# Hajjar and Gourley 1997a & Hajjar, Gourley, and Olson 1997b

In these two companion papers, the authors presented the development and verification of a 3D concentrated plasticity finite element formulation for CFT beam-column members. The beam-column finite element was adopted 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 macro model developed in this work employed a concentrated plasticity approach to account for material nonlinearity. The frame analysis program was based on the direct stiffness approach and an incremental updated Lagrangian formulation. Perfect bond between the steel and concrete was assumed. The stiffness matrix consisted of three components, including an elastic stiffness matrix, a geometric stiffness matrix, and a plastic reduction stiffness matrix. 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 elastic flexural and axial rigidity terms were taken as the superposition of the concrete core and steel tube strengths. The contribution of the concrete core to the elastic torsional rigidity was neglected. The geometric stiffness matrix was taken from the literature and was formulated assuming small strains, moderate displacements, and moderate rotations.

The cyclic stress-strain behavior of the CFT beam-column cross-section was simulated with a bounding surface model in three-dimensional stress-resultant space. The bounding surface formulation consisted of two limit surfaces at each element end. The inner loading surface represented the initiation of inelasticity, while the outer bounding surface represented the condition when the limiting stiffness of the CFT beam-column was reached. Both surfaces had the capability of expanding, contracting, and translating to capture typical characteristics of short CFT members under cyclic loading. For both the loading surface and bounding surface, the cross-section strength formula derived by Hajjar and Gourley (1996) was utilized. The CFT cross-section was assumed to be a work-hardening type material and thus both the loading and bounding surfaces were convex.

In this formula, the location of the centroids of the loading surface and bounding surface were traced with the use of a back-force vector that defined the location of the surface centroids relative to the stress-resultant-space origin. Once the load point reached the loading surface, it was assumed to remain on that surface, and the corresponding plastic deformations were assumed to be perpendicular to the loading surface, as an associated flow rule was used. The loading and bounding surfaces were assumed to harden isotropically and then kinematically. Isotropic hardening was based on the amount of accumulated plastic work. Both surfaces kinematically hardened in the same direction, although their rates of hardening were different. The rate of hardening became equal when the two surfaces contacted to each other. The kinematic hardening was assumed to be in the direction of the vector from the point on the bounding surface intersected by the incremental force vector, and its conjugate point on the loading surface.

Three cyclic tests and eight monotonic tests were chosen for the calibration of the bounding surface plasticity model. The tests included both proportional and non-proportional loading. The test setups used for calibration purposed included monotonically loaded beam-columns subjected to eccentric load, monotonically loaded beam-columns in single curvature bending with constant axial force, and cyclically loaded beam-columns in reverse curvature bending with constant axial force. In normalized force space, the behavior of CFT sections is sensitive to cross-section sizes and material strengths. Thus the parameters related to the material model were intended to be kept constant or varied with simple functions in terms of nominal strength of the concrete core normalized with respect to the nominal strength of the CFT cross section (P_{co}/P<usb>o). This ratio accounted implicitly for the changes in both the D/t ratio and f’_{c}/f_{y} ratios.

For the verification of the analytical work, additional tests including monotonic and cyclic loading were selected. The analysis program was then executed for these tests while adhering to the calibrated parameters. It was found that the maximum error in the load-deflection response was less than 10% in most of the cases.

## References

Hajjar, J. F. and Gourley, B. C. (1996). “Representation of Concrete-Filled Tube Cross-Section Strength,” Journal of Structural Engineering, ASCE, Vol. 122, No. 11, November, pp. 1327-1336.

Hajjar, J. F. and Gourley, B. C. (1997a). “A Cyclic Nonlinear Model for Concrete-Filled Tubes. I. Formulation,” Journal of Structural Engineering, ASCE, Vol. 123, No. 6, June, pp. 736-744.

Hajjar, J. F., Gourley, B. C., and Olson, M. C. (1997b). “A Cyclic Nonlinear Model for Concrete-Filled Tubes. II. Verification,” Journal of Structural Engineering, ASCE, Vol. 123, No. 6, June, pp. 745-754.