresses was implemented in two sequential stages. In the first stage, an initial stress data file was created based on the spatial distribution characteristics of the residual stresses and the type of finite elements employed. This data file was then imported into the computational model. Following this, displacement constraints were applied to all nodes of the structure to carry out the initial phase of the analysis. The objective of this step was to achieve an equilibrium state in which the only internal forces present were stresses, with no resultant displacements, this condition represents the state of the member when subjected to the initial stresses at each integration point under fully constrained conditions. During this stage, the constrained reaction forces at all nodes were recorded and saved in a separate data file. The second stage constituted the formal computational analysis, in which both the initial stress data file (containing the integration point stresses) and the constrained reaction force information obtained from the first stage were reloaded into the model. Through this second analysis, the equilibrium state of the structure, incorporating the effects of residual stresses, was determined, thereby effectively representing the initial stress state of the member under residual stress influence 17. To streamline the modeling and computational procedures, the residual stresses within each cross-sectional mesh region were simplified by using the average stress value across the region, which was then uniformly applied to all integration points within that mesh (as illustrated in Fig. 5). The resulting stress distribution after this application is depicted in Fig. 9. Although the residual stress distribution is simplified as regionally uniform in this study for computational efficiency, this simplification may underestimate the localized amplification of stress near welds or stiffener zones. Future refinement of the model could incorporate gradient residual stress fields or measured stress profiles to better reflect real-world distribution.
The nonlinear buckling analysis was performed with comprehensive consideration of initial geometric imperfections. In this analysis, a concentrated load was applied at the primary connection nodes of the structural coupling system, utilizing a static analysis approach that incorporates large deformation effects. As illustrated in Fig. 7, the precise distribution pattern of this concentrated load is clearly depicted. The numerical solution process employs the arc-length method, with calculation convergence significantly enhanced through optimized configuration of both the load sub-step divisions and the maximum allowable equilibrium iterations. The reference load magnitude was determined based on the eigenvalue associated with the structure's first modal shape, and the actual analysis was conducted by applying a load equal to 1.2 times the critical buckling load value in the negative direction along the longitudinal z-axis at the main coupling node located at the column top.
λ Verification of the FE model
Fig. 10
Stress-strain curve and extreme point stress contour under initial torsional imperfection.
The finite element model was validated by comparing its predictions against the derived theoretical solutions (Section 2) for cruciform axial compression members. The simulation results were systematically compared and analyzed with the theoretical predictions derived from the direct analysis method. The test members were uniformly numbered according to the rule 
indicates the member length. For example, the specimen labeled T(20)-1.43°-3 corresponds to the following specific parameters: a width b of 120 mm, a thickness t of 6 mm (calculated from the given b/t ratio of 20, where t = 6 mm), a length 
of 3000 mm (the number 3 in the label signifies 3 meters), and a maximum initial torsion angle of 1.43°.
, where each parameter is defined as follows:
denotes the b/t ratio of the member (in this study, the width b was fixed at 120 mm),
represents the maximum initial torsion angle of the member, and
A
Fig. 11
Stress-strain curve and extreme point stress contour under initial torsional imperfection and residual stress.
Ultimate stresses considering initial torsional imperfections ( ) | Ultimate stresses considering initial torsion and residual stress defects ( ) | |
|---|---|---|
Theoretical value | 201.26 | 177.97 |
FE value | 191.40 | 176.60 |
/ | 0.951 | 0.992 |
The theoretical values are derived from Equations. (17) and (24), and their corresponding simulation curves are presented in Figures. 10 (a) and 11(a). The specific numerical values are comprehensively detailed in Table 1. The computational results demonstrate that the error is consistently maintained within a 5% margin, thereby confirming the high accuracy and reliability of the finite element simulation method employed in this study. The 5% error threshold is widely accepted in structural engineering finite element simulations for validating models against theoretical or experimental benchmarks, as seen in established practices from AISC and other researchers 45. Furthermore, for structural members exhibiting varying degrees of initial defects, the respective force distribution patterns are visualized through the force cloud diagrams illustrated in Figures. 10 (b) and 11 (b), providing an intuitive representation of the stress characteristics under different defect conditions.
