Combined Elastic Modulus Theory of Double Circular Sandwiched CFST

Literature Overview

The paper by Xiong Ping, Wang Sheng, and Wen Tao, published in the Journal of Huaqiao University in 2019 (Vol. 40, No. 1, pp. 48-55), presents a theoretical framework for calculating the combined elastic modulus of double circular sandwiched concrete-filled steel tube (CFST) members. The study employs thick-walled cylinder spatial axisymmetric elasticity methods to account for the deformation compatibility between the steel tubes and the concrete layers. This work addresses a specialized composite section type that combines the advantages of multiple steel tubes with concrete infill, offering enhanced stiffness, ductility, and energy absorption capacity for structural applications.

Core Technical Findings

The authors established a calculation model for the combined elastic modulus of sandwiched CFST members by considering the deformation compatibility between the steel tubes and concrete after loading. The influence of steel ratio, concrete strength grade, and Poisson's ratio on the combined modulus was analyzed, and equal combined modulus curves were derived.

Parameter	Effect on Combined Elastic Modulus	Trend
Total steel ratio	Increases combined modulus	Monotonically increasing
Inner vs. outer steel ratio distribution	No effect when total ratio is constant	Independent of distribution
Concrete strength grade	Increases combined modulus	Higher grade → higher modulus
Poisson's ratio of steel and concrete	Increases combined modulus	Higher Poisson's ratio → higher modulus
Elastic modulus of steel and concrete	Increases combined modulus	Higher modulus → higher combined modulus
Confinement effect	Comparable to conventional CFST	Similar confining pressure

A key finding is that when the total steel ratio is constant, the combined modulus is independent of how the steel is distributed between the inner and outer tubes. This suggests that the overall stiffness of the sandwiched section is governed by the total steel content rather than its spatial distribution.

Interpretation of Technical Points

Thick-Walled Cylinder Elasticity Approach

The thick-walled cylinder spatial axisymmetric elasticity method is a classical approach in elasticity theory that provides exact solutions for the stress and displacement fields in cylindrical bodies under axisymmetric loading. The method assumes that the stress and displacement fields are independent of the axial coordinate (plane strain condition) or that the axial strain is constant (generalized plane strain condition). For the double circular sandwiched CFST member, the model consists of concentric cylindrical layers: an outer steel tube, an outer concrete layer, an inner steel tube, and an inner concrete core.

The deformation compatibility conditions at each interface require that the radial displacements and radial stresses are continuous across the interfaces. These conditions, combined with the equilibrium equations and constitutive relations for each layer, yield a system of equations that can be solved to determine the stress and displacement fields throughout the cross-section. The combined elastic modulus is then defined as the ratio of the axial stress to the axial strain of the composite section under uniaxial loading.

Influence of Design Parameters

The finding that the combined modulus increases with the total steel ratio is expected, as steel has a significantly higher elastic modulus (approximately 200 GPa) compared to concrete (approximately 25-35 GPa for typical structural grades). The independence of the combined modulus from the distribution of steel between the inner and outer tubes, when the total ratio is constant, is a notable result that simplifies the design optimization of sandwiched CFST sections. This means that engineers can choose the inner and outer tube dimensions based on other criteria (such as buckling resistance, fabrication convenience, or aesthetic considerations) without compromising the overall stiffness of the section.

The influence of Poisson's ratio on the combined modulus is also significant. Higher Poisson's ratios for both steel and concrete lead to greater lateral strain under axial loading, which in turn enhances the confinement effect and increases the combined modulus. This is consistent with the well-established principle that confinement improves the mechanical behavior of concrete under multiaxial stress states.

Equal Combined Modulus Curves

The derivation of equal combined modulus curves provides a valuable design tool for engineers. These curves represent loci of design parameters that yield the same combined elastic modulus, enabling engineers to optimize the section design by selecting combinations of steel ratio, concrete grade, and tube dimensions that achieve the target stiffness.

