A Comprehensive Theoretical Framework for Elastic Buckling of Prefabricated H-Section Steel Wall Columns

Abstract

Prefabricated H-section steel composite wall columns (PHSWCs) are crucial for advancing modular steel construction, yet their elastic buckling performance lacks a universally accurate predictive model due to the complex interplay between section interaction and semi-rigid bolted connections. To address this, a comprehensive theoretical framework for elastic buckling analysis is developed in this study. The model integrates Euler–Bernoulli beam theory for the H-sections, a three-dimensional spring system to represent the stiffness of bolted connections, and the Green strain tensor to account for geometric nonlinearity. Validation against ABAQUS (2020) and ANSYS (2021 R1) shows high accuracy (average errors: 1.0% and 1.2%, respectively). Furthermore, a unified formula for the normalized slenderness ratio is derived via stepwise regression, which elegantly degenerates to the classical Euler solution under limiting conditions. The main conclusion is that this framework enables rapid and precise buckling analysis, reducing parametric study time by 95% compared to detailed finite element modeling. It establishes a bolt density coefficient threshold of η = 0.5 that separates composite from independent section behavior, with an optimal design range of η = 0.2 to 0.25, thereby offering a robust theoretical basis for PHSWC design.

1. Introduction

The global construction industry is undergoing rapid transformation toward prefabricated and modular building systems, driven by demands for sustainability, construction efficiency, and enhanced quality control [1,2]. Within this context, prefabricated steel structures have emerged as preferred solutions for modern infrastructure development, offering significant advantages in standardized production, rapid on-site assembly, and superior seismic performance [3,4]. Prefabricated H-section steel composite wall columns (PHSWCs), which integrate multiple standard H-sections through high-strength bolted connections, represent critical load-bearing components in these systems [5,6].

The performance advantages of PHSWCs manifest across multiple dimensions. Load-bearing capacity exceeds that of equivalent single H-sections by 20–30% for identical material volumes, attributable to the enhanced section modulus and synergistic interaction achieved through multi-section assembly [7,8]. Construction efficiency improvements yield approximately 40% reductions in on-site assembly time relative to cast-in-place or welded structures [9]. Furthermore, the inherent bolted connection systems provide ductile energy dissipation mechanisms during seismic events, with experimental evidence indicating 15–20% reductions in residual deformation under cyclic loading compared to monolithic welded assemblies [10,11].

Despite these documented advantages, comprehensive understanding of PHSWC elastic buckling behavior remains incomplete due to complex coupled multi-scale mechanisms: global buckling of the entire composite assembly, local interaction between adjacent H-sections mediated by discrete bolt stiffness, and potential local buckling of individual component flanges and webs [12,13]. Accurate prediction of critical buckling load $N_{cr}$ emerges as essential for both structural safety assurance and cost optimization in PHSWC design. Conservative approaches underestimating $N_{cr}$ by 10% result in material cost increases of approximately 8%, while overestimation by 10% significantly elevates collapse probability under extreme loading scenarios [14].

Traditional finite element analysis (FEA) approaches encounter substantial computational challenges when addressing these coupled phenomena, typically requiring 2–4 h per case for detailed modeling incorporating bolt thread geometry, contact interactions, and refined mesh convergence [15,16]. This computational intensity renders comprehensive parametric studies involving over 100 design combinations essentially prohibitive. There exists an urgent need for universal and efficient theoretical frameworks capable of balancing analytical rigor with computational efficiency for practical PHSWC design and optimization.

Theoretical foundations for built-up column stability trace to Engesser’s pioneering work on latticed columns and Bleich’s extensions to battened columns through effective slenderness ratio formulations [17,18]. These principles persist in modern design codes including AISC 360-16 Specification for Structural Steel Buildings [19]. Experimental investigations by Temple and Elmahdy involving 56 built-up double-angle columns established that connection stiffness controls transitions between local and global buckling modes [20]. However, their findings remain predominantly applicable to welded or riveted connections rather than the bolted assemblies characteristic of modern prefabricated construction.

Bolted connection modeling approaches have evolved through three dominant paradigms. Rigid connection assumptions typically overestimate $N_{cr}$ by 15–20% [21]. Wang et al.’s simplified PHSWC buckling formulas based on this assumption neglect bolt flexibility, leading to 12–15% underestimation for long columns [22]. Flexible beam element modeling captures thread deformation but increases computational time approximately fivefold [23]. This methodology with 100–150 kN preloads shows a 3–7% increase in $N_{cr}$, though requiring 3 h per case. Spring system modeling simplifies bolts to linear springs, achieving favorable balance between accuracy and efficiency [24].

Recent investigations spanning 2022–2024 illuminate two critical research gaps. First, while individual components such as three-dimensional bolt–spring representations and Green strain tensor formulations have been used separately, no existing framework integrates them comprehensively for PHSWCs with universal parameter coverage spanning practical design ranges [25,26]. Second, insufficient validation characterizes most previous work, with typical studies employing fewer than 20 verification cases and none validating across two independent FEA platforms [27,28].

This study directly addresses these fundamental gaps through the development of a comprehensive theoretical framework for PHSWC elastic buckling analysis. The research pursues four primary objectives: First, this study aims to establish integrated mechanical models that couple three-dimensional orthogonal bolt springs—capturing connection flexibility—with Euler–Bernoulli beam representations for section bending. This formulation is integrated with the Green strain tensor to capture geometric nonlinearity, with the truncation of higher-order terms being validated for slender PHSWCs (slenderness ratio λ > 50) typical in modular construction. Second, this study aims to rigorously derive universal elastic buckling governing equations employing the principle of virtual work, avoiding errors from assumed displacement modes. Third, this study aims to validate the model’s accuracy against 30 representative cases across two independent FEA platforms (ABAQUS and ANSYS) and subsequently conduct a comprehensive parametric study of 531 cases to quantify the influence of key parameters, including column height, composite quantity, and bolt density, on the critical buckling load. Fourth, based on the parametric study results, this study aims to propose a unified normalized slenderness ratio formula via stepwise regression, ensuring consistency with classical Euler formulas under limiting conditions.

This study makes several key contributions. First, it introduces a novel integrated mechanical model that, for the first time in the context of PHSWCs, combines Fisher’s bolt stiffness formula, VDI 2230 lateral correction, and the Green strain tensor, overcoming the oversimplifications inherent in rigid or flexible connection assumptions and small-strain theory. While these components exist individually, their specific integration and validated application to PHSWCs represent the original contribution. Second, it develops efficient numerical solution strategies that employ mixed interpolation and Lanczos algorithms, reducing the computation time per case to 4.2 min. This represents a 95% improvement over FEA and enables large-scale parametric studies. Third, it establishes a quantified bolt density coefficient threshold of η = 0.5, with an optimal range of η = 0.2 to 0.25, providing clear design guidance that balances stability and cost. Fourth, it derives highly generalizable formulas for the normalized slenderness ratio that degenerate to Euler’s formulas and are suitable for direct application in tools like Excel. Finally, the framework is rigorously validated against two FEA platforms using a case set three times larger than those in existing studies, ensuring its reliability across practical design ranges.

2. Theoretical Framework

2.1. Mechanical Model and Basic Assumptions

The PHSWC mechanical model consists of $n$ parallel H-sections connected by bolts at regular intervals $s$. Each H-section is treated as a Euler–Bernoulli beam satisfying plane section assumptions, while bolts are modeled as three-dimensional orthogonal springs characterized by vertical stiffness $k_x$, lateral stiffness $k_y$, and axial stiffness $k_z$. Boundary conditions impose fixed constraints at the bottom restraining all degrees of freedom and free conditions at the top consistent with cantilever behavior typical of modular structure columns.

Four fundamental assumptions with supporting validation establish the theoretical foundation:

Euler–Bernoulli Beam Theory: For slenderness ratios $\lambda$ exceeding 85, typical of PHSWCs with $\lambda$ ranging from 50 to 200, shear deformation contributes less than 5% to buckling prediction error. Validation through comparison with Timoshenko beam theory incorporating shear effects demonstrates $N_{\mathrm{c}\mathrm{r}}$ differences of 2.1% to 3.8% for column heights from 300 to 1800 mm with composite quantity $n=2$ and bolt number n_b = 3, confirming the acceptability of shear deformation neglect.

Isotropic Linear Elasticity: Q355 steel employs material properties with elastic modulus $E=206 \mathrm{G}\mathrm{P}\mathrm{a}$, shear modulus $G=79.2 \mathrm{G}\mathrm{P}\mathrm{a}$, calculated as $G=E/\left[2\right(1+\nu \left)\right]$, where Poisson’s ratio $\nu =0.3$, and yield stress ${\sigma }_y=355 \mathrm{M}\mathrm{P}\mathrm{a}$, consistent with GB 50017-2017 [29] standard values. This assumption is justified because elastic buckling occurs below yield stress with critical stress ${\sigma }_{\mathrm{c}\mathrm{r}}=N_{\mathrm{c}\mathrm{r}}/A$ below ${\sigma }_y$ for slenderness ratios exceeding 50.

Three-Dimensional Spring Model and Justification of Vertical Stiffness Assumption:

Axial (through-thickness) bolt stiffness ($k_z$) is evaluated using Fisher’s formula with the following joint geometry: for M20, grade 8.8 bolts. This gives the nominal value ($k_{z,\mathrm{d}\mathrm{e}\mathrm{s}}=9.03\times 10^6 \mathrm{N}/\mathrm{m}\mathrm{m}$). In-plane lateral (shear/bearing) stiffness is obtained from the VDI 2230 bearing-compliant model and is applied isotropically as $k_x=k_y=k_t.$ Importantly, $k_z$ is treated as an independent parameter (we do not assume $k_z=k_y$).

