Moment-Curvature Full Curve Analysis of Steel Tube Confined High-Strength Concrete Members Using the Strip Method

Literature Overview

The paper by Qian Jiaru, Wang Gang, Zhao Zuozhou, and Kang Zhao, published in Industrial Construction in 2004, Volume 34, Issue 8, pages 70–72, presents a comprehensive study on the flexural behavior of steel tube confined high-strength concrete (HTSC) members. Funded by the Beijing Natural Science Foundation (Grant No. 8002012) and the Ministry of Education Doctoral Point Fund (Grant No. 20000000345), the research was conducted at Tsinghua University. This work is of particular importance because it provides a practical analytical framework—the strip method—for calculating the full moment-curvature relationship of composite steel-concrete sections, which is essential for nonlinear structural analysis and performance-based design.

Experimental Program and Methodology

Twelve steel tube confined high-strength concrete specimens were subjected to flexural loading tests. The specimens were designed to capture the complete flexural behavior from initial cracking through yield to ultimate failure. The primary measurement outputs included load-displacement responses, strain distributions across the section depth, and curvature development throughout the loading history.

The Strip Method Approach

The strip method is a classical analytical technique that divides the cross-section into discrete horizontal strips, each assumed to be in a plane (Bernoulli-Euler hypothesis). For each strip, the stress-strain relationship of the respective material (steel tube or concrete) is applied based on the strain at that location. The moment-curvature relationship is then obtained by integrating the stress distribution across the entire cross-section.

The key steps in the strip method calculation are:

1. Section discretization – Divide the cross-section into n horizontal strips of height Δy.
2. Strain assignment – For a given curvature φ, calculate the strain in each strip as ε_i = φ × y_i (where y_i is the distance from the neutral axis).
3. Stress determination – Using the constitutive laws for steel (elastic-perfectly plastic or with strain hardening) and high-strength concrete (with confinement enhancement), determine σ_i for each strip.
4. Equilibrium check – Verify that Σσ_i × A_i = 0 (axial force equilibrium) and Σσ_i × A_i × y_i = M (moment equilibrium).
5. Iterative solution – Adjust the neutral axis position until equilibrium is satisfied for the given curvature.

Key Analytical Results

The authors simplified the full moment-curvature curve into a three-segment piecewise linear model, which is practical for engineering design applications:

The three-segment linear model provides closed-form equations for sectional flexural stiffness at each stage and a corresponding formula for the ultimate flexural capacity of the section. This simplification allows engineers to incorporate the nonlinear behavior of steel tube confined HTSC members into practical design procedures without requiring full nonlinear finite element analysis for every design iteration.

Technical Significance and Standards Context

The research addresses a critical gap in the design methodology for composite steel-concrete members. Traditional elastic design methods significantly overestimate flexural stiffness in the post-cracking stage and underestimate ductility. The three-segment model bridges this gap by providing:

• A first elastic stiffness (EI)_1 based on the uncracked section properties.
• A post-cracking stiffness (EI)_2 that accounts for concrete cracking and partial steel yielding.
• A plastic stiffness (EI)_3 that represents the fully developed plastic hinge condition.

This approach aligns with the performance-based design philosophy that has since become mainstream in seismic-resistant structural design, particularly as codified in ASCE 41 and similar standards.

Engineering Practice Applications

For practicing engineers, this study offers several actionable insights:

• Performance-based design – The three-segment moment-curvature model can be directly implemented in pushover analysis or nonlinear static analysis software to predict the seismic performance of steel tube confined HTSC columns and beams.
• Stiffness estimation – The initial elastic stiffness is critical for serviceability limit state checks (deflection, vibration), while the plastic stiffness governs ultimate capacity and drift capacity.
• Material selection optimization – The confinement effect of the steel tube on high-strength concrete is captured in the model, allowing engineers to optimize the steel tube thickness relative to the concrete strength grade for cost-effective design.
• Comparison with conventional RC – The model highlights the superior ductility and energy dissipation capacity of steel tube confined HTSC members compared to conventional reinforced concrete sections of similar dimensions.

Study Insights and Reflections

The elegance of this work lies in its balance between analytical rigor and practical applicability. The strip method, while computationally straightforward, captures the essential nonlinear behavior of the composite section. The simplification to a three-segment linear model is particularly valuable for engineers who need to perform hand calculations or incorporate nonlinear behavior into iterative design processes. However, I would note that the model assumes uniform confinement across the section, which may not fully capture the localized effects near loading points or at member connections. Additionally, the study focuses on monotonic loading; cyclic loading behavior, which is critical for seismic applications, may exhibit different stiffness degradation patterns due to the Bauschinger effect in the steel tube and fatigue-related deterioration of the confined concrete. Nevertheless, this work laid important groundwork for the subsequent development of fiber-based analytical models used in modern structural analysis software.