# Hajjar, Schiller, and Molodan 1998a & Hajjar, Molodan, and Schiller 1998b

In these two papers, the authors presented the development and verification of a 3D distributed plasticity finite element formulation for CFT beam-column members. The beam-column finite element was implemented into a 3D fully nonlinear general purpose frame analysis program for unbraced frames having wide flange steel girders framing into CFT beam-columns. The finite element formulation was calibrated and verified using worldwide experimental results with a wide range of material and cross section properties.

## Analytical Study

The fiber model developed in this work discretized each end cross section of the element into a grid of steel and concrete fibers. Numerical integration through the cross section was used to compute cross section rigidities and stress-resultants, and integration along the length allowed for modeling of distributed plasticity throughout the element. Geometric nonlinearity was also included through the use of a co-rotational formulation assuming small strains, moderate displacements, and moderate rotations. The CFT fiber element model neglects shear deformation due to bending (i.e., Euler-Bernoulli beam theory is used), nonlinearity due to torsion, and lateral-torsional and flexural-torsional buckling, as these modes of failure rarely occur, even in slender CFT beam-columns.

The CFT formulation adds three translational DOFs at each element end, resulting in an 18 DOF finite element, to permit the concrete core to translate axially relative to the steel tube. The element is thus able to track slip in an arbitrarily-oriented CFT. A set of nonlinear springs permits transfer of force between the steel and concrete, and the formulation is able to capture behavior ranging from perfect bond to immediate slip. Calibration and verification of the bond strength and nonlinear slip stiffness (yielding a cyclic bilinear load-slip relation with little stiffness after loss of bond) were performed based on results of a number of tests of steel members framing into CFTs with simple shear tabs, and of CFTs in flexure in which the concrete core was allowed to slip through the ends of the member (all tests had no shear studs inside the tube). With this model, the concentration of plasticity and/or nonlinear slip in a CFT connection in a braced or unbraced frame may be tracked accurately.

The steel and concrete constitutive formulations of the fiber model were each based on stress-space bounding surface plasticity formulations. The steel formulation models the rounded shape of the stress-strain curve found in cold-formed tube steel, cyclic hardening, and ratcheting behavior. The concrete formulation captures strength and stiffness degradation by means of a cumulative damage parameter. The concrete formulation also models fibers which cycle into tension and then back into compression by simulating crack opening and closure. The uniaxial stress-strain behavior of the constitutive models were calibrated to simulate the cyclic behavior of ASTM A500 cold-formed tube steel and normal strength concrete. In addition, for the steel formulation, the different stress-strain behavior exhibited in the corners and flanges of cold-worked steel tubes was modeled. For the concrete formulation, the post-failure plastic modulus of the stress-strain response was calibrated to be substantially increased from the softening modulus established for unconfined concrete. The resulting more gradual loss of strength, calibrated to match the results from CFT flexural experiments conducted by Tomii and Sakino (1979a) that yielded moment-curvature-thrust data, represents the added ductility seen in rectangular CFTs as a result of the moderate confinement exhibited in these members. The beam-column fiber model was verified against over 30 experimental results of CFT beam-columns and subassemblages subjected to monotonic and cyclic loading. Correlation was found to be strong prior to local buckling occurring in the CFTs, as the formulation does not account explicitly for local buckling.

## References

Hajjar, J. F., Schiller, P. H., and Molodan, A. (1998a). “A Distributed Plasticity Model for Concrete-Filled Steel Tube Beam-Columns with Interlayer Slip,” Engineering Structures, Vol. 20, No. 8, August, pp. 663-676.

Hajjar, J. F., Molodan, A., and Schiller, P. H. (1998b). “A Distributed Plasticity Model for Cyclic Analysis of Concrete-Filled Steel Tube Beam-Columns and Composite Frames,” Engineering Structures, Vol. 20, Nos. 4-6, April-June, pp. 398-412.