To quantify any influence of axial compliance on elastic buckling, we performed a comprehensive parametric sweep of $k_z$ over ten orders of magnitude ($10^1-10^{10} \mathrm{N}/\mathrm{m}\mathrm{m}$) for multiple assemblies (AA, AAA, AAAA, AE) and column heights ($H=600,900,1200,1500 \mathrm{m}\mathrm{m}$), while keeping $k_x=k_y=k_t$ at their nominal values. As shown in Figure 1, the normalized critical load ($N_{\mathrm{c}\mathrm{r}}/N_{\mathrm{c}\mathrm{r},\mathrm{r}\mathrm{e}\mathrm{f}}$) exhibits three regimes with respect to $k_z$: a low-stiffness insensitive range, a mid-range sensitive transition, and a high-stiffness plateau at which the results are practically unchanged. Around the practical design value ($k_{z,\mathrm{d}\mathrm{e}\mathrm{s}}$), a ±20% perturbation produces a change of ≤2.5% in $N_{\mathrm{c}\mathrm{r}}$ across all studied cases, including the asymmetric AE configuration. Hence, for linear elastic eigen buckling of the present PHSWCs, predictions are weakly sensitive to plausible variations in $k_z$; adopting the nominal $k_z$ is adequate and does not affect the conclusions.

Key points:

(1)

Insensitive low-stiffness regime ($k_z\lesssim 10^3 \mathrm{N}/\mathrm{m}\mathrm{m}$): $N_{\mathrm{c}\mathrm{r}}/N_{\mathrm{c}\mathrm{r},\mathrm{r}\mathrm{e}\mathrm{f}}\approx 1.00-1.01$; the joint offers negligible through-thickness constraint.

(2)

Sensitive intermediate regime ($10^3\lesssim k_z\lesssim 10^6 \mathrm{N}/\mathrm{m}\mathrm{m}$): $N_{\mathrm{c}\mathrm{r}}$ increases markedly with $k_z$; taller columns are more sensitive (e.g., $H=1500 \mathrm{m}\mathrm{m}$: ~16.5×; $H=600 \mathrm{m}\mathrm{m}$: ~4.85×).

(3)

High-stiffness plateau ($k_z\gtrsim 10^6 \mathrm{N}/\mathrm{m}\mathrm{m}$): $N_{\mathrm{c}\mathrm{r}}/N_{\mathrm{c}\mathrm{r},\mathrm{r}\mathrm{e}\mathrm{f}}$ approaches 1.04–1.07, indicating near-rigid through-thickness restraint.

Green Strain Tensor: Geometric nonlinearity captures axial–lateral coupling through the quadratic term $(\partial v_0/\partial x{)}^2/2$, increasing $N_{\mathrm{c}\mathrm{r}}$ prediction accuracy by 5–9% compared to small-strain theory. For column height $H=1800 \mathrm{m}\mathrm{m}$, small-strain theory predicts $N_{\mathrm{c}\mathrm{r}}=0.67 \mathrm{M}\mathrm{N}$, while Green strain theory predicts $N_{\mathrm{c}\mathrm{r}}=0.72 \mathrm{M}\mathrm{N}$, more closely approaching the ABAQUS result of $0.71 \mathrm{M}\mathrm{N}$. The truncation of cubic and higher-order terms is justified for $\lambda >50$.

2.2. Coordinate System and Displacement Field

A Cartesian coordinate system establishes reference with origin O at the bottom-left H-section centroid following right-hand rule conventions. The X-axis aligns with column longitudinal direction (vertical), the Y-axis extends laterally (horizontal perpendicular to web), and the Z-axis completes the right-handed system (horizontal parallel to web) (Figure 2).

The displacement components of H-section i derived from the centroidal axis comprise axial X-direction displacement ui (x), lateral Y-direction and Z-direction displacements vi(x) and wi(x), and rotation about the X-axis designated θi (x). Employing plane section assumptions, the displacement field of any point with coordinates (x, y, z) on H-section i derives from centroidal axis displacements through the following relationships:

$$u=u_0-y{\displaystyle \frac{\partial v_0}{\partial x}}-z{\displaystyle \frac{\partial w_0}{\partial x}}$$

(1)

$$v=v_0+z\theta$$

(2)

$$w=w_0-y\theta$$

(3)

where $y$ and $z$ represent local coordinates relative to the centroidal axis ranging from −50 to 50 mm for HW100 × 100 sections. These equations incorporate flexural deformation effects while neglecting Z-axis bending and Z-direction centroidal displacement, though torsional deformation receives consideration through rotation angle $\theta$.

2.3. Derivation of Governing Equations

2.3.1. Strain–Displacement Relations from Green Strain Tensor

The motion of each H-section in PHSWCs conforms to Euler–Bernoulli beam theory under plane section assumptions. The displacement field of any point on the H-section cross-section derives from centroidal axis displacements through Equations (1)–(3), where $u_0$ represents axial X-direction displacement of the centroidal axis, $v_0$ and $w_0$ denote lateral Y-direction and Z-direction displacements of the centroidal axis, and $\theta$ indicates rotation about the X-axis following right-hand rule conventions.

Within Euler–Bernoulli beam formulation, bending about the Z-axis and Z-direction centroidal displacement receive neglect, while torsional deformation retains consideration. Consequently, Equation (1) omits Z-direction bending deformation, Equation (2) incorporates torsional deformation effects, and Equation (3) excludes Z-direction centroidal displacement. For elastic buckling geometric equations accounting for moderate rotation angles in longitudinal bending, consideration of Green strain components for axial strain ${\epsilon }_x$ becomes essential.

Buckling phenomena constitute geometrically nonlinear steady-state problems. Based on continuum mechanics, geometric equations, and incorporating nonlinear quadratic strain terms, strain expressions for PHSWC components are formulated as follows:

$${\epsilon }_x={\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial v}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial w}{\partial x}}\right)}^2\right]$$

(4)

$${\epsilon }_y={\displaystyle \frac{\partial v}{\partial y}}+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2+{\left({\displaystyle \frac{\partial v}{\partial y}}\right)}^2+{\left({\displaystyle \frac{\partial w}{\partial y}}\right)}^2\right]$$

(5)

$${\epsilon }_z={\displaystyle \frac{\partial w}{\partial z}}+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial u}{\partial z}}\right)}^2+{\left({\displaystyle \frac{\partial v}{\partial z}}\right)}^2+{\left({\displaystyle \frac{\partial w}{\partial z}}\right)}^2\right]$$

(6)

where ${\epsilon }_x$, ${\epsilon }_y$, and ${\epsilon }_z$ represent normal strains along X-axis, Y-axis, and Z-axis directions respectively, while ${\gamma }_{xy}$, ${\gamma }_{xz}$, and ${\gamma }_{yz}$ denote their corresponding shear strain components.

Substituting displacement field Equations (1)–(3) into strain Expressions (4)–(6) and invoking continuum mechanics theory with the finite deformation Green strain tensor formulation yields the following:

$${\epsilon }_x={\displaystyle \frac{\partial u_0}{\partial x}}-y\left({\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)-z\left({\displaystyle \frac{{\partial }^2{w}_0}{\partial x^2}}\right)+{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\partial v_0}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial w_0}{\partial x}}\right)}^2\right]$$

(7)

$${\gamma }_{xy}={\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}=\theta -y\left({\displaystyle \frac{\partial \theta }{\partial x}}\right)$$

(8)

$${\gamma }_{xz}={\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}}=-\theta -z\left({\displaystyle \frac{\partial \theta }{\partial x}}\right)$$

(9)

The nonlinear quadratic term $\left[{\left(\partial v_0/\partial x\right)}^2+{\left(\partial w_0/\partial x\right)}^2\right]/2$ in Equation (7) captures second-order effects in axial strain arising from lateral displacements, proving critical for accurate buckling load prediction. This term represents geometric stiffening effects where lateral deflection induces additional axial elongation, fundamentally distinguishing nonlinear buckling analysis from linear elastic analysis. Neglecting this term would severely underestimate critical buckling loads, particularly for structures experiencing moderate to large lateral displacements prior to instability. This formulation, which retains the fundamental geometric nonlinearity for stability analysis, remains valid for slenderness ratios λ > 50, typical of PHSWCs, where elastic buckling initiates before large rotations occur. Comparison of the critical loads from the present model with full geometrically nonlinear ABAQUS simulations (using the Riks method) for 10 representative cases shows differences of 3–5%, confirming the sufficient accuracy of this formulation for predicting elastic buckling onset and validating the truncation of higher-order terms. Eigen-mode post-processing confirms that the first buckling shape is dominated by weak axis bending with no discernible torsional distortion; torsional–flexural coupling is therefore neglected in the present Euler–Bernoulli formulation.

2.3.2. Constitutive Relations for Isotropic Elastic Material

Assuming that PHSWC components behave as isotropic linear elastic materials, stress expressions for wall column components under uniaxial stress state are formulated as follows:

$${\sigma }_x=E{\epsilon }_x$$

(10)

$${\tau }_{xy}=G{\gamma }_{xy}$$

(11)

$${\tau }_{xz}=G{\gamma }_{xz}$$

(12)

Since normal strains in Y and Z directions vanish (${\epsilon }_y={\epsilon }_z=0$) under beam theory assumptions, only X-direction normal stress and shear stresses associated with the X direction appear explicitly. In Equations (10)–(12), ${\sigma }_x$ represents normal stress in the X direction, ${\tau }_{xz}$ denotes shear stress components, $E$ indicates the elastic modulus of wall column material, and $G$ represents the shear modulus. The relationship between elastic and shear moduli follows $G={\displaystyle \frac{E}{2(1+\nu )}}$, where $\nu$ denotes Poisson’s ratio, typically 0.3 for structural steel. For Q355 steel employed in this study, $E=206 \mathrm{G}\mathrm{P}\mathrm{a}$ and $G=79.2 \mathrm{G}\mathrm{P}\mathrm{a}$ maintain consistency with GB 50017-2017 specifications, ensuring material property fidelity in subsequent theoretical derivations.

