Variational Model for Interactive Buckling of Thin-Walled I-Section Members with Nonlinear Materials

Abstract: Thin-walled cold-formed members are highly susceptible to the interaction of multiple instability modes. Consequently, research into the mechanical mechanisms of interactive buckling for various cross-sectional shapes and material types requires further refinement and development. This study establishes a precise variational model for slender thin-walled I-section members, comprehensively accounting for material nonlinearity, geometric nonlinearity, initial global geometric imperfections, and the connection stiffness between flanges and webs. Using this model, we analyze interactive buckling behavior, post-buckling equilibrium paths, ultimate load-carrying capacity, and local buckling wavelengths. The results indicate that the nominal yield strength of the material significantly affects the post-buckling capacity, while initial global imperfections primarily influence the ultimate capacity rather than the post-buckling behavior. Furthermore, the local buckling wavelength of the plates decreases as the post-buckling load declines. The proposed variational model was validated against existing experimental data, showing high consistency. This model elucidates the full-process mechanical mechanism of nonlinear global-local interactive buckling in thin-walled members, providing a theoretical foundation for further research and structural design.

1. Introduction

Thin-walled members, particularly those formed through cold-working processes, are widely used in civil engineering due to their high strength-to-weight ratio. However, their slender nature makes them prone to various instability modes, including global buckling (Euler buckling), local buckling of individual plate elements, and distortional buckling. In many practical scenarios, these modes do not occur in isolation but interact with one another—a phenomenon known as interactive buckling. This interaction often leads to a significant reduction in the member's load-carrying capacity and a more sudden, brittle failure mode compared to isolated buckling.

While traditional linear elastic theories provide a baseline for understanding these phenomena, they often fail to capture the complex behavior of members made from nonlinear materials (such as stainless steel or high-strength aluminum alloys) or those exhibiting significant geometric nonlinearities. Furthermore, the influence of initial geometric imperfections and the actual rotational restraint provided at the junction of the flange and web are critical factors that must be integrated into a unified analytical framework.

Cold-formed thin-walled members exhibit three fundamental buckling modes: local buckling of the plates, distortional buckling of the cross-section, and global buckling of the member. Under certain conditions, coupling effects between different buckling modes can occur within the member, significantly impacting its load-carrying capacity and post-buckling performance. In recent years, many scholars have investigated the interactive buckling performance of thin-walled members with various cross-sectional forms, including lipped channel sections and angle sections. In most literature, the material properties are assumed to follow an ideal elasto-plastic model or a multi-linear constitutive model without a yield plateau. However, the stress-strain curves of nonlinear metallic materials, such as stainless steel, are smooth and continuous without a distinct yield plateau.

Given the limited analytical research on the interactive buckling of thin-walled members made of nonlinear metallic materials, and the lack of analytical methods accounting for both global initial imperfections and material nonlinearity, this work employs the Rayleigh-Ritz method to establish a variational model. This model is applicable to slender thin-walled I-section members with simply supported ends undergoing global-local interactive buckling failure modes. The model comprehensively accounts for geometric nonlinearity, material nonlinearity (Ramberg-Osgood model), global initial imperfections, and varying connection stiffnesses between the flanges and the web.

2. Material Constitutive Model

The classical continuous constitutive model for nonlinear metallic materials is the Ramberg-Osgood model. In its original form, the model is expressed as:

$$\epsilon = \frac{\sigma}{E} + K \left( \frac{\sigma}{E} \right)^n$$

where $E$ represents the initial Young's modulus, $K$ is the material hardening coefficient, and $n$ is the strain hardening exponent. Following subsequent modifications, the classical expression was established as:

$$\epsilon = \frac{\sigma}{E} + 0.002 \left( \frac{\sigma}{\sigma_{0.2}} \right)^n$$

In this formulation, $\sigma_{0.2}$ represents the nominal yield strength (the stress at $0.2\%$ plastic strain). This model is widely utilized in structural engineering to describe the stress-strain behavior of materials that exhibit significant non-linear characteristics without a distinct yield point, such as aluminum alloys and stainless steel.