Connection with Engineering Practice

Application Scenarios

Double circular sandwiched CFST members are particularly suited for applications where high stiffness, ductility, and energy absorption are required. These include:

Seismic isolation structures: The sandwiched configuration provides multiple layers of steel and concrete that can absorb seismic energy through plastic deformation and friction.
Blast-resistant structures: The multiple layers of steel and concrete provide enhanced resistance to blast loading, with the concrete layers absorbing energy through crushing and the steel tubes providing confinement and ductility.
Heavy-duty columns: The increased stiffness and load-carrying capacity of sandwiched CFST columns make them suitable for heavy industrial structures and high-rise buildings.

Design Considerations

Design Parameter	Recommended Range	Rationale
Total steel ratio	15-30%	Optimal balance of stiffness and material efficiency
Concrete grade	C40-C60	Adequate strength and workability for filling
Inner tube diameter ratio	0.4-0.7 of outer tube	Ensures adequate concrete layer thickness
Concrete layer thickness	≥ 50 mm	Ensures adequate cover and confinement
Poisson's ratio assumption	0.3 for concrete, 0.3 for steel	Standard values for elastic analysis

Quality Control and Testing

The quality of double circular sandwiched CFST members depends on the proper execution of several construction activities:

Steel tube fabrication: Both the inner and outer steel tubes should be fabricated to precise dimensional tolerances to ensure proper concentricity and adequate concrete layer thickness.
Concrete filling: The concrete filling process should be carefully controlled to ensure complete filling of the annular spaces without voids or honeycombing. Vibration and grouting techniques should be used to achieve full compaction.
Interface bonding: The bond between the steel tubes and the concrete should be verified through pull-off tests or ultrasonic testing to ensure adequate composite action.
Elastic modulus verification: The combined elastic modulus of the fabricated member should be verified through a non-destructive or semi-destructive test, such as a low-amplitude cyclic loading test.

Comparison with Conventional CFST

Property	Conventional CFST	Double Circular Sandwiched CFST
Combined elastic modulus	Moderate	Higher (due to additional steel)
Ductility	High	Very high (multiple failure modes)
Energy absorption	Good	Excellent (multiple layers)
Fabrication complexity	Low	Moderate to high
Cost	Lower	Higher
Confinement effect	Standard	Comparable (as reported by authors)

The authors report that the confinement effect of sandwiched CFST is essentially the same as that of conventional CFST, which is an important finding for design purposes. This means that the well-established design provisions for conventional CFST columns can be adapted for sandwiched CFST members with appropriate modifications.

Key Questions and Reflections

This study provides a theoretical framework for the combined elastic modulus of double circular sandwiched CFST members, but several practical aspects remain to be investigated. First, the theoretical model assumes perfect bonding between the steel tubes and the concrete, which may not be achieved in practice due to construction imperfections. Second, the model is based on linear elasticity, which is appropriate for the elastic modulus calculation but does not capture the nonlinear behavior under high loading. Third, the long-term behavior of sandwiched CFST members, including the effects of concrete creep and steel relaxation on the combined modulus, is not addressed. Fourth, the effect of the inner steel tube on the buckling behavior of the outer steel tube is not considered, which is important for slender members.

Study Insights and Implications

The practical significance of this study lies in its provision of a rigorous theoretical framework for the elastic analysis of double circular sandwiched CFST members. The use of thick-walled cylinder elasticity theory provides an exact solution for the combined elastic modulus, which serves as a benchmark for more approximate methods. The finding that the combined modulus is independent of the steel distribution between the inner and outer tubes, when the total ratio is constant, simplifies the design optimization process and provides flexibility in section design. The equal combined modulus curves offer a practical tool for selecting design parameters that achieve the target stiffness. For engineers considering the use of sandwiched CFST members in structural applications, this study provides the theoretical basis for predicting the elastic behavior of these members and for making informed design decisions. The study also highlights the potential of sandwiched CFST members as a versatile structural system that combines high stiffness, ductility, and energy absorption capacity, making them suitable for demanding structural applications.