Parametric analysis and discussion
A systematic nonlinear analysis of cruciform axial compression members was carried out in this study to comprehensively examine the stress behavior of these structural elements. The investigation encompasses the effects of parameter variations, including the b/t ratio, initial torsional imperfections, residual stresses, and length-to-thickness ratio, on the ultimate load-carrying capacity of the members. By systematically altering these parameters, the study seeks to elucidate their specific influence mechanisms on the overall structural strength and stability. The findings aim to establish a theoretical foundation and offer practical optimization recommendations for engineering design applications.
b/t ratio
Through a systematic variation of the parameter t while maintaining the parameter b at a constant value 120 mm, the width-thickness ratio of the structural member can be effectively modified. This approach allows for an in-depth investigation into how alterations in the width-thickness ratio impact the torsional instability bearing capacity of the member. The precise parameter values employed in this experimental study are detailed in Table 2.
Specimens | b/mm | t/mm | l/mm | |
|---|---|---|---|---|
T (10)-1.43°-3 | 120 | 12 | 1.43 | 3000 |
T (15)-1.43°-3 | 8 | |||
T (20)-1.43°-3 | 6 | |||
T (25)-1.43°-3 | 4.8 | |||
T (30)-1.43°-3 | 4 |
Fig. 12
Effect of b/t ratio on ultimate bearing capacity: (a) Stress vs Strain curve, (b) Relationship between torsion angle and ultimate stress, and (c) Theoretical (according to Section2) and simulated prediction of stress law changes.
The simulation results are illustrated in Fig. 12. Under low load conditions, the stress-strain relationship predominantly exhibits linear behavior, indicative of the initial elastic deformation stage. As the load level increases, this relationship gradually becomes nonlinear, signifying the onset of the plastic deformation stage. Notably, the member b/t ratio significantly influences this transition process. Specifically, larger b/t values cause the material system enters the nonlinear stress-strain phase earlier, highlighting the critical role of the b/t ratio in accelerating the stiffness degradation and the progression toward plastic deformation.
When the applied load reaches the critical limiting value, the stress-strain curves display distinct descending segments, indicating a reduction in load-carrying capacity (Specified by stress in the figure). This phenomenon is most pronounced within the b/t interval of 15–20. Precisely, for small b/t ratios (typically less than 15), the ultimate stress exceeds the yield strength, indicating that failure is primarily governed by the inherent strength characteristics of material. In this scenario, the load-bearing capacity is chiefly determined by the strength properties of material. When the b/t ratio reaches a critical value of approximately 18.5, the load reduction peaks at about 10%, exemplifying the significant influence of the b/t ratio on the critical failure state. At an b/t ratio of approximately 21 (
= 1.15), the ultimate load approaches the theoretical ideal torsional buckling load, suggesting that the member’s load capacity at this stage is mainly controlled by buckling behavior. Beyond this point, as the b/t ratio continues to increase, the measured ultimate load surpasses the ideal torsional buckling load, indicating that the member experiences a substantial decline in load capacity post-buckling. After buckling, the overall load-carrying capacity enhances again due to the post-buckling strength of the plate. However, it is important to note that the initial torsional effects predominantly detrimentally impact the member’s deformation performance. This effect is particularly evident in wider and thicker members and may induce unintended deformation patterns during load application, which could compromise the overall structural stability.
λ Initial torsion angle
The analysis of the b/t ratio parameters indicates that when
is set to 1.0 (which corresponds to a b/t ratio b/t of approximately 18.5), the ultimate load-bearing capacity of the component exhibits the most pronounced decline (The findings reported by Dobrić et al.46 indicated that the measured value in their study was 1.2.). Based on this finding, a specific b/t value within this range was selected to conduct a more detailed investigation into how variations in the initial torsion angle parameter affect the ultimate load capacity. The particular parameters chosen for this analysis are presented in Table 3.
Specimen | b/mm | t/mm | φ0m/o | l/mm |
|---|---|---|---|---|
T(18.5)-1°-3 | 120 | 6.5 | 1 | 3000 |
T(18.5)-1.43°-3 | 1.43 | |||
T(18.5)-2°-3 | 2 | |||
T(18.5)-3°-3 | 3 | |||
T(18.5)-4°-3 | 4 | |||
T(18.5)-5°-3 | 5 | |||
T(18.5)-10°-3 | 10 | |||
T(18.5)-15°-3 | 15 |
Fig. 13
Effect of initial torsional angle on ultimate bearing capacity: (a) Stress vs Strain curve, (b) Relationship between torsion angle and ultimate stress, and (c) Theoretical and simulated prediction of stress law changes.
