Three-Dimensional Degenerated Laminated Curved Beam Element for Stability Analysis of Steel Tube Concrete Arches

Literature Overview

The paper by Wang Xiaogang, published in the Chinese Journal of Computational Mechanics in 2001, presents a specialized finite element formulation for the stability analysis of steel tube concrete (STC) arch structures. Building upon a previously developed three-dimensional degenerated beam element, the author constructs a 12-20 node three-dimensional degenerated laminated curved beam element using the equivalent numerical integration method. The formulation accounts for geometric nonlinearity and is specifically designed for the linear elastic stability analysis of laminated beams and arches. The engineering application is demonstrated through the calculation of in-plane and out-of-plane buckling stability factors for the Shaoxing Light Textile Bridge, which features STC arch members.

Theoretical Framework and Element Formulation

The development of specialized finite elements for STC structures is motivated by the unique mechanical behavior of these composite members. An STC arch member consists of a steel tube and a concrete core, each with distinct material properties and deformation characteristics. The interaction between the steel and concrete components creates a complex stress state that requires careful modeling for accurate stability analysis.

The key features of the proposed element formulation include:

Feature	Description	Engineering Significance
12-20 node configuration	Variable node count allows flexibility in mesh refinement	Enables accurate representation of curved geometries and stress concentrations
Degenerated laminated curved beam	Accounts for the layered nature of STC members (steel tube + concrete core)	Captures the composite behavior and interfacial interaction between steel and concrete
Equivalent numerical integration	Simplifies the integration of complex constitutive matrices	Improves computational efficiency while maintaining accuracy
Geometric nonlinearity	Includes large displacement effects in the formulation	Essential for accurate buckling analysis where pre-buckling deformations are significant
Linear elastic stability analysis	Determines the critical buckling load under elastic conditions	Provides the upper bound for stability assessment and identifies the buckling mode

The concept of a "laminated" beam element is particularly appropriate for STC arches because the steel tube and concrete core can be treated as distinct layers with different elastic moduli, Poisson's ratios, and thermal expansion coefficients. The degenerated formulation allows the curved beam element to be derived from a more general element formulation, maintaining mathematical consistency while reducing computational complexity.

Stability Analysis Methodology

The stability analysis of STC arches involves determining the critical buckling load and the corresponding buckling mode. The proposed element formulation enables the following analysis capabilities:

In-plane buckling: The arch buckles within the plane of its curvature, typically involving a combination of axial compression and bending.
Out-of-plane buckling: The arch buckles perpendicular to its plane of curvature, typically involving lateral-torsional instability.
Geometric nonlinear effects: The formulation accounts for the change in geometry due to pre-buckling deformations, which is essential for accurate critical load prediction.

The finite element formulation for linear elastic stability analysis typically involves the following steps:

Step	Description	Mathematical Basis
Discretization	Divide the arch into finite elements	The proposed 12-20 node degenerated laminated curved beam element
Stiffness matrix assembly	Assemble element stiffness matrices into the global stiffness matrix	Based on the principle of virtual work or minimum potential energy
Geometric stiffness	Compute the geometric stiffness matrix accounting for pre-buckling stresses	Related to the initial stress state in the structure
Eigenvalue problem	Solve the generalized eigenvalue problem for the critical load	(K + λKg)φ = 0, where K is the elastic stiffness, Kg is the geometric stiffness, and λ is the load multiplier
Buckling mode extraction	Identify the buckling mode shape corresponding to the lowest eigenvalue	Provides insight into the failure mechanism

Engineering Application: Shaoxing Light Textile Bridge

The Shaoxing Light Textile Bridge serves as a practical engineering case study for validating the proposed element formulation. The bridge features STC arch members that are subjected to complex loading conditions, including dead load, live load, and wind load. The stability analysis provides the in-plane and out-of-plane buckling stability factors, which are essential for the design verification of the arch members.

The application of this specialized element to a real bridge structure demonstrates the practical utility of the formulation. The calculated stability factors can be directly compared with the design requirements specified in relevant codes, such as GB 50396-2014 for STC structures and the relevant bridge design codes. If the calculated stability factors are below the required minimum values, design modifications are necessary, such as increasing the cross-sectional dimensions, adding lateral bracing, or modifying the arch geometry.

Comparison with Alternative Analysis Methods

Analysis Method	Advantages	Limitations
Proposed degenerated laminated curved beam element	Captures composite behavior; accounts for geometric nonlinearity; efficient for curved geometries	Requires specialized formulation; limited to linear elastic stability
Shell element model	Captures full 3D stress state; can model local buckling	Computationally expensive; requires fine mesh; may be overkill for global stability
Simplified beam element (non-laminated)	Simple and efficient	Does not capture the composite behavior of STC members; may overestimate stability
Analytical methods	Provides closed-form solutions for simple geometries	Limited to idealized geometries and loading conditions; not applicable to complex bridge structures

The proposed element formulation offers a balanced approach between accuracy and computational efficiency, making it suitable for the stability analysis of STC arch structures in engineering practice.

Study Insights and Reflections

This research represents an important contribution to the finite element analysis of STC structures, particularly for arch configurations that are increasingly used in modern bridge engineering. The development of specialized elements that capture the composite behavior of STC members is essential for accurate stability analysis, as the interaction between the steel tube and concrete core significantly influences the buckling behavior.

The use of the equivalent numerical integration method is a practical innovation that simplifies the implementation of the element formulation while maintaining computational accuracy. This approach is particularly valuable for engineering applications where computational efficiency is important, such as parametric studies and optimization of arch geometry and cross-section.

For practitioners in steel pipe manufacturing and structural engineering, this study highlights the importance of accurate stability analysis in the design of STC arch structures. The stability factors calculated using the proposed element should be incorporated into the design process to ensure that the arch members have adequate buckling resistance under all relevant loading conditions. The findings also underscore the value of developing specialized finite element formulations tailored to specific structural systems, as these can provide more accurate and efficient analysis results than general-purpose elements. The Shaoxing Light Textile Bridge case study demonstrates that the proposed formulation can be successfully applied to real engineering structures, providing reliable stability predictions that support safe and economical design decisions.