# Chen and Chen 1973

The elasto-plastic behavior of pin-ended, circular and square concrete-filled steel tubes was studied analytically. Three types of stress-strain relationships were investigated: 1) uniaxial state of stress in both the steel and concrete; 2) uniaxial stress state in the steel and triaxial stress state in the concrete, assuming the triaxial effect increases the concrete's ductility only; and 3) uniaxial steel stress state and triaxial concrete stress state, assuming the triaxial effect increases both ductility and strength in the concrete. Interaction curves relating axial force, end moment, and slenderness ratio were presented for the maximum load carrying capacity of a beam-column and compared to test results from other investigators. The three main objectives of the paper were to: 1) develop theoretical column curves for combined axial load and bending taking into account the effect of concrete confinement; 2) demonstrate that an axially loaded CFT column may be accurately predicted by assuming a certain amount of eccentricity in axial load application; and 3) present interaction curves for simply-supported CFT columns under unsymmetric loads.

## Theoretical Discussion

The authors stated that all columns must be treated as beam-columns because imperfections always exist and loads are never applied concentrically. This implies analyzing columns as deflection problems rather than problems analyzed by the eigenvalue or tangent-modulus approach. The moment-curvature relationship for a constant axial load was developed first under the following assumptions: the concrete has no tensile strength; the steel has an elastic-perfectly plastic stress-strain relationship; plane sections remain plane in bending; and complete interaction between the steel and concrete exists. For each given axial load, three sets of moment-thrust-curvature (M-P-Φ) curves were developed corresponding to the three aforementioned stress-strain relationships. The theoretical method used to obtain interaction curves was the Column Curvature Curve (CCC) method. Axially loaded members that were initially curved, eccentrically loaded, and contained residual stresses in the steel tube were considered. To account for these imperfections, the authors assumed an initial applied load eccentricity of 0.001*L for circular sections and 0.002L for square sections. The theoretical curves for axially loaded columns showed that triaxial effects were negligible for values of L/D > 15 for circular columns and L/D > 20 for square columns. Test results obtained by Knowles and Park were compared to the theoretical values and good correlation existed for ratios of L/D > 15, especially for large eccentricities. Columns with L/D ratios less than 15 were influenced by the effect of triaxial confinement of the concrete. An accurate prediction of the ultimate axial load for square columns using the CCC method was obtained by using the concrete stress-strain curve assuming triaxial stresses with an increase in both strength and ductility. Because the triaxial effects were much greater for the circular columns than the square columns, the theoretical estimates for the circular columns were quite conservative for low L/D ratios. The circular columns were more accurately modeled by amplifying the value of f'c used in the initial stress-strain relationship. For columns with combined axial load and moment, the authors found that the unconfined stress-strain curve for concrete was adequate for columns with L/D >15 under both symmetric and unsymmetric loading. For columns with L/D < 15, the effects of column instability may be neglected and the confined stress-strain curve of the concrete should be used.

## References

Chen, W. F. and Chen C. H. (1973). “Analysis of Concrete-Filled Steel Tubular Beam-Columns,” Memoires, IABSE, Vol. 33, No. II, pp. 37-52.