As shown in the simulation results illustrated in Fig. 13 (a) and (b), the maximum load-bearing capability of the structural component decreases as the initial torsion angle becomes larger. A clearly nonlinear relationship is observed between the decline in load capacity due to the initial torsion angle and the parameter lambda. More specifically, the detrimental effect on load capacity becomes more pronounced as lambda increases up to a certain threshold, after which it starts to gradually lessen (refer to Fig. 13 (c)). When
, this weakening effect reaches its maximum level: within an initial torsion angle range of 0–5°, the ultimate load capacity of the structural member decreases by about 20.3% compared to the reference value. When the initial torsion angle is further increased to the range of 5–10°, there is an additional reduction of approximately 6.6% relative to the baseline. Extending the initial torsion angle to the 10–15° range leads to a further relative decrease of around 6.6%. Beyond this, in the 10–15° initial torsion angle range, the additional decline in load capacity is reduced to about 3%. These results suggest that an initial torsion angle within the 0–5° range has the most substantial negative impact on the ultimate load-bearing capacity. In contrast, when lambda takes on either very small or very large values, the effect of the initial torsion angle on the ultimate load capacity of the structural member becomes relatively less significant. This finding is consistent with the conclusions reported in existing literature 18. These results suggest that current design codes such as GB 50017 and Eurocode 3 may need to consider incorporating limits or correction factors related to initial torsional imperfections. For example, the slenderness ratio limit (λ) could be adjusted for cruciform sections when initial torsional angle exceeds 3°, or a new design curve accounting for φ₀ could be proposed.
λ Residual Stress
Based on the previously referenced literature 47, the welding cross-shaped residual stress values were initially set using a relatively conservative approach. In this study, those values have been further adjusted by incorporating the data provided in literature 48, which allows for an amplification of the residual stress magnitudes. This modification is intended to simulate more severe and unfavorable operational conditions, thereby facilitating a comparative analysis under heightened stress scenarios. Regarding the selection of specimen parameters, the analysis is founded on the basic parameters outlined in Table 3. Particular attention has been given to two specific residual compressive stress conditions: one equivalent to 0.1 times the yield strength (0.1
) and another to 0.25 times the yield strength (0.25
). Previous research has demonstrated that within a certain range, the initial torsion angle can lead to the most pronounced reduction in the ultimate load-carrying capacity. Consequently, the current investigation concentrates solely on the initial torsion angle values that fall within this critical range, with the aim of examining the combined influence of internal residual stresses. The outcomes of these related simulation analyses are presented in Table 4.
Specimen | ||||||
|---|---|---|---|---|---|---|
NR | 217 | 212 | 206 | 199 | 192 | 187 |
Rs0.1 | 205 | 200 | 196 | 189 | 183 | 179 |
Rs0.25 | 184 | 181 | 177 | 172 | 168 | 164 |
Fig. 14
The variation patterns of FEA of initial torsion specimens considering the effect of residual stresses.
As illustrated in Fig. 14, residual stress exerts a notable influence on the ultimate stress of the structural member, and this influence is strongly associated with the initial torsion angle. Based on the legend provided, NR denotes the scenario where no residual stress is present, RS 0.1 refers to the condition in which residual stress is considered at 0.1 times the yield strength (0.1
), and RS0.25 represents the condition where residual stress is taken as 0.25 times the yield strength (0.25
). The data presented in the figure clearly demonstrate that, for a given initial torsion angle, the ultimate stress of the member progressively declines as the level of residual stress increases, from the NR condition to RS 0.1, and then to RS 0.25. For instance, when the initial torsion angle is set at 1°, the ultimate stress measures 217 MPa under the no residual stress condition (NR), reduces to 205 MPa when residual stress equivalent to 0.1
(RS 0.1) is considered, and further drops to 184 MPa under the condition of 0.25
residual stress (RS 0.25). This trend clearly indicates that the presence of residual stresses leads to a significant reduction in the ultimate load-bearing capacity of the member, and this detrimental effect becomes more pronounced as the magnitude of the residual stress increases.