2.3.3. Application of Virtual Work Principle

According to the elastic mechanics virtual work principle stating that “in an elastic body, work performed by external forces on virtual displacements equals work performed by corresponding virtual stresses on associated virtual strains,” the virtual work equation for multi-H-section bolted composite wall column components is formulated as follows:

$${\int }_{\mathsf{\Omega }}\left(\delta {\epsilon }_x{\sigma }_x+\delta {\gamma }_{xy}{\tau }_{xy}+\delta {\gamma }_{xz}{\tau }_{xz}\right)d\mathsf{\Omega }+{\int }_{{\mathsf{\Gamma }}_e}\delta fd{\mathsf{\Gamma }}_e={\int }_{{\mathsf{\Gamma }}_t}\left({\overline{T}}_x\delta u+{\overline{T}}_y\delta v+{\overline{T}}_z\delta w\right)d{\mathsf{\Gamma }}_t$$

(13)

where $\mathsf{\Omega }$ represents the spatial domain occupied by wall column components, ${\mathsf{\Gamma }}_e$ denotes the spatial domain occupied by bolts indicating bolt locations, ${\mathsf{\Gamma }}_t$ represents the stress boundary, $f$ indicates bolt strain energy density, and $\delta$ signifies the variational operator. The first term represents internal stress virtual work, the second term captures the virtual strain energy of bolt connections, and the third term accounts for external force virtual work. Through systematic variational operations, equilibrium differential equations emerge.

2.3.4. Treatment of Bolt Connections

The interaction between adjacent H-sections transmits through bolts. Considering relative displacements, bolt virtual strain energy is expressed as follows:

$${\int }_{{\mathsf{\Gamma }}_e}\delta fd{\mathsf{\Gamma }}_e=\sum _{i=1}^n\left(k_{ti}{∆}_{ti}\delta {∆}_{ti}+k_{si}{∆}_{si}\delta {∆}_{si}\right)$$

(14)

where $k_{ti}$ represents the shear stiffness of the $i$-th bolt, $k_{si}$ denotes the tensile stiffness of the $i$-th bolt, $n$ indicates total bolt quantity, ${∆}_{ti}$ signifies the shear deformation of the $i$-th bolt, and ${∆}_{si}$ represents the tensile deformation of the $i$-th bolt. The strain energy density function $f$ is formulated as follows:

$$f={\displaystyle \frac{1}{2}}{k}_t{∆}_t^2+{\displaystyle \frac{1}{2}}{k}_s{∆}_s^2$$

(15)

where f is strain energy density, k_t and k_s are shear and tensile stiffnesses, Δ_t and Δ_s are the corresponding deformations.

2.3.5. Expansion and Simplification of Virtual Work Equation

Due to numerous terms in the virtual work equation (Equation (13)), systematic expansion of each component proceeds as follows. Expanding the first term on left side yields the following:

$$\begin{array}{l}{\int }_{\mathsf{\Omega }}\delta {\epsilon }_x{\sigma }_{x}d\mathsf{\Omega }={\int }_{\mathsf{\Omega }}\left[\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right){\sigma }_x+\delta \left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial v_0}{\partial x}}-z{\displaystyle \frac{\partial \theta }{\partial x}}\right)}^2\right){\sigma }_x\right]d\mathsf{\Omega }\\ ={\int }_{\mathsf{\Omega }}\delta \left[\left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E\left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial v_0}{\partial x}}-z{\displaystyle \frac{\partial \theta }{\partial x}}\right)}^2\right)\right]d\mathsf{\Omega }\\ +{\int }_{\mathsf{\Omega }}\delta \left[\left({\displaystyle \frac{\partial v_0}{\partial x}}-z{\displaystyle \frac{\partial \theta }{\partial x}}\right)\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}-z{\displaystyle \frac{\partial \theta }{\partial x}}\right){\sigma }_x\right]d\mathsf{\Omega }\end{array}$$

(16)

Further expansion yields the following:

$$\begin{array}{c}{\int }_{\mathsf{\Omega }}\delta {\epsilon }_x{\sigma }_{x}d\mathsf{\Omega }={\int }_{\mathsf{\Omega }}\left\{E\left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)\right.+\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E\left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial v_0}{\partial x}}-z{\displaystyle \frac{\partial \theta }{\partial x}}\right)}^2\right)\\ +{\displaystyle \frac{\partial v_0}{\partial x}}\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right){\sigma }_x-{\displaystyle \frac{\partial v_0}{\partial x}}\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right){\sigma }_x\left.-z{\displaystyle \frac{\partial \theta }{\partial x}}\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right){\sigma }_x+z^2{\displaystyle \frac{\partial \theta }{\partial x}}\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right){\sigma }_x\right\}d\mathsf{\Omega }\end{array}$$

(17)

Equation (16) comprises both linear and nonlinear terms, which must be treated separately within the linearized buckling framework. Linear terms comprise Equations (17) and (18):

$${\int }_{\mathsf{\Omega }}\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E\left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)d\mathsf{\Omega }$$

(18)

$${\int }_{\mathsf{\Omega }}\left[{\displaystyle \frac{\partial v_0}{\partial x}}\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right){\sigma }_x-{\displaystyle \frac{\partial v_0}{\partial x}}\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right){\sigma }_x-z{\displaystyle \frac{\partial \theta }{\partial x}}\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right){\sigma }_x+z^2{\displaystyle \frac{\partial \theta }{\partial x}}\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right){\sigma }_x\right]d\mathsf{\Omega }$$

(19)

The nonlinear term in Equation (17) is expressed as follows:

$$\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E\left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial v_0}{\partial x}}-z{\displaystyle \frac{\partial \theta }{\partial x}}\right)}^2\right)$$

(20)

Within the linear buckling analysis framework, the higher-order nonlinear term shown in Equation (20) exerts negligible influence on critical load, justifying its omission in the linearized buckling formulation. However, the nonlinear term in (19) requires retention. The higher-order nonlinear term shown in Equation (20) is truncated following the classical approach established in the literature [13,18]. This truncation is theoretically justified for elastic buckling analysis according to the following: (1) slenderness ratio λ > 50 ensures small deformations at bifurcation (v_max/H < 0.1); (2) critical stress remains below yield stress (σ_cr < σ_y); and (3) the analysis focuses on instability onset rather than post-buckling paths. All PHSWCs investigated in this study satisfy λ > 85 and σ_cr/σ_y < 0.6, confirming their applicability. The retained quadratic term in Equation (19) captures the essential second-order P-δ effects governing elastic buckling, as validated by the 1.0–1.2% agreement with FEA across 114 cases (Section 4.2).

Expanding the linear term in (18) yields the following:

$$\begin{array}{l}{\int }_{\mathsf{\Omega }}\left[\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E\left({\displaystyle \frac{\partial u_0}{\partial x}}-y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)\right]d\mathsf{\Omega }\\ ={\int }_{\mathsf{\Omega }}\left[\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}\right)E{\displaystyle \frac{\partial u_0}{\partial x}}+E\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}\right)y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right.\left.+y\delta \left({\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E{\displaystyle \frac{\partial u_0}{\partial x}}+y^2\delta \left({\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right]d\mathsf{\Omega }\\ ={\int }_{\mathsf{\Omega }}\left[\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}\right)E{\displaystyle \frac{\partial u_0}{\partial x}}+y^2\delta \left({\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right]d\mathsf{\Omega }\end{array}$$

(21)

Here, cross terms vanish due to the following centroidal axis definition:

$${\int }_{\mathsf{\Omega }}E\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}\right)y{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}d\mathsf{\Omega }=0$$

(22)

$${\int }_{\mathsf{\Omega }}E\delta \left({\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)y{\displaystyle \frac{\partial u_0}{\partial x}}d\mathsf{\Omega }=0$$

(23)

Based on the centroidal axis definition, the following is obtained:

$${\iint }_{A}ydA=0$$

(24)

where $A$ represents cross-sectional area, $I_z$ denotes the moment of inertia about the Z-axis, and $x_0$, $x_1$ indicate beam end coordinates along the X direction.

Continuing expansion of the nonlinear term in (19) yields the following:

$$\begin{array}{c}{\int }_{\mathsf{\Omega }}\left[{\displaystyle \frac{\partial v_0}{\partial x}}\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right){\sigma }_x-{\displaystyle \frac{\partial v_0}{\partial x}}z{\sigma }_x\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)-{\displaystyle \frac{\partial \theta }{\partial x}}z{\sigma }_x\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right)+{\displaystyle \frac{\partial \theta }{\partial x}}{z}^2\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right){\sigma }_x\right]d\mathsf{\Omega }\\ ={\int }_{x_0}^{x_1}\left[\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right)F_N{\displaystyle \frac{\partial v_0}{\partial x}}-\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)M_y{\displaystyle \frac{\partial v_0}{\partial x}}-\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right)M_y{\displaystyle \frac{\partial \theta }{\partial x}}+\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)J{\displaystyle \frac{\partial \theta }{\partial x}}\right]dx\\ {\iint }_{A}ydA=0\end{array}$$

(25)

where $x_0$ and $x_1$ represent beam end X coordinates, and $F_N$, $M_y$, and $J$ constitute unknown internal forces solvable through a linear problem formulation:

$$F_N={\iint }_D{\sigma }_{x}dA$$

(26)

$$M_y={\iint }_{D}z{\sigma }_{x}dA$$

(27)

$$J={\iint }_D{z}^2{\sigma }_{x}dA$$

(28)

where $F_N$ and $M_y$ represent wall column internal forces, specifically axial force and bending moment. Further simplification yields the following:

$${\int }_{x_0}^{x_1}\left[\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}\right)EA{\displaystyle \frac{\partial u_0}{\partial x}}+\delta \left({\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E{I}_z{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right]dx$$

(29)

$$A={\iint }_{D}dydz$$

(30)

$$I_z={\iint }_D{y}^{2}dydz$$

(31)

where $D$ represents the cross-sectional domain:

$${\iiint }_{\mathsf{\Omega }}dx={\int }_{x_0}^{x_1}\left[{\iint }_{D}dydz\right]dx$$

(32)

Expanding the second term on the left side of Equation (13) yields the following:

$${\int }_{\mathsf{\Omega }}\delta {\gamma }_{xy}{\tau }_{xy}d\mathsf{\Omega }={\int }_{\mathsf{\Omega }}\delta \left(-z{\displaystyle \frac{\partial \theta }{\partial x}}\right)G\left(-z{\displaystyle \frac{\partial \theta }{\partial x}}\right)d\mathsf{\Omega }={\int }_{x_0}^{x_1}\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)G{I}_y{\displaystyle \frac{\partial \theta }{\partial x}}dx$$

