Equivalent Shear Modulus of Concrete-Filled Steel Tubes via Finite Element Method

Literature Overview

This 2009 paper by Tu Yongqing and Wang Sijun from Beihang University presents a finite element method (FEM) approach for determining the equivalent shear modulus of concrete-filled steel tubes (CFST) under shear loading. The authors propose simplified calculation formulas for both circular and square CFST sections and compare their results with other established methods.

Core Technical Content

The Problem of Shear Modulus in CFST Members

In structural analysis, the shear modulus governs the lateral deflection of beams and the torsional response of members. For CFST sections, the composite behavior of steel and concrete creates a complex shear stress distribution that does not conform to simple homogenization models. The equivalent shear modulus must account for:

The different shear moduli of steel and concrete
The composite action and interfacial bond
The confinement effect on concrete behavior
The shear stress distribution across the section

FEM Methodology

The authors developed a FEM model that applies pure shear loading to CFST members and computes the equivalent shear modulus from the ratio of applied shear stress to the resulting shear strain. The approach involves:

Modeling the steel tube and concrete core as separate materials with appropriate constitutive relationships
Applying boundary conditions that simulate pure shear (no normal stress, uniform shear stress on the cross-section)
Extracting the average shear strain from the displacement field
Computing the equivalent shear modulus as G_eq = τ / γ_avg

Key Parameters Affecting Equivalent Shear Modulus

Parameter	Effect on G_eq	Notes
Steel tube thickness-to-diameter ratio (t/D)	Increases G_eq	Thicker walls contribute more shear stiffness
Concrete strength (f_c)	Moderately increases G_eq	Higher strength concrete has higher G_c
Steel grade (f_y)	Slight increase in G_eq	Steel G is fixed by elastic modulus
Section shape (circular vs. square)	Square sections show lower G_eq	Due to corner stress concentrations
Confinement effect	Enhances concrete contribution	Lateral restraint increases effective G_c

Simplified Formulas

For circular CFST:

G_eq ≈ G_c + (G_s - G_c) × (2t/D) × f(α)

For square CFST:

G_eq ≈ G_c + (G_s - G_c) × (4t/D) × f(β)

Where G_c and G_s are the shear moduli of concrete and steel respectively, and f(α), f(β) are correction factors accounting for composite action and confinement.

Engineering Practice Implications

Frame analysis accuracy: In multi-story steel-concrete composite frames using CFST columns, the shear deformation of columns affects the overall lateral stiffness and period. Using an incorrect shear modulus (e.g., that of plain concrete or plain steel) can lead to significant errors in predicting drift and internal forces under lateral loads.
Design code provisions: Chinese codes (GB 51248-2016) provide simplified methods for calculating the equivalent elastic modulus of CFST, but the shear modulus provisions are less well-established. This study's simplified formulas offer a more rigorous basis for design calculations.
Comparison with other methods: The authors note that different methods yield different results because:

Empirical formulas often assume uniform shear stress distribution
Some methods neglect the confinement effect on concrete shear stiffness
The FEM approach captures the actual stress distribution and interaction

Practical application: For preliminary design, engineers can use the simplified formulas provided. For critical structures or detailed analysis, the full FEM approach should be employed, preferably validated against experimental data.

Critical Reflections

The study makes an important contribution by providing a systematic methodology for determining the equivalent shear modulus. However, several aspects merit further consideration:

The analysis is limited to the elastic range. Under cyclic loading (e.g., seismic), the shear modulus degrades due to cracking in the concrete and yielding in the steel. A degraded shear modulus model would be needed for nonlinear time-history analysis.
The interfacial bond between steel and concrete is assumed to be perfect. In reality, debonding can occur under high shear or cyclic loading, reducing the effective composite action.
For square and rectangular CFST, the corner regions experience complex stress states that may not be fully captured by standard FEM discretization. Mesh refinement studies should be conducted to ensure convergence.

The practical recommendation is that engineers designing CFST structures for shear-critical members (such as shear walls or short columns) should use the FEM-derived equivalent shear modulus rather than code-simplified values, particularly when the section dimensions and material properties fall outside the range of experimental data used to calibrate empirical formulas.