Specifically, within regions exhibiting residual compressive stress, the combined effect of externally applied stress and the inherent residual compressive stress can induce localized plastic deformation in the material prior to the attainment of conventional yield thresholds under overall loading conditions. Conversely, in areas characterized by residual tensile stress, the externally applied stress acts to partially counterbalance the pre-existing residual tensile stress. This results in a reduction of the net tensile stress experienced by the material in those regions, thereby retarding the initiation of the yielding process within that localized zone. However, from the standpoint of the overall structural behavior, the presence of residual stresses causes an uneven distribution of internal stresses across the member. This non-uniformity diminishes the overall stiffness and stability of member, which in turn leads to a significant reduction in its ultimate stress capacity. Additionally, by examining how the ultimate stress varies under different initial torsion angles, it becomes evident that the ultimate stress exhibits a decreasing trend as the initial torsion angle increases, regardless of whether residual stresses are present. This phenomenon occurs because an initial torsion angle introduces additional torsional effects upon loading, which amplifies deformation and promotes stress concentration within the member, thereby further decreasing its ultimate load-carrying capacity.
In summary, residual stresses markedly reduce the ultimate stress of structural members by altering the initial internal stress state, resulting in an uneven distribution of stresses and a consequent decline in overall stiffness and stability, which is consistent with the findings of Chen et al. 49. The greater the level of residual stress, the more pronounced this reduction becomes. Furthermore, an increase in the initial torsion angle exacerbates stress concentration and deformation within the member, leading to an additional decrease in its ultimate stress capacity.
λ Slenderness ratio
The allowable limit for the aspect ratio of cruciform members, as stipulated by the relevant code 12, should not be less than 5.07b/t. Nevertheless, finite element simulation analyses reveal that even when this limit is not satisfied, torsional instability may still arise in cruciform members possessing an initial torsion angle. Here, the term 'initial torsion angle' refers to the definition provided in GB 55006 50, which specifies that the torsional deformation of the column should not exceed h/250 (with h representing the width of the cross section) and that the absolute value of the deformation must not surpass 5 mm 51. For the purposes of this study, the most critical working condition is adopted for calculation—specifically, the maximum horizontal deflection of the column section is set to 5 mm. By systematically adjusting the width and thickness parameters of the member, the aspect ratio is varied, and the relationship between the length-to-thickness ratio and the b/t ratio is examined. This approach allows for the determination of the critical conditions under which torsional and flexural instabilities occur in the members. Additional relevant details are presented in Table 5, where the parameter k denotes the ratio of the aspect ratio to the b/t ratio.
l/mm | b/mm | t/mm | b/t | 5.07 b/t | k | |
|---|---|---|---|---|---|---|
714 | 33 | 6 | 5 | 31.04 | 25.35 | 6.21 |
2700 | 63 | 6 | 10 | 51.87 | 50.7 | 5.19 |
5928 | 93 | 6 | 15 | 77.65 | 76.05 | 5.18 |
12408 | 123 | 6 | 20 | 123.15 | 101.4 | 6.16 |
20856 | 153 | 6 | 25 | 166.28 | 126.75 | 6.65 |
1428 | 66 | 12 | 5 | 25.89 | 25.35 | 5.18 |
2700 | 63 | 6 | 10 | 51.87 | 50.7 | 5.19 |
4604 | 62 | 4 | 15 | 92.84 | 76.05 | 6.19 |
6531 | 61.5 | 3 | 20 | 124.69 | 101.4 | 6.23 |
8052 | 61.2 | 2.4 | 25 | 153.54 | 126.75 | 6.14 |
Fig. 15
The relationship between length-to-slender ratio and b/t ratio for an initial torsion angle under consideration.
The results of the modal analysis indicate that when the slenderness ratio (
) is less than or equal to 5.18, the initial torsional state of the member can significantly reduce its overall stable load-carrying capacity if torsional instability occurs. A critical observation is that the coefficient k, which serves as the primary quantitative metric for evaluating the influence of initial torsional deformation, is not an invariant constant but rather exhibits dynamic variability in response to changes in the member's length-to-slenderness ratio (defined as the ratio of member length to its radius of gyration). As illustrated in Fig. 15, empirical data demonstrate that the k-value typically follows a distinct ascending trend as the length-to-slenderness ratio increases. This discovery highlights a pivotal practical consideration in engineering structural design: the initial torsion angle of a member exerts a substantial and non-negligible influence on the critical load-carrying capacity of cruciform members under combined torsional and flexural instability scenarios. Notably, current structural design codes remain deficient in providing explicit, actionable guidelines for addressing the critical factor of initial torsional effects. Consequently, engineering practitioners must systematically incorporate the impact of initial torsional states into structural analysis and design computations during engineering practice, ensuring rigorous consideration of this critical factor to maintain structural safety and design reliability.