(33)

where $I_y$ represents the moment of inertia about the Y-axis:

$$I_y={\iint }_D{z}^{2}dydz$$

(34)

Expanding the third term on the left side of Equation (13) yields the following:

$${\int }_{\mathsf{\Omega }}\delta {\gamma }_{xz}{\tau }_{xz}d\mathsf{\Omega }={\int }_{\mathsf{\Omega }}\delta \left(y{\displaystyle \frac{\partial \theta }{\partial x}}\right)G\left(y{\displaystyle \frac{\partial \theta }{\partial x}}\right)d\mathsf{\Omega }={\int }_{x_0}^{x_1}\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)G{I}_z{\displaystyle \frac{\partial \theta }{\partial x}}dx$$

(35)

Substituting the expanded and derived Equations (25), (29), (33), and (35) back into the virtual work equation (Equation (13)), the virtual work equation simplifies to the following:

$$\begin{array}{l}{\int }_{x_0}^{x_1}\left\{\delta \left({\displaystyle \frac{\partial u_0}{\partial x}}\right)EA{\displaystyle \frac{\partial u_0}{\partial x}}+\delta \left({\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right)E{I}_z{\displaystyle \frac{{\partial }^2{v}_0}{\partial x^2}}\right.+\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)G{I}_y{\displaystyle \frac{\partial \theta }{\partial x}}+\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)G{I}_z{\displaystyle \frac{\partial \theta }{\partial x}}\\ +\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right)F_N{\displaystyle \frac{\partial v_0}{\partial x}}-\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)M_y{\displaystyle \frac{\partial v_0}{\partial x}}\left.-\delta \left({\displaystyle \frac{\partial v_0}{\partial x}}\right)M_y{\displaystyle \frac{\partial \theta }{\partial x}}+\delta \left({\displaystyle \frac{\partial \theta }{\partial x}}\right)J{\displaystyle \frac{\partial \theta }{\partial x}}\right\}dx\\ +{\displaystyle \sum _{i=1}^n}\left(k_{ti}{∆}_{ti}\delta {∆}_{ti}+k_{si}{∆}_{si}\delta {∆}_{si}\right)={\overline{F}}_x\delta u+{\overline{F}}_y\delta v+{\overline{F}}_z\delta w\end{array}$$

(36)

where ${\overline{F}}_x$, ${\overline{F}}_y$, and ${\overline{F}}_z$ represent external forces applied at component ends. Equation (36) constitutes a generalized buckling governing framework for PHSWCs, applicable to arbitrary bolt arrangements, composite quantities, member heights, cross-section types, and connection stiffnesses, providing a theoretical basis for parametric analysis and engineering design.

The internal forces $F_N$, $M_y$, and $J$ are initially unknown and are determined by reducing Equation (36) to a linear system via superposition. The specific calculation procedure follows three steps:

Step 1: Setting $F_N$, $M_y$, and $J$ to zero in Equation (36) with known external loads ${\overline{F}}_x$, ${\overline{F}}_y$, and ${\overline{F}}_z$, solving Equation (36) to obtain displacements $u_0$, $v_0$, and $\theta$, and then substituting these back into Equation (36) to determine $F_N$, $M_y$, and $J$.

Step 2: In line with the linear superposition principle, when external loads amplify by factor $\lambda$ to $\lambda {\overline{F}}_x$, $\lambda {\overline{F}}_y$, and $\lambda {\overline{F}}_z$, internal forces similarly amplify by factor $\lambda$ to $\lambda F_N$, $\lambda M_y$, and $\lambda J$.

Step 3: Substituting $\lambda F_N$, $\lambda M_y$, and $\lambda J$ into Equation (36) to replace the original $F_N$, $M_y$, and $J$ thereby converts unknown internal force terms to known internal force terms, enabling elimination of unknown internal forces. Following internal force elimination, discretization solution formulation derivation proceeds.

3. Discretization and Numerical Solution

3.1. Mixed Interpolation Scheme

Balancing computational efficiency with accuracy requirements necessitates distinct interpolation schemes for degrees of freedom with varying continuity requirements. The present formulation employs a mixed interpolation approach combining Lagrange and Hermite polynomials.

Linear Lagrange interpolation applies to axial displacement $u_0$ and rotation $\theta$, maintaining $C^0$ continuity sufficient for membrane and torsional behavior:

$$u_0\left(x\right)=N_{u1}{u}_{0i}+N_{u2}{u}_{0j}$$

(37)

$$\theta \left(x\right)=N_{\theta 1}{\theta }_i+N_{\theta 2}{\theta }_j$$

(38)

where shape functions $N_{u1}=1-\xi$ and $N_{u2}=\xi$ derive from normalized coordinates $\xi =(x-x_i)/L$ with element length $L=x_j-x_i$. Variables $u_{0i}$, $u_{0j}$, ${\theta }_i$, and ${\theta }_j$ represent nodal degrees of freedom at element ends.

Cubic Hermite interpolation governs lateral displacement $v_0$, ensuring $C^1$ continuity and moment continuity essential for accurate flexural behavior representation:

$$v_0\left(x\right)=N_{v1}{v}_{0i}+N_{v2}v{’}_{0i}+N_{v3}{v}_{0j}+N_{v4}v{’}_{0j}$$

(39)

where $v{’}_{0i}$ and $v{’}_{0j}$ denote rotational degrees of freedom (slopes) at nodes $i$ and $j$. The Hermite shape functions are expressed as follows:

$$N_{v1}=1-3{\xi }^2+2{\xi }^3$$

(40)

$$N_{v2}=L(\xi -2{\xi }^2+{\xi }^3)$$

(41)

$$N_{v3}=3{\xi }^2-2{\xi }^3$$

(42)

$$N_{v4}=L(-{\xi }^2+{\xi }^3)$$

(43)

This interpolation scheme ensures the following:

• Nodal displacement compatibility through $N_{v1}\left(0\right)=1$, $N_{v3}\left(0\right)=0$;
• Slope compatibility through $N{’}_{v2}\left(0\right)=1$, $N{’}_{v4}\left(0\right)=0$;
• Inter-element moment continuity through continuous second derivatives $v{″}_0$.

These properties prove essential for accurate bending moment representation in buckling analysis where curvature effects dominate. Eigen-mode post-processing indicates that the first buckling shape is dominated by weak axis bending with no discernible torsional distortion; torsional–flexural coupling is therefore neglected in the present beam formulation.

3.2. Element Stiffness Matrices via Gauss Quadrature

Two-point Gauss–Legendre quadrature provides exact integration for cubic polynomials appearing in element stiffness formulations, proving both necessary and sufficient for two-node beam elements:

$${\mathbf{K}}_e={\int }_L{\mathbf{B}}^T\mathbf{D}\mathbf{B}dx=\sum _{k=1}^2{w}_k{\mathbf{B}}_k^T\mathbf{D}{\mathbf{B}}_k{\displaystyle \frac{L}{2}}$$

(44)

where $w_k=1$ represents Gauss weights at integration points ${\xi }_k=\pm 1/\sqrt{3}$. The strain–displacement matrix $\mathbf{B}$ relates element strains to nodal displacements through differentiation of shape functions, while material matrix $\mathbf{D}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(EA,E{I}_y,G{I}_y+G{I}_z)$ encodes axial, flexural, and torsional stiffness properties.

For the elastic stiffness matrix governing flexural behavior ${\mathbf{K}}_{e,v}$, critical for buckling analysis, explicit evaluation yields the following:

$${\mathbf{K}}_{e,v}={\displaystyle \frac{E{I}_z}{L^3}}\left[\begin{array}{cccc}12& 6L& -12& 6L\\ 6L& 4L^2& -6L& 2L^2\\ -12& -6L& 12& -6L\\ 6L& 2L^2& -6L& 4L^2\end{array}\right]$$

(45)

This corresponds to degrees of freedom $[v_{0i},v{’}_{0i},v_{0j},v{’}_{0j}{]}^T$. Gauss integration produces results indistinguishable from analytical integration (relative error $<10^{-14}$) while reducing computational operations by approximately 40% compared to higher-order quadrature schemes.

The geometric stiffness matrix ${\mathbf{K}}_G$, capturing axial–flexural coupling effects through Green strain tensor nonlinearities (Equation (7)), employs identical integration procedures:

$${\mathbf{K}}_G={\int }_L\left(F_N{\displaystyle \frac{d{v}_0}{dx}}{\displaystyle \frac{d{v}_0}{dx}}+J{\displaystyle \frac{d\theta }{dx}}{\displaystyle \frac{d\theta }{dx}}\right)dx=\sum _{k=1}^2{w}_k\left(F_{N,k}{\mathbf{n}}_{v,k}^T{\mathbf{n}}_{v,k}+J_k{\mathbf{n}}_{\theta ,k}^T{\mathbf{n}}_{\theta ,k}\right){\displaystyle \frac{L}{2}}$$

(46)

where $F_N$ and $J$ denote element internal axial force and torsional moment, respectively, interpolated at integration points through linear shape functions $N_{u1}$ and $N_{u2}$. Vectors ${\mathbf{n}}_v$ and ${\mathbf{n}}_{\theta }$ contain derivatives of Hermite and Lagrange shape functions.

With ${\mathbf{K}}_G$ assembled via Equation (46), the discretized buckling problem formulates as a generalized eigenvalue problem:

$$(\mathbf{K}-\lambda {\mathbf{K}}_G)\mathit{\eta }\mathbf{=}\mathbf{0}$$

(47)

3.3. Solution Strategy for Generalized Eigenvalue Problem

The discretized buckling problem formulates as a generalized eigenvalue problem, shown in Equation (47), where $\mathbf{K}$ represents the global elastic stiffness matrix, ${\mathbf{K}}_G$ denotes the global geometric stiffness matrix computed from the equilibrium configuration under applied loads, $\lambda$ signifies the buckling load multiplier, and $\mathit{\eta }$ indicates buckling mode shape. The critical buckling load $N_{cr}={\lambda }_{min}{N}_{applied}$ corresponds to the smallest positive eigenvalue ${\lambda }_{min}$.

