Stress-Strain Relationship Model of Steel-Concrete Members Under Cyclic Loading

Literature Overview

The paper by Zhang Wenfu and colleagues, published in the Journal of Daqing Petroleum Institute in 2010, presents a constitutive model for steel-concrete members subjected to cyclic loading conditions. The model is developed specifically for steel-concrete frame-braced systems, where the braces undergo significant inelastic deformation during seismic events. The work addresses the need for accurate material-level constitutive relationships that can be incorporated into component-level restoring force models, which are essential for nonlinear seismic analysis of steel-concrete composite structures.

Steel Stress-Strain Model Under Cyclic Loading

The steel stress-strain relationship under cyclic loading is modeled using a bilinear kinematic hardening model, which captures the Bauschinger effect and the progressive degradation of stiffness observed in real structural steel under repeated loading. The model is defined by the following parameters:

The kinematic hardening model is particularly appropriate for the steel tubes used in steel-concrete braces because it accounts for the shift in the yield surface during reverse loading, which is a critical phenomenon in low-cycle fatigue behavior. The Bauschinger effect, where the reverse yield stress is lower than the initial yield stress, is captured by the translation of the yield surface in stress space.

From a welding metallurgy perspective, the cyclic behavior of the steel is significantly influenced by the welding process used to fabricate the steel tube. For example, the heat-affected zone (HAZ) of an electric resistance welded (ERW) or high-frequency welded (HFW) pipe exhibits different cyclic hardening behavior compared to the base metal, due to the microstructural changes induced by the welding thermal cycle. The HAZ of an ERW pipe typically experiences a microstructure consisting of fine-grained ferrite and pearlite, which may exhibit higher yield strength but lower ductility compared to the normalized base metal. This microstructural heterogeneity can lead to localized stress concentrations and premature fatigue crack initiation at the weld seam under cyclic loading.

Concrete Skeleton Curves Under Confined and Unconfined Conditions

The paper presents separate skeleton curves for confined and unconfined concrete, which is essential for accurately modeling the behavior of concrete encased within a steel tube. The confined concrete skeleton curve is derived using the Mander model, which accounts for the lateral confinement pressure provided by the steel tube.

The confined concrete parameters are as follows:

Where ρ_s is the volumetric confinement ratio, E_s is the elastic modulus of the steel tube, and E_c is the elastic modulus of the concrete.

The key insight from this model is that the confinement effect significantly enhances both the peak strength and the ductility of the concrete. For a typical steel tube with a diameter-to-thickness ratio of 50 and a steel ratio of 10 percent, the confined concrete strength can be 1.5 to 2.0 times the unconfined strength, and the peak strain can increase from 0.002 to 0.005 or higher. This enhanced ductility is critical for the seismic performance of steel-concrete braces, as it allows the member to undergo large inelastic deformations without catastrophic failure.

Hysteresis Rules and Cyclic Degradation

The paper establishes loading and unloading rules for the concrete component of the constitutive model, which govern the shape of the hysteresis loops during cyclic loading. The hysteresis rules are based on the following principles:

1. During loading, the stress follows the ascending branch of the skeleton curve until the peak strength is reached.
2. Upon unloading, the stress decreases linearly to the opposite side of the stress-strain space, with the unloading stiffness determined by the secant modulus.
3. Upon reloading in the reverse direction, the stress follows a path that passes through the previous extremum point, accounting for the residual strain and the Bauschinger effect.
4. The stiffness degrades with each loading cycle, with the degradation rate depending on the strain amplitude.

The hysteresis energy dissipation capacity is a critical parameter for seismic design. The equivalent viscous damping ratio, which quantifies the energy dissipation per cycle, can be estimated from the hysteresis loop area. For steel-concrete braces, the equivalent viscous damping ratio typically ranges from 10 to 25 percent, depending on the steel ratio, the confinement level, and the strain amplitude.

Application to Steel-Concrete Frame-Braced Systems

The constitutive model developed in this paper is intended for use in component-level restoring force models of steel-concrete braces. The restoring force model captures the global behavior of the brace, including axial stiffness, bending stiffness, and the interaction between axial force and bending moment. The material-level constitutive relationships are incorporated into the fiber section model, where each fiber represents a portion of the cross-section with its own stress-strain behavior.

The fiber section model is discretized into multiple layers, with each layer assigned a material type (steel or concrete) and the corresponding constitutive law. The section forces are computed by integrating the fiber stresses over the cross-section, and the section deformations are obtained from the fiber strains. This approach allows for a realistic representation of the nonlinear behavior of the steel-concrete brace, including the gradual yielding of the steel tube, the cracking and crushing of the concrete, and the interaction between the steel and concrete components.

Reflections and Implications for Welding and Material Engineering

The constitutive model presented in this paper has several implications for the welding and material engineering of steel-concrete composite structures:

First, the model assumes that the steel tube behaves as a homogeneous material, which is an idealization that may not hold for welded pipes. The weld seam in an ERW or HFW pipe introduces a region of microstructural heterogeneity that can affect the cyclic behavior of the member. Engineers should consider the location of the weld seam relative to the critical sections of the brace, and may need to apply correction factors to the constitutive model to account for the reduced fatigue resistance of the weld zone.

Second, the confinement effect on the concrete is directly related to the integrity of the steel tube. Any defect in the steel tube, such as a lack of fusion at the weld seam, a crack in the HAZ, or a geometric discontinuity, can compromise the confinement pressure and lead to premature concrete failure. Therefore, rigorous non-destructive testing of the steel tube, including ultrasonic testing (UT) of the weld seam per GB/T 11345 or ISO 17640, is essential to ensure the structural integrity of the composite member.

Third, the cyclic degradation of the steel stress-strain behavior is influenced by the welding residual stresses present in the fabricated member. Residual stresses from the welding process can superimpose on the applied cyclic stresses, accelerating fatigue crack initiation and propagation. Post-weld stress relief treatment, such as low-temperature annealing or thermal stress relief, may be necessary for critical applications to reduce the residual stress level and improve the fatigue performance of the member.

The constitutive model developed in this paper provides a valuable tool for the seismic design of steel-concrete frame-braced systems, but its accuracy depends on the fidelity of the material properties used in the model. Welding engineers and material scientists must ensure that the material properties assigned to the model reflect the actual behavior of the fabricated member, including the effects of the welding process on the microstructure and mechanical properties of the steel tube.