Conclusions
This study systematically investigates the influence of initial torsional imperfections on the buckling resistance of axially loaded cruciform section members through theoretical derivations and nonlinear finite element simulations, yielding the following key conclusions:
1.
A comprehensive theoretical framework considering initial torsional defects, residual stresses, and elastoplasticity is established. This framework, including derived equilibrium equations and ultimate stress expressions, effectively describes the torsional instability behavior and ultimate bearing capacity characteristics of cruciform sections under axial compression, filling the theoretical gap in existing design codes (e.g., GB 50017, Eurocode 3, AISC 360) that neglect initial torsional imperfections.
2.
The FE model developed in this study, which accurately incorporates initial torsional defects and residual stress distributions, is validated by comparing simulation results with theoretical predictions (with errors within 5%). This model provides a reliable tool for analyzing the combined effects of multiple imperfections on the stability of cruciform section members.
3.
Parametric analyses reveal critical influence mechanisms: (i) The b/t ratio significantly affects the transition from elastic to plastic deformation, with a critical value of approximately 18.5 where the ultimate load reduction peaks, and post-buckling strength contributions become notable for larger b/t ratios. (ii) Initial torsional angles within 0–5° exert the most significant detrimental effect on ultimate capacity (20.3% reduction), with the influence weakening beyond this range. (iii) Residual stresses amplify the reduction in ultimate stress, with a more pronounced effect as residual stress levels increase (e.g., 0.25
residual stress leads to a greater reduction than 0.1
), and their coupling with initial torsion exacerbates stress concentration. (iv) The slenderness ratio (
) interacts with b/t, where a k-value (ratio of slenderness ratio to b/t) below 5.18 indicates heightened susceptibility to torsional instability induced by initial torsion.
4.
Based on the parametric results, a resistance reduction factor
is proposed for initial torsional angles (
) in the critical range of 0–5°. This factor may be incorporated into existing column buckling curves to account for initial torsion, particularly for cruciform sections with b/t ≈ 18.5 and
.
5.
This research enhances the understanding of torsional and flexural-torsional buckling behavior of cruciform sections under initial imperfections. However, the current study focuses on Q235 steel and specific defect ranges; future work should extend to other steel grades, complex initial defect combinations (e.g., simultaneous initial flexural and torsion), and dynamic loading conditions to further improve the universality of the findings for engineering design. Furthermore, future investigations will incorporate energy-based methods to quantitatively analyze the dissipation pathways during torsional buckling, providing a more fundamental understanding of the instability mechanisms.
References
1.
Guarracino, F. & Simonelli, M. G. The torsional instability of a cruciform column in the plastic range: Analysis of an old conundrum. Thin Wall Struct. 113, 273–286. 10.1016/j.tws.2016.11.007 (2017).
2.
Dabrowski, R. On torsional stability of cruciform columns. J. Constr. Steel Res. 9, 51–59 (1988).
3.
Lu, L. et al. Global Buckling Investigation of the Flanged Cruciform H-shapes Columns (FCHCs). Appl. Sci. Basel. 11, 11458 (2021).
4.
Asghari, A. & Pavir, A. Evaluation of the shear strength of flanged cruciform steel columns in the panel zone of moment-resisting frames. Structures 73 10.1016/j.istruc.2025.108283 (2025).
5.
Ngian, S. P., Tahir, M. M., Sulaiman, A. & Siang, T. C. Wind-moment design of semi-rigid un-braced steel frames using cruciform column (CCUB) section. Int. J. Steel Struct. 15, 115–124 (2015).
6.
Tahir, M. M., Shek, P. N., Sulaiman, A. & Tan, C. S. Experimental investigation of short cruciform columns using universal beam sections. Constr. Build. Mater. 23, 1354–1364. 10.1016/j.conbuildmat.2008.07.014 (2009).
7.
Lu, W. B., Chen, M., Shi, Y. & Li, B. S. Numerical simulation and specification provisions for cruciform cold-formed steel built-up columns. Structures 51, 484–497. 10.1016/j.istruc.2023.03.043 (2023).
8.
Luo, B. et al. Axial compressive bearing capacity of high-strength concrete-filled Q690 square steel tubular stub column. Constr. Build. Mater. 413 10.1016/j.conbuildmat.2023.134859 (2024).