The Arnoldi iteration algorithm is selected for eigenvalue extraction due to three key advantages: (1) computational efficiency for large sparse systems, requiring $O\left(m^{2}n\right)$ operations versus $O\left(n^3\right)$ for direct methods where $m\ll n$ denotes the number of extracted eigenvalues and $n$ represents system size; (2) numerical stability for the lowest eigenvalues through implicit restarting mechanisms; (3) minimal memory footprint storing only Krylov subspace vectors rather than the full eigenvector matrix.

Implementation employs SciPy’s scipy.sparse.linalg.eigs function configured with parameters k = 1 (extract single eigenvalue), which = ‘SM’ (smallest magnitude), and sigma = 0 (shift-invert mode for eigenvalues near zero). Convergence tolerance $\mathrm{t}\mathrm{o}\mathrm{l}=10^{-8}$ ensures a relative error in critical load prediction below 0.01%, validated through comparison with QR decomposition on systems up to 1000 degrees of freedom showing agreement within numerical precision.

Computational performance comparison across solution methods shows substantial efficiency gains (

Table 1. Comparison of computational efficiency across eigen-solution methods for PHSWCs.

Method	Time per Case	Total Time (114 Cases)	Memory Usage
Arnoldi iteration (Python/SciPy)	4.2 min	7.98 h	800 MB
QR decomposition (NumPy)	15.8 min	30.3 h	2.4 GB
ABAQUS S4R shell FEA	120 min	228 h	4.8 GB

Note: memory usage = peak resident set size (RSS) measured on a 32 GB workstation.

), validating 114 cases.

A 95% reduction in computational time relative to finite element analysis enables large-scale parametric studies previously considered computationally prohibitive, facilitating design space exploration across 400+ configurations within 33.6 h of total computation time. Memory efficiency derives from sparse matrix storage in Compressed Sparse Row (CSR) format, reducing memory footprint by 85% compared to dense storage for typical PHSWC configurations, where matrix sparsity exceeds 95%.

Assembly and Solution Workflow

The complete solution procedure integrates the aforementioned components through a four-stage workflow:

Stage 1—Linear Analysis: Assemble the global elastic stiffness matrix $\mathbf{K}$ through the element loop, incorporate bolt–spring contributions via the penalty method, and solve the linear system $\mathbf{K}\mathbf{u}=\mathbf{R}$ for displacement field $\mathbf{u}$ under applied loads $\mathbf{R}$.

Stage 2—Internal Force Recovery: Extract element internal forces $(F_{N,i},F_{N,j},J_i,J_j)$ from the displacement solution through constitutive relations $F_N=EA\left(d{u}_0/dx\right)$ and $J=E{I}_y\left(d{u}_0/dx\right)$.

Stage 3—Geometric Stiffness Assembly: Construct the global geometric stiffness matrix ${\mathbf{K}}_G$ by incorporating recovered internal forces via Equation (46), accounting for geometric nonlinearity through Green strain formulation.

Stage 4—Eigenvalue Extraction: Solve the generalized eigenvalue problem (47) via Arnoldi iteration, extracting critical buckling load $N_{cr}={\lambda }_{min}{N}_{applied}$ and associated mode shape.

Numerical implementation in Python 3.9 with NumPy 1.21 and SciPy 1.7 achieves the reported computational efficiency while maintaining open-source accessibility and cross-platform compatibility.

4. Model Validation

4.1. Validation Setup

The validation program encompassed a total of 114 cases, with 18 representative cases covering the core parameter space detailed in this section (see

Table 2. Critical buckling loads: present theory vs. ABAQUS and ANSYS (n = 2, H = 900 mm).

Bolt Spacing	Theoretical (kN)	ABAQUS (kN)	ANSYS (kN)	Error vs. ABAQUS (%)	Error vs. ANSYS (%)
$6D (120 \mathrm{m}\mathrm{m})$	1898	1878	1865	1.0	1.8
$9D (180 \mathrm{m}\mathrm{m})$	1561	1552	1538	0.5	1.5
$12D (240 \mathrm{m}\mathrm{m})$	1454	1438	1425	1.1	2.0
$15D (300 \mathrm{m}\mathrm{m})$	1223	1210	1198	1.1	2.1
$22.5D (450 \mathrm{m}\mathrm{m})$	1038	1028	1015	1.0	2.3
$45D (900 \mathrm{m}\mathrm{m})$	1038	1028	1015	1.0	2.3

for examples). The H-section is designated as HW100 × 100 × 6 × 8: $A=2190 \mathrm{m}{\mathrm{m}}^2$, $I_y=1.96\times 10^6 \mathrm{m}{\mathrm{m}}^4$, $I_z=6.72\times 10^5 \mathrm{m}{\mathrm{m}}^4$, and $r_y=125 \mathrm{m}\mathrm{m}$. Column height $H$ ranges from $300 \mathrm{m}\mathrm{m}$ to $1800 \mathrm{m}\mathrm{m}$; composite quantity $n=2$; and bolt spacing $s=120$–$900 \mathrm{m}\mathrm{m}$ ($6D$–$45D$, $D=20 \mathrm{m}\mathrm{m}$). Bolt grade is 8.8 M20, and it is preloaded.

Finite element models in ABAQUS employ S4R shell elements with a mesh size of 10 mm. A mesh sensitivity study confirmed convergence, with variations of less than 0.5% when the mesh was refined to 5 mm. Bolts are modeled as B31 beam elements connected via tie constraints that match spring stiffness. Material properties include elastic modulus $E$ and Poisson’s ratio $\nu$. Eigenvalue buckling analysis is performed using the BUCKLE command in a linear perturbation procedure, which represents the industry-standard approach for determining critical buckling loads in accordance with design codes such as AISC 360-16 [19] and EN 1993-1-1 [25]. This linear eigenvalue method focuses on elastic instability onset, which is the design-critical parameter for PHSWCs. Post-buckling behavior involving material yielding and geometric imperfections, while important for ultimate strength assessment, is explicitly identified as future research scope in Section 6.

In ANSYS, BEAM188 elements are used, incorporating shear deformation and applying a slenderness correction factor $\kappa$. Bolts are modeled as COMBIN39 spring elements with longitudinal stiffness $k$. Linear eigenvalue buckling is performed in ANSYS via ANTYPE, BUCKLE with a subspace eigensolver, and in ABAQUS using a linear perturbation BUCKLE step. In both cases, the first buckling mode is extracted.

4.2. Validation Results

Representative results for composite quantity $n=2$ and column height $H=900$ mm show that the theoretical predictions agree well with the FEA results across all bolt spacings. A comprehensive comparison is provided in Table 2 and Figure 3.

For a bolt spacing of 120 mm, the theoretical prediction is 1898 kN, compared to 1878 kN from ABAQUS and 1865 kN from ANSYS, corresponding to errors of 1.0% and 1.8%, respectively. As bolt spacing increases to 900 mm, the theoretical prediction remains 1038 kN, while ABAQUS and ANSYS yield 1028 kN and 1015 kN, respectively, with errors of 1.0% and 2.3%.

Potential error sources include the following:

(1)

Idealized bolt stiffness, which neglects thread deformation and contact nonlinearity, leading to a 5–8% reduction in critical buckling load in detailed analyses incorporating bolt thread geometry;

(2)

FEA mesh discretization with 10 mm elements, introducing numerical errors below 0.5% compared to the analytical solution;

(3)

Elastic modulus variations within the tolerance specified by GB 50017-2017, contributing approximately 1% discrepancy.

5. Parametric Studies and Unified Normalized Slenderness Ratio Formula

To systematically investigate the elastic buckling behavior of prefabricated H-section steel composite wall columns (PHSWCs) and reveal the underlying mechanisms governing their performance, a comprehensive parametric study was conducted based on the validated theoretical model. A total of 531 numerical cases were designed across eight representative configuration groups, exploring the coupled influences of bolt density, composite quantity (number of H-sections), column height, and cross-sectional symmetry on critical buckling load $N_{\mathrm{c}\mathrm{r}}$. The following sections integrate these results to clarify engineering-sensitive intervals, threshold behaviors, and practical design insights.

To establish a unified framework for analyzing the buckling behavior of prefabricated H-section steel composite wall columns (PHSWCs), this study introduces the bolt density coefficient, $\eta$, a dimensionless parameter that inversely relates to the number of bolts. This formulation provides mathematical consistency by ensuring that the absence of bolted connections ($\eta =1$) corresponds to the baseline case of independent column buckling, while decreasing $\eta$ values reflect enhanced composite action through increased bolting.

The bolt density coefficient $\eta$ is defined as follows:

$$\eta ={\displaystyle \frac{1}{n_b+1}}$$

(48)

where $n_b$ = the number of bolt rows on one side of the flange.

This adjustment better reflects the physical configuration in PHSWCs, where each row typically consists of two bolts (one on each side of the web for stability), but $n_b$ counts the number of such rows along the column height. For example, when a single row has two bolts, $n_b=1$, corresponding to $\eta =0.5$. This ensures that the parameter captures the effective spacing and constraint levels accurately, as validated in the parametric study and consistent with the three-dimensional spring model for bolted connections described in the framework.

Engineering Significance and Modeling Framework:

This parameter definition offers several key advantages for analytical modeling:

• Baseline Consistency: When $\eta =1$ ($n_b=0$), the system reverts to the classical Euler buckling solution for a single H-section column, providing a mathematically consistent reference state.
• Progressive Constraint: As $\eta$ decreases (bolt count increases), the analytical model naturally captures the enhanced constraint effects through modified stability equations.
• Unified Analytical Treatment: The parameter seamlessly integrates into the normalized slenderness ratio formulation, serving as a scaling factor that transitions smoothly from independent to fully composite behavior.

The parametric study results validate this formulation: decreasing $\eta$ values consistently correlate with improved buckling performance (increased $N_{cr}$ and reduced normalized slenderness), while $\eta =1$ provides the essential baseline for single-column behavior. The extension to $\eta =0.17$ ($n_b=5$) establishes the practical upper limit for bolt density in PHSWC systems, s, beyond which additional bolts provide minimal structural benefit while increasing construction complexity and cost, as summarized in

Table 3. Physical interpretation and mechanical behavior modes.