3. Variational Modeling

This study is based on the Rayleigh-Ritz method. First, continuous displacement functions for the global member and local plates are assumed. Subsequently, the strain energy and total potential energy formulas of the member are derived. The equilibrium differential equations and integral boundary conditions are then derived using the Euler-Lagrange equations and the stationary condition of the functional.

3.1 Global and Local Displacement Functions

The continuous displacement function for the I-section slender member is constructed by splicing T-section components. For slender members, the critical stress for global buckling about the weak axis is typically smaller than the critical stress for local plate buckling. It is assumed that the web does not undergo local buckling. During the loading process, the member first undergoes global buckling, causing the flange on one side of the web to be in compression and the flange on the other side to be in tension. Subsequently, local buckling occurs in the compression flange, resulting in global-local modal coupling.

The out-of-plane displacement function $w_f(x, z)$ for the compression flange is assumed to follow:

$$w_f(x, z) = f \cdot \sin\left(\frac{m\pi z}{l}\right) \cdot \gamma(x)$$

3.2 Total Potential Energy

The total potential energy $\Pi$ of the system is expressed as the sum of the internal strain energy $U$ and the potential energy of the external loads $V$:

$$\Pi = U + V$$

The internal strain energy $U$ accounts for the membrane and bending energies of the flanges and the web, considering the coupling between global displacement and local plate deformations. Material nonlinearity is incorporated by using the Ramberg-Osgood relationship within the strain energy density integrals.

3.3 Connection Stiffness

A key feature of this model is the inclusion of the connection stiffness between the flange and the web. Rather than assuming a perfectly rigid or pinned connection, we introduce a rotational spring constant $k_{\phi}$ to simulate the actual constraint conditions. The strain energy of the torsional springs is expressed as:

$$U_{sp} = \frac{1}{2} \int_{0}^{L} k_{\phi} \left( \left. \frac{\partial w_f}{\partial x} \right|_{x=0} \right)^{2} dz$$

4. Results and Discussion

The variational equations are solved numerically to obtain the load-displacement curves and the evolution of buckling modes.

4.1 Influence of Material Yield Strength

The analysis reveals that the nominal yield strength of the material plays a dominant role in the post-buckling regime. As the yield strength increases, the member exhibits a higher residual capacity after the onset of local buckling. However, the transition from elastic to plastic behavior significantly softens the response, leading to a more complex interaction between material yielding and geometric instability.

4.2 Influence of Initial Imperfections

Initial global geometric imperfections primarily affect the peak load or the ultimate carrying capacity of the member. The ultimate bearing capacity of the component is inversely proportional to the amplitude of the initial global imperfection. For instance, when the amplitude of the global initial imperfection reaches $L/1000$, the ultimate capacity decreases by approximately $15\%$.

4.3 Local Buckling Wavelengths

The model also tracks the change in the local buckling wavelength. It is observed that the local buckling wavelength of the plate elements tends to decrease as the post-buckling load drops. This "wavelength shortening" is a characteristic feature of the nonlinear interaction between the global bending of the member and the localized wrinkling of its constituent plates.

5. Model Validation

To verify the accuracy of the proposed variational model, simulations were conducted to replicate experimental tests on thin-walled stainless steel members (e.g., specimen I404-L3000). The experimental ultimate load was approximately $24.71\text{ kN}$, while the variational model predicted $25.36\text{ kN}$, a relative error of only $2.6\%$. In the deep post-buckling stage, the error in bearing capacity remained low ($2.7\%$), significantly outperforming standard finite element simulations which showed errors up to $16.1\%$ due to the omission of initial geometric imperfections.

6. Conclusion

A variational model for the interactive buckling of thin-walled I-section slender members was established using the Rayleigh-Ritz method. The model successfully accounts for global initial imperfections, material nonlinearity (Ramberg-Osgood), and geometric nonlinearity.

The results demonstrate that:
1. Post-buckling capacity is positively correlated with the material's nominal yield strength.
2. Local buckling wavelengths decrease as the post-buckling load declines.
3. Global initial imperfections significantly reduce ultimate capacity but have a smaller impact on the far post-buckling path.
4. The model provides high accuracy compared to experimental data, with ultimate load errors within $5\%$.

This model provides a theoretical foundation for the structural design of thin-walled members made of nonlinear materials and facilitates efficient parametric analysis compared to purely numerical methods.