9.
Hutchinson, J. & Budiansky, B. in Buckling of Structures: Symposium Cambridge/USA, June 17–21, 1974. 98–105 (Springer).
10.
Keintjem, M., Suwondo, R. & Suangga, M. Efficiency Assessment of Cruciform Steel Columns: Balancing Axial Capacity and Weight. 15, 21342–21347 (2025).
11.
Kumar, P. A. & Anupriya, B. Performance assessment of Cruciform steel column: FEM simulation. Mater. Today. 64, 1043–1047 (2022).
12.
GB50017-2017. Standard for design of steel structures (China Architecture & Building, 2017).
13.
EN 1993-1-1:2010. Eurocode 3: design of steel structures: part 1–1:general rules and rules for buildings (British Standard Institution, 2010).
14.
AISC 360. Specification for Structural Steel Buildings (American Institute of Steel Construction, 2016).
15.
Dinis, P. & Camotim, D. in Proc. SSRC Annual Stability Conference, Pittsburgh, USA.
16.
Dobrić, J. et al. Buckling strengths of cold-formed built-up cruciform section columns under axial compression. Thin Wall Struct. 200 10.1016/j.tws.2024.111879 (2024).
17.
Cao, X. et al. Local buckling of high strength steel welded cruciform-section columns under axial compression. Structures 56 10.1016/j.istruc.2023.104941 (2023).
18.
Chen, G. & Trahair, N. S. J. E. Inelastic torsional buckling strengths of cruciform columns. Eng. Struct. 16, 83–90 (1994).
19.
Trahair, N. S. Strength design of cruciform steel columns. Eng. Struct. 35, 307–313. 10.1016/j.engstruct.2011.11.026 (2012).
20.
Trahair, N. S. Shear effect on cruciform post-buckling. Eng. Struct. 49, 24–26. 10.1016/j.engstruct.2012.10.017 (2013).
21.
Makris, N. Plastic torsional buckling of cruciform compression members. J. Eng. Mech-asce. 129, 689–696 (2003).
22.
Schurig, M. & Bertram, A. The torsional buckling of a cruciform column under compressive load with a vertex plasticity model. Int. J. Solids Struct. 48, 1–11 (2011).
23.
Naderian, H. R., Ronagh, H. R. & Azhari, M. Torsional and flexural buckling of composite FRP columns with cruciform sections considering local instabilities. Compos. Struct. 93, 2575–2586. 10.1016/j.compstruct.2011.04.020 (2011).
24.
Yu, S., Ren, X., Zhang, J. & Sun, Z. Numerical simulation on the excavation damage of Jinping deep tunnels based on the SPH method. Geomech. Geophys. Geo. 9, 1 (2023).
25.
Xiang, Z., Yu, S. & Wang, X. Modeling the hydraulic fracturing processes in shale formations using a meshless method. Water 16, 1855 (2024).
26.
Hu, X., Yu, S., Gao, Y., Yu, J. & Dong, J. Experimental and meshless numerical simulation on the crack propagation processes of marble SCB specimens. Eng. Fract. Mech. 308, 110354 (2024).
27.
John, S. K., Cascardi, A., Verre, S. & Nadir, Y. RC-columns subjected to lateral cyclic force with different FRCM-strengthening schemes: experimental and numerical investigation. Bull. Earthq. Eng. 23, 1561–1590. 10.1007/s10518-025-02100-5 (2025).
28.
Yu, S., Hu, X. & Liang, Z. Exploring the elliptic fissure cracking mechanisms from the perspective of sand 3D printing technology and Meshfree numerical strategy. Int. J. Solids Struct. 310, 113216 (2025).
29.
Zhang, Q. et al. Investigating the interaction mechanisms between fissures and layers of SCB specimens using a novel layer 3D printing technology and DEM. Theor Appl. Fract. Mech, 105044 (2025).
30.
Toupin, R. A. Saint-Venant's principle. Arch. Ration. Mech. Anal. 18, 83–96 (1965).
31.
Jones, R. M. Deformation theory of plasticity (Bull Ridge Corporation, 2009).
32.
Chen, S. Design Theory of Steel Structures (Science, 2016).
33.
Stability, E. C. f. C. S. C. o. Manual on stability of steel structures part 2.2 mechanical properties and residual stresses. (European Convention for Constructional Steelwork, (1976).