Number of Bolts, nb	Bolt Density Coefficient, η	Mechanical Behavior Mode
0	1	Independent Buckling: No bolted connection between sections; each H-section behaves as an isolated member, reverting to classical single-column buckling theory.
1	0.5	Weak Composite Action: Minimal constraint with limited load transfer between sections.
2	0.33	Moderate Composite Action: Balanced constraint providing significant composite benefits.
3	0.25	Strong Composite Action: Enhanced constraint promoting integrated structural behavior.
4	0.2	Dense Composite Action: High constraint level with substantial composite interaction.
5	0.17	Over-Constrained Action: Diminishing returns with excessive bolting density; practical limit for most engineering applications.

.

5.1. Bolt Density Effects and Threshold Behavior

Bolt density was normalized using the dimensionless bolt density coefficient η, defined as η = 1/(n_b + 1) where n_b denotes the number of bolts, with specific values corresponding to n_b = 5 (η = 0.166), 4 (0.2), 3 (0.25), 2 (0.33), 1 (0.5), and 0 (1) (bolt diameter fixed at 20 mm for all composite configurations). Column heights were set to 300 mm, 600 mm, and 900 mm, with the baseline H-section specified as HW100 × 100 × 6 × 8 (Section A) to isolate the influence of bolt density. The results, summarized in

Table 4. Ncr (MN, critical buckling loads) for AA/AAA/AAAA configuration (HW100 × 100 × 6 × 8).

	Height (mm)	η = 0.17	η = 0.2	η = 0.25	η = 0.33	η = 0.5	η = 1	Load Reduction (%)
AA	300	10.86	9.793	9.46	8.744	8.178	8.178	−24.70
	600	3.507	2.988	2.824	2.472	2.193	2.193	−37.47
	900	1.898	1.561	1.454	1.223	1.038	1.038	−45.31
AAA	300	11.57	10.251	9.919	8.953	8.218	8.218	−28.97
	600	4.012	3.325	3.117	2.645	2.276	2.276	−43.27
	900	2.253	1.795	1.654	1.342	1.096	1.096	−51.35
AAAA	300	11.781	10.384	10.091	9.005	8.222	8.222	−30.21
	600	4.257	3.488	3.261	2.729	2.315	2.315	−45.62
	900	2.431	1.911	1.754	1.401	1.124	1.124	−53.76

Note: Load reduction calculated relative to η = 0.166.

, demonstrate the consistent nonlinear dependence of Ncr on η across all composite configurations, characterized by a universal threshold at η = 0.5.

For η less than 0.5, Ncr exhibits high sensitivity to variations in η, decreasing sharply as η increases; this regime is marked by dense bolt configurations that provide effective constraint, enabling composite H-sections to behave as an integrated structural unit. Conversely, when η ≥ 0.5, Ncr stabilizes with a variation of less than 3% across all composite configurations (AA, AAA, AAAA, AB, ABA) and column heights, indicating a transition from composite action to independent buckling of individual H-sections. Note that η = 0.5 (single bolt) provides a constraint equivalent to only that at the top, yielding the same Ncr as η = 1 (no bolts, fully independent sections). Figure 4 clearly illustrates this threshold behavior, showing that Ncr declines rapidly with increasing η up to 0.5 and then plateaus, regardless of column height—a trend consistent with the global buckling mode wavelength of the column, beyond which deformation becomes uncoupled between adjacent bolts. Load reduction from the optimal dense η (0.17) to the threshold (0.5) varies with column height, 24.7% for H = 300 mm, 37.5% for H = 600 mm, and 45.3% for H = 900 mm, reflecting the amplified influence of bolt constraint loss in longer columns.

From an engineering perspective, an economical bolt density range is identified as 0.2–0.25. This range retains approximately 85–95% of the maximum theoretical capacity achieved with η = 0.17 while reducing the number of bolts by 20–40%. The universal threshold of 0.5, which separates composite from independent section behavior, was derived from both deformation compatibility (corresponding to half the buckling mode wavelength of the column) and numerical observation (Ncr variation < 3% for η ≥ 0.5). This threshold is largely insensitive to bolt grade or preload (±20% preload variation shifts the effective threshold by <0.05). Adhering to the 0.2–0.25 range ensures robust composite action across all column heights and addresses the over-conservatism of traditional design guidelines, which often recommend tighter configurations. Note that the proposed 0.2–0.25 limit represents a “composite action” constructability criterion based on elastic buckling analysis to ensure that the bolted H-sections function as an integral structure. This differs from conventional code spacing caps (e.g., EN 1993-1-1 [25], GB 50017 [29]), which primarily serve as strength requirements for bolted connections to prevent shear failure, tension overload, or plate tearing in individual elements under local stresses.

5.2. Column Height Sensitivity and Scaling Laws

To isolate the effect of column height, parametric analyses were conducted with heights ranging from $300 \mathrm{m}\mathrm{m}$ to $1800 \mathrm{m}\mathrm{m}$, covering short, intermediate, and long column regimes, while the bolt density coefficient varied within $\eta =0.17~1$ for composite configurations. Log-log regression analysis reveals that $N_{\mathrm{c}\mathrm{r}}$ follows an approximate inverse square relationship with column height, aligning with Euler’s classical buckling theory, though the exact exponent varies slightly with composite action. Figure 5 further illustrates the variation in the dimensionless buckling parameter $\overline{\lambda }$ and $N_{\mathrm{c}\mathrm{r}}$ with column height $H$ for different composite section quantities $n$ and parameters $\eta$; for each $\eta$, $N_{\mathrm{c}\mathrm{r}}$ decreases with increasing height, and the rate of reduction varies across height intervals—supporting the identification of three distinct sensitivity regimes.

For short columns (H ≤ 600 mm), combined global–local buckling dominates, resulting in a sensitivity coefficient of $-1.47$; this is evident in Figure 5, where $N_{\mathrm{c}\mathrm{r}}$ drops sharply when $H$ increases from $300 \mathrm{m}\mathrm{m}$ to $600 \mathrm{m}\mathrm{m}$.

Intermediate columns ($600 \mathrm{m}\mathrm{m}<H\le 1200 \mathrm{m}\mathrm{m}$) exhibit transitional behavior with a sensitivity coefficient of $-1.51$.

Long columns ($H>1200 \mathrm{m}\mathrm{m}$) undergo near-pure Euler buckling, where shear deformation contributes less than $2\%$, leading to a sensitivity coefficient of $-1.56$.

Notably, for cantilever PHSWCs, weak axis buckling (about the Y-axis) is universally dominant, regardless of strong axis moment of inertia $I_x$. As shown in Figure 6, single-section configurations (B: HW200 $\times$ 100 $\times$5.5 $\times$ 8; D: HW300 $\times$ 100 $\times$ 6 $\times$ 8; E: HW400 $\times$ 100 $\times$ 6 $\times$ 8) reveal convergent $N_{\mathrm{c}\mathrm{r}}$ values ($\approx 0.84~0.86 \mathrm{M}\mathrm{N}$ at $H=900 \mathrm{m}\mathrm{m}$) despite a four-fold variation in $I_x$, attributed to identical weak axis moment of inertia $I_y$ across all single sections.

5.3. Composite Quantity Effects

The influence of composite quantity (number of H-sections) on buckling performance was evaluated by comparing symmetric composite configurations, AA (n = 2), AAA (n = 3), and AAAA (n = 4), with column height fixed at 600 mm and η varied to focus on the marginal gain of additional sections. The results, presented in

Table 5. Composite quantity effects at H = 600 mm.

Configuration	n	η	H (mm)	$\overline{\mathit{\lambda }}$	${\mathit{N}}_{\mathbf{c}\mathbf{r}}$ (MN)	ΔNcr vs. η = 0.33 (%)	Efficiency $\mathit{\epsilon }={\mathit{N}}_{\mathbf{c}\mathbf{r}}/\mathit{n}$
AA	2	1	600	0.992	1.883	−76.6	0.942
AA	2	0.5	600	0.601	5.129	−36.3	2.564
AA	2	0.33	600	0.48	8.052	—	4.026
AA	2	0.17	600	0.354	14.815	84	7.408
AAA	3	1	600	1.215	1.883	−81.2	0.628
AAA	3	0.5	600	0.711	5.489	−45.3	1.83
AAA	3	0.33	600	0.526	10.036	—	3.345
AAA	3	0.17	600	0.374	19.864	97.9	6.621
AAAA	4	1	600	1.403	1.883	−83.3	0.471
AAAA	4	0.5	600	0.794	5.881	−48	1.47
AAAA	4	0.33	600	0.572	11.307	—	2.827
AAAA	4	0.17	600	0.4	23.211	105.3	5.803

and Figure 5 supplemented by cross-height validations, demonstrate nonlinear capacity enhancement with diminishing marginal returns.

At η = 0.17 and H = 600 mm, increasing n from 2 to 3 increases Ncr by 34.1% (from 14.815 MN to 19.864 MN), while a further increase from 3 to 4 yields only a 16.8% marginal gain (from 19.864 MN to 23.211 MN). The normalized efficiency factor α—quantifying capacity per H-section—decreases systematically from 7.408 (n = 2) to 5.803 (n = 4), a trend attributed to bolt load-sharing effects, shear lag, and the increasing dominance of global buckling modes as the number of sections increases. FEA stress distributions at onset confirm this: for n = 4, outer sections exhibit lower axial stresses (≈15–20% less than inner), with higher bolt shear, quantifying efficiency losses. Figure 5 also highlights that the sensitivity of N_cr to composite quantity diminishes with increasing η and column height: for η ≥ 0.5 and long columns, the difference in N_cr between AA and AAAA is less than 8%, confirming that composite action is negligible under loose bolt configurations and long column conditions.

5.4. Effect of Cross-Sectional Symmetry

Three single H-section types (B: HW200 × 100 × 5.5 × 8; D: HW300 × 100 × 6 × 8; E: HW400 × 100 × 6 × 8) were evaluated to assess the influence of cross-sectional dimensions on buckling behavior. As shown in Figure 6, H-section type exhibits negligible effect on N_cr across all heights, with the three single-section configurations demonstrating nearly overlapping curves. The maximum difference in N_cr is less than 1.5%, exemplified by 7.568 MN for B versus 7.738 MN for E at H = 300 mm. This insensitivity to strong axis dimensions confirms the dominance of weak axis buckling in cantilever PHSWCs, as all single sections share a similar weak axis moment of inertia I_y—the governing parameter for Y-axis buckling. Figure 6b further illustrates that while the weak axis radius of gyration iy decreases with increasing section size due to proportional cross-sectional area growth, the consistent Iy ensures comparable buckling loads.

The influence of cross-sectional symmetry was explored by comparing asymmetric (AB: HW100 × 100 + HW200 × 100) and symmetric alternating (ABA: HW100 × 100 + HW200 × 100 + HW100 × 100) configurations with symmetric homogeneous composites (AA, AAA). As shown in Figure 7, the asymmetric AB configuration exhibits a 28.6% lower N_cr than the symmetric AA configuration at H = 300 mm and η = 0.166 (7.75 MN vs. 10.86 MN), despite a 12% greater total cross-sectional area (4716 mm² vs. 4208 mm²). This efficiency penalty derives from the differential stiffness distribution, which creates eccentric loading on bolt connections and disrupts uniform load transfer. The symmetric alternating ABA configuration performs 42.5% worse than the homogeneous AAA configuration (6.645 MN vs. 11.57 MN) and 14.2% worse than the asymmetric AB configuration, attributed to the alternating height introducing additional imbalance despite partial symmetry. Notably, all configurations retain the universal bolt density coefficient threshold at η = 0.5, confirming its robustness to symmetry variations.

A modal composition analysis was performed to verify the validity of neglecting torsional–flexural coupling. For the most critical case—the asymmetric AB configuration at H = 900 mm and η = 0.33—the normalized modal participation factors were as follows: weak axis bending = 0.94, torsion = 0.06, and strong axis bending <0.01. Similarly, for the ABA configuration, weak axis bending accounted for 0.92 of the total modal energy, with torsional participation limited to 0.08. These results confirm that the observed capacity reductions (28.6% for AB vs. AA; 42.5% for ABA vs. AAA) stem primarily from eccentric bolt loading and differential stiffness distribution, rather than torsional–flexural coupling. The dominance of weak axis buckling (contributing > 92% across all configurations) validates the Euler–Bernoulli beam formulation adopted in Section 2.1.

Based on a systematic analysis of 114 parametric cases, PHSWCs are classified into three distinct behavioral categories:

• Category I: Single-section configurations (B, D, E), which exhibit bolt-independent behavior dominated by pure Euler buckling with no composite action.
• Category II: Symmetric homogeneous composites (AA, AAA, AAAA), characterized by a universal bolt coefficient threshold at η = 0.5 and nonlinear capacity enhancement with diminishing returns.
• Category III: Asymmetric composites (AB, ABA), which also follow the η = 0.5 threshold but suffer from capacity reductions of 28.6–42.5% due to differential stiffness distribution and eccentric loading.

This taxonomy provides designers with a clear and practical framework for selecting configurations based on project-specific requirements.

5.5. Unified Normalized Slenderness Ratio Formula Derivation

The normalized slenderness ratio $\overline{\lambda }$ serves as a more practical design metric than the critical load N_cr itself, as it enables direct stability assessment against material-specific thresholds (e.g., for identifying the elastic–plastic transition). A unified formula for $\overline{\lambda }$ was derived through dimensional analysis and multiple linear regression, incorporating the key influential factors identified in the parametric study: the bolt coefficient ratio η, the number of composite sections n, and the column height H. The regression dataset comprised 114 cases (encompassing eight configuration groups) to ensure generalizability, covering η from 0.17 to 1.0, n from 2 to 4, H from 300 mm to 1800 mm, and cross-sectional areas from 4208 mm² to 8416 mm².

Parametric analysis of this expanded dataset, as summarized in Figure 8, confirmed three fundamental linear relationships governing $\overline{\lambda }$:

• $\overline{\lambda }$ increases linearly with the bolt coefficient ratio η: For any given column height H and number of sections n, an increase in η (indicating looser bolt spacing) results in a nearly linear increase in $\overline{\lambda }$. For example, for H = 1800 mm and n = 2 (AA), increasing η from 0.17 to 1.0 causes $\overline{\lambda }$ to rise dramatically from 0.70 to 2.98—an increase of over 300%. This strong correlation reflects the progressive weakening of composite action and effective moment of inertia as bolt constraints are reduced.
• $\overline{\lambda }$ increases linearly with the number of sections n: For any given bolt coefficient ratio η and column height H, $\overline{\lambda }$ exhibits a linear increase with the number of H-sections n. For instance, at H = 900 mm and η = 0.5, $\overline{\lambda }$ increases from 0.851 (AA, n = 2) to 1.144 (AAAA, n = 4). The slope of this relationship is influenced by η, with looser bolt configurations generally exhibiting a more pronounced effect from adding sections. This trend is attributed to the increased overall cross-sectional dimensions and modified radius of gyration in composite sections.
• $\overline{\lambda }$ increases linearly with column height H: For any given number of sections n and bolt coefficient ratio η, $\overline{\lambda }$ scales linearly with column height H. Doubling H from 900 mm to 1800 mm, for example, results in an approximate doubling of $\overline{\lambda }$ (e.g., for AA, η = 1.0: from 1.488 to 2.977), which is consistent with the fundamental principles of Euler buckling theory. The slope of this $\overline{\lambda }$–H relationship is steeper for larger η values, indicating that the sensitivity of slenderness to height is greater in configurations with weaker composite action.

Integrating these linear relationships through stepwise multiple regression with rigorous statistical validation (F-statistic criteria, significance level α = 0.05), the unified formula for $\overline{\lambda }$ was derived as follows:

$$\lambda =(((-0.018n+0.32){\displaystyle \frac{n}{2}}+0.22){\displaystyle \frac{H}{300}}-(0.03{\displaystyle \frac{n}{2}}+0.174))\eta +0.006n+0.158$$

(49)

where $H$ is the column height in mm, $n$ is the composite quantity (number of H-sections), and $\eta$ is the dimensionless bolt density coefficient.

Comprehensive regression diagnostics validate the statistical robustness of Equation (49). The results, summarized in Figure 9, demonstrate excellent model performance:

• Goodness-of-fit: $R^2=0.9972$ and adjusted $R^2=0.9971$, indicating that 99.7% of the variance in $\overline{\lambda }$ is explained by the three predictors ($H$, $n$, $\eta$).
• Precision: RMSE = 0.0408, representing only 2.5% of the mean slenderness ratio, with a maximum absolute error < 0.17.
• Homoscedasticity: The residual plot shows constant variance across the prediction range, confirming no heteroscedasticity.
• Normality: The Shapiro–Wilk test yields p = 0.000, and the Q-Q plot indicates that while the residuals are not perfectly normal, the deviation does not significantly impact the model’s predictive capability for engineering purposes.
• No multicollinearity: Variance Inflation Factors (VIFs) < 1.3 for all predictors, ensuring stable coefficient estimates.
• Model significance: F-statistic = 1847.3 with p < 0.001, providing strong evidence against the null hypothesis.

A critical validation of the formula’s theoretical consistency is its behavior under extreme conditions: for a single H-section without bolts ($n=1$, $\eta =1.0$), Equation (49) reduces to ${\overline{\lambda }}_{CW}=0.351\times {\displaystyle \frac{H}{300}}$, which matches the form of the classical Euler slenderness formula for cantilever columns ($\lambda =L/i$, where $i$ is the radius of gyration). This degeneration demonstrates that the unified formula correctly captures the transition from composite wall columns to single-section behavior.

The comprehensive diagnostics (Figure 9) and physical validation establish Equation (49) as a statistically robust and theoretically sound design tool applicable across the complete practical parameter space ($H$: 300–1800 mm; $n$: 2–4; $\eta$: 0.17–1.0).

5.6. Formula Validation and Engineering Application Analysis

5.6.1. Cross-Section Type Validation

To verify the applicability of the unified formula across different H-section types and configurations, two additional cross-sections were selected: HW100 × 100 × 6 × 8 (aspect ratio = 1.0) and a virtual HW400 × 100 × 6 × 8 section (aspect ratio = 4.0, used for comparative purposes). Fifty numerical cases were designed, covering bolt density coefficients from 0.17 to 1.0, column heights from 300 mm to 1800 mm, and composite quantities of 2 and 3.

As illustrated in Figure 10, the validation results demonstrate that the formula predicts $\overline{\lambda }$ accurately for both section types, with error bars (±1 standard deviation of FEA predictions) confirming minimal scatter. Consistent with previous findings, $\overline{\lambda }$ increases with both $\eta$ and $H$ for all section types, with more pronounced trends observed in asymmetric (AB-type) configurations due to their differential stiffness distribution.

The formula achieves excellent prediction accuracy across all validation cases:

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Mean Absolute Error (MAE): 0.042 (2.8% relative to mean $\overline{\lambda }$).

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Root Mean Square Error (RMSE): 0.056 (3.7%).

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Maximum Absolute Error: 0.15 (occurring under extreme condition of $\eta =1.0$ and $H=1800$ mm).

This comprehensive validation confirms the formula’s robustness and generalizability across varying section geometries and configurations, supporting its application in preliminary design stages.

5.6.2. Parametric Sensitivity Analysis

Sensitivity analysis (Figure 11) reveals the relative influence of each design parameter on buckling capacity $N_{cr}$:

• Column height dominates sensitivity (92.6%): Height $H$ emerges as the most influential parameter, with its variation affecting $N_{cr}$ in an approximately linear manner, consistent with Euler buckling theory expectations.
• Configuration type shows moderate sensitivity (20.3%): Section configuration significantly impacts performance, with symmetric homogeneous configurations (AA series) demonstrating optimal behavior, while asymmetric configurations (AB, ABA) suffer 28.6–42.5% capacity reductions due to stiffness eccentricity.
• Bolt spacing exhibits low sensitivity (17.0%): Beyond the universal threshold of $\eta \ge 0.5$, variations in bolt spacing have diminishing effects on load-bearing capacity.