34.
Jang, D. Y., Liou, J. & Cho, U. Study of residual stress distribution in the machined stainless steel components. Tribol T. 37, 594–600 (1994).
35.
Yuan, H., Wang, Y., Shi, Y. & Gardner, L. Residual stress distributions in welded stainless steel sections. Thin Wall Struct. 79, 38–51 (2014).
36.
Stolarski, T., Nakasone, Y. & Yoshimoto, S. Engineering analysis with ANSYS software (Butterworth-Heinemann, 2018).
37.
Yuan, H. X., Wang, Y. Q., Shi, Y. J. & Gardner, L. Residual stress distributions in welded stainless steel sections. Thin Wall Struct. 79, 38–51. 10.1016/j.tws.2014.01.022 (2014).
38.
Quach, W., Teng, J. G. & Chung, K. F. Three-stage full-range stress-strain model for stainless steels. J. Struct. Eng. 134, 1518–1527 (2008).
39.
de Araujo, R. R. et al. Experimental and numerical assessment of stayed steel columns. J. Constr. Steel Res. 64, 1020–1029. 10.1016/j.jcsr.2008.01.011 (2008).
40.
Barsanescu, P. D. & Comanici, A. M. von Mises hypothesis revised. Acta Mech. 228, 433–446 (2017).
41.
Rabi, M., Shamass, R. & Cashell, K. A. Description of the constitutive behaviour of stainless steel reinforcement. Case Stud. Constr. Mater. 20 10.1016/j.cscm.2024.e03013 (2024).
42.
Al-Thairy, H. & Wang, Y. C. A numerical study of the behaviour and failure modes of axially compressed steel columns subjected to transverse impact. Int. J. Impact Eng. 38, 732–744. 10.1016/j.ijimpeng.2011.03.005 (2011).
43.
Elflah, M., Theofanous, M. & Dirar, S. Behaviour of stainless steel beam-to-column joints-part 2: Numerical modelling and parametric study. J. Constr. Steel Res. 152, 194–212. 10.1016/j.jcsr.2018.04.017 (2019).
44.
Chen, X. & Liu, Y. Finite element modeling and simulation with ANSYS Workbench (CRC, 2018).
45.
Trahair, N. S., Bradford, M., Nethercot, D. & Gardner, L. The behaviour and design of steel structures to EC3 (CRC, 2017).
46.
Dobrić, J. et al. Design of short-to-intermediate slender built-up flanged cruciform columns. Thin Wall Struct. 212 10.1016/j.tws.2025.113181 (2025).
47.
Cao, X. et al. Residual stresses of 550 MPa high strength steel welded cruciform sections: Experimental and numerical study. Structures 44, 579–593. 10.1016/j.istruc.2022.08.024 (2022).
48.
Chen, J. Stability of Steel Structure Theory and Design (Science, 2014).
49.
Chen, Q., Zhang, L. & Zhao, O. Compressive behaviour and capacities of S690 high strength steel welded π-shaped and cruciform section stub columns. Thin Wall Struct. 203 10.1016/j.tws.2024.112194 (2024).
50.
GB 55006 – 2021. General Standard for Steel Structures (Architecture & Building, 2021).
51.
GB 50205 – 2001. Code for Acceptance of Construction Quality of Steel Structures (Architecture & Building, 2001).
A
Data Availability
Data supporting the findings of this study are available from Mr. Li [Zhengchao Li] at xiyuan1988@126.com upon reasonable request.
A
Funding
Declaration
The authors greatly appreciate the financial support from the Key Laboratory of Structural Engineering Open Subjects of Jiangsu Province, China (Grant No. ZD1204) and the Natural Science Foundation of China (51378326).
A
Author Contribution
Zhengchao Li: Methodology, Writing - review & editing, Visualization, Supervision, original draft, Validation, Formal analysis. Zheng Li: Methodology, Writing - review & editing, Visualization, Supervision, Validation, Formal analysis. Long Jin: Methodology, Writing - review & editing, Visualization, Supervision, Validation, Formal analysis. Qicai Li: Conceptualization, Methodology, Writing - review & editing, Visualization, Supervision, Project administration, Resources.
Competing interests
The authors declare no conflicts of interest.
Statement
Data supporting the findings of this study are available from Mr. Li [Zhengchao Li] at xiyuan1988@126.com upon reasonable request.