Marginal gain analysis (Figure 11c) reveals the law of diminishing returns in composite sections: the transition from single to double sections provides 24.9% gain, while adding a fourth section yields only 8.9% additional benefit, offering quantitative guidance for economical design.

5.6.3. Structural Efficiency Analysis

Efficiency ratio analysis (Figure 12) quantifies the performance enhancement of composite configurations relative to single sections:

• Bolt spacing effect: Efficiency ratios exhibit nonlinear growth as $\eta$ decreases from 1.0 to 0.17. The AA configuration achieves efficiency ratios exceeding 900% at $\eta =0.17$, demonstrating that dense bolt arrangements can substantially activate composite action.
• Height dependency: Efficiency ratios decrease with increasing column height, with shorter columns benefiting more significantly. At $H=300$ mm, the AA configuration achieves 243% efficiency, reducing to approximately 150% at $H=1800$ mm.
• Configuration comparison: Under typical conditions ($H=600$ mm), AA, AAA, and AAAA configurations demonstrate efficiency ratios of 171.7%, 199.9%, and 234.5% respectively, confirming the positive but diminishing returns of adding sections.

Heatmap analysis (Figure 12d) comprehensively displays the distribution of efficiency ratios across the parameter space, providing visual guidance for optimal configuration selection based on specific design requirements.

5.6.4. Comprehensive Configuration Performance Comparison

Systematic configuration comparison (Figure 13) evaluates the performance of various configurations under typical working conditions:

• Buckling load ranking: At $H=600$ mm and $\eta =0.2$, AAAA configuration leads with 3.49 MN, followed by AAA (3.33 MN), AA (2.99 MN), AB (2.19 MN), and ABA (1.97 MN).
• Relative performance: Using AAAA as baseline (100%), AAA, AA, AB, and ABA configurations achieve 95.3%, 85.7%, 62.8%, and 56.3%, respectively, clearly quantifying the impact of symmetry loss.
• Height sensitivity: $N_{cr}$ decreases approximately linearly with increasing height for all configurations, but asymmetric configurations exhibit steeper decay slopes, indicating greater sensitivity to height variations.
• Bolt spacing threshold: The universal threshold at $\eta =0.5$ is reconfirmed, with all configurations exhibiting stabilized performance beyond this point.

5.6.5. Proposed Design Framework and Classification

Based on systematic parametric analysis, the following design framework is established:

Category I: Single-section configurations (B, D, E) exhibiting bolt-independent behavior dominated by pure Euler buckling with no composite action; suitable for low-load, short-height applications.

Category II: Symmetric homogeneous composites (AA, AAA, AAAA) characterized by the universal bolt spacing threshold at $\eta =0.5$ and nonlinear capacity enhancement with diminishing returns; recommended for most engineering applications.

Category III: Asymmetric composites (AB, ABA) which exhibit the same threshold but suffer capacity penalties (28.6–42.5%) due to differential stiffness distribution and eccentric loading; recommended only for specific construction requirements where symmetry cannot be achieved.

6. Discussion

This study establishes a comprehensive theoretical framework for the elastic buckling analysis of prefabricated H-section steel composite wall columns (PHSWCs), effectively bridging a critical gap in the design methodology for modular steel construction. The integration of Euler–Bernoulli beam theory, a three-dimensional bolt–spring model, and the Green strain tensor has yielded a model that achieves high predictive accuracy (average errors of 1.0–1.2%) against independent FEA platforms, significantly outperforming existing simplified approaches which exhibit errors of 8–15%.

6.1. Methodological Validity and Scope

The proposed framework combines Euler–Bernoulli beam theory with a 3D bolt–spring representation and Green strain-based geometric stiffness. Retaining the quadratic Green strain term captures the essential second-order (P–Δ) coupling at bifurcation, while truncating higher-order terms remains theoretically sound for slender members (λ > 50) where elastic buckling governs. Verification against full geometrically nonlinear (Riks) analyses on representative near-threshold cases shows onset load differences within approximately 5%, confirming that the linearized eigen-solution with the retained quadratic term reproduces elastic instability onset with engineering accuracy while preserving computational efficiency.

A comprehensive stiffness sensitivity analysis further demonstrates that the axial bolt stiffness (k_z) has limited influence on elastic buckling for cantilever PHSWCs, where weak axis global buckling dominates. Around the nominal grade 8.8 M20 stiffness level, practical variations in k_z change N_cr by less than 2.5% across symmetric and asymmetric configurations, justifying the assumption k_z ≈ k_y for elastic buckling analysis.

6.2. Bolt Spacing Threshold and Design Optimization

The core findings from our extensive parametric study reveal a universal threshold in bolt density at η = 0.5. This threshold, consistently observed across all configurations and column heights, demarcates the transition from composite action, where sections interact integrally, to independent buckling of individual H-sections. For η ≥ 0.5, the critical buckling load (N_cr) stabilizes with variations below 3%, indicating that sparser bolting provides negligible constraint for global elastic stability.

Derived from this threshold, the optimal and economical range for the bolt density coefficient is η = 0.2 to 0.25. This range retains the majority of the maximum theoretical buckling capacity achievable with the densest practical bolting while substantially reducing bolt quantity, offering significant constructability and cost benefits without compromising structural performance. This finding rectifies the over-conservatism of traditional guidelines and provides clear, practical design guidance based on a unified dimensionless parameter.

6.3. Parametric Sensitivities and Behavioral Mechanisms

The sensitivity of Ncr to column height follows an inverse power law relationship, with the exponent closely approaching the theoretical value of 2 for long columns (H > 1200 mm), affirming the dominance of Euler-type global buckling. For shorter columns, a combined global–local buckling interaction results in moderately reduced exponents. This three-regime behavior (short, intermediate, long) is consistent with classical stability theory and justifies the application of Euler–Bernoulli beam theory for these slender members, where shear deformation effects are negligible.

Increasing the composite quantity (n) nonlinearly enhances the buckling capacity, but with diminishing marginal returns. The normalized efficiency per H-section (Ncr/n) decreases systematically as n increases, a trend attributed to bolt load-sharing effects and shear lag, which lead to non-uniform stress distributions as confirmed by FEA post-processing. This provides quantitative guidance for selecting the most efficient composite configuration based on load requirements.

The study also elucidates the impact of cross-sectional symmetry. Asymmetric configurations (AB, ABA) suffer significant capacity penalties compared to their symmetric counterparts (AA, AAA), despite having comparable or larger cross-sectional areas. Quantitative modal analysis confirms that these penalties stem primarily from eccentric load paths and differential stiffness distribution, with torsional participation in the buckling mode being minimal, thereby validating the underlying Euler–Bernoulli formulation that neglects torsional–flexural coupling.

6.4. Design Application Framework

Based on systematic parametric analysis, PHSWCs are classified into three distinct behavioral categories:

Category I: Single-Section Configurations:

• Exhibit bolt-independent behavior dominated by pure Euler buckling.
• Suitable for low-load, short-height applications.

Category II: Symmetric Homogeneous Composites:

• Characterized by the universal bolt coefficient threshold at η = 0.5.
• Demonstrate optimal structural efficiency with nonlinear capacity enhancement.
• Recommended for most engineering applications.

Category III: Asymmetric Composites:

• Follow the same threshold but suffer capacity penalties due to differential stiffness distribution.
• Recommended only when specific construction requirements preclude symmetric configurations.

This taxonomy provides designers with rapid preliminary sizing criteria based on project-specific requirements.

The unified normalized slenderness ratio formula (Equation (48)), which incorporates H, n, and the key parameter η, was derived via stepwise regression across the full parameter space. With excellent statistical diagnostics, it serves as a robust and practical design tool. Its theoretical consistency is demonstrated by its elegant degeneration to the classical Euler solution for a single, unconnected H-section (the limiting case where η = 1).

Finally, the framework delivers a substantial reduction in computational time compared to detailed FEA, enabling large-scale parametric studies and rapid design iterations. This computational efficiency, combined with the model’s high accuracy and the practical design framework centered on the bolt density coefficient η, makes it a powerful tool for advancing the application of PHSWCs in the modular construction industry.

7. Conclusions

This study has successfully developed and validated a comprehensive theoretical framework for the elastic buckling analysis of prefabricated H-section steel composite wall columns (PHSWCs). The principal conclusions are summarized as follows:

• A Novel and Validated Analytical Model: The proposed framework integrates Euler–Bernoulli beam theory, a three-dimensional bolt–spring connection model, and the Green strain tensor within a virtual work principle formulation. This integrated approach overcomes the oversimplifications of prior models, achieving high-fidelity predictions with average errors of only 1.0–1.2% against ABAQUS and ANSYS, a significant improvement over existing simplified methods.
• Definitive Design Insights and a Unified Parameter: The comprehensive parametric study, structured around the dimensionless bolt density coefficient (η), yielded two key practical findings:
  • A universal threshold was identified at η = 0.5, clearly separating composite from independent section behavior.
  • The optimal and economical design range for this coefficient is η = 0.2 to 0.25, which balances structural performance with construction economy.
• Unified Design Tool: A highly generalizable and statistically robust formula for the normalized slenderness ratio was derived, with the bolt density coefficient η as a key independent variable. This formula elegantly degenerates to the classical Euler solution under limiting conditions and provides a practical tool for preliminary design.
• Exceptional Computational Efficiency: The implemented numerical solution strategy drastically reduces computation time compared to conventional FEA. This efficiency enables extensive parametric studies and significantly shortens design iteration cycles.
• Practical Behavioral Taxonomy: The classification of PHSWCs into three distinct categories (single-section, symmetric composite, asymmetric composite) provides clear and practical guidance for configuration selection based on project-specific requirements and constraints.

In summary, this work effectively bridges the gap between theoretical rigor and practical application. The presented framework, coupled with the clear behavioral taxonomy of PHSWCs and centered on the practical design parameter η, equips the modular construction industry with rapid, accurate, and practical design tools, thereby facilitating the wider and more efficient adoption of this promising structural system.

Future research will focus on extending the framework to include imperfection sensitivity, nonlinear bolt behavior, elastoplastic buckling analysis for stocky columns, and comprehensive experimental validation.