# Zhong and Miao 1988

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A standard test for short CFTs was developed to provide a basis to correlate results from different sources and to provide stress-strain input for design formulations. In lieu of experimental results, the authors also developed an analytical method to determine key points on the stress-strain curve which could be used for in computing the ultimate design strength of the CFTs.

## Experimental Study, Discussion, and Results

The main goal of the standard test was to accurately obtain the longitudinal stress-strain relationships in the steel and the concrete for use in a design method. This required a test on short columns that would not fail by buckling. Columns having an L/D ratio ranging from 2.0 to 5.0 and a ratio of steel to concrete area, *, of 0.115 and 0.211 were selected for study. From their tests, the authors concluded that a standard test should use an L/D of 3.0 to 3.5. Tubes with a ratio of 3.5 showed no unloading, remained essentially straight throughout the test, and had constant strain through the cross-section. This was not the case, however, for specimens with L/D greater than 4.0. The lower limit of 3.0 was to avoid significant end effects. They also recommended using plate hinges for end supports (the type of support condition did not markedly affect the results and the plate hinges were the simplest to construct).

## Theoretical Discussion

The analysis of the stress-strain relationship showed three stages: elastic, elasto-plastic, and strain hardening (for α=0.06) or perfectly-plastic stage (α= 0.05). Unloading occurred if α fell below 0.04. Formulas are presented to calculate these points and the elastic modulus of the combined section. The deformation behavior of the CFT for (for α=.04) closely resembles that of a plain steel tube with good ductility and energy absorbing capacity.

## Design Formulation

Two semi-empirical approaches were suggested for determining the ultimate load capacity of a CFT. The first method used the theory of plasticity. The yield strength was taken as the sum of the individual steel and concrete strengths. The yield strength of the steel tube was determined using the theory of plasticity, as given by the following formula:

The yield strength of concrete was given by:

where σr is the confining pressure:

The μ' value was determined empirically based on experiments by:

Summing the two values of P resulted in the yield strength. The second method used the theory of elasticity. This method incorporated a similar summation of the strengths of the steel and concrete. It assumed that both materials were elastic-perfectly plastic, a valid assumption since confined concrete will behave in a manner much like steel. The steel was governed by the von Mises' yield criterion with plastic flow occurring when the steel met the yield surface. At this point on the surface, the longitudinal stress will decrease and the hoop stress will increase, due to the expanding concrete. Before the yielding of the steel, the strength of the CFT was found by simply summing the Mohr's strength of each material. By a similar formulation used above, stresses could be found. After yield, the confinement pressure was obtained assuming compatibility of the longitudinal and circumferential strains of the steel and concrete.

## References

Zhong, S.-T. and Miao, R.-Y. (1988). “Stress-Strain Relationship and Strength of Concrete Filled Tubes,” Composite Construction in Steel and Concrete, Proceedings of the Engineering Foundation Conference, Buckner, C. D. and Viest, I. M. (eds.), Henniker, New Hampshire, 7-12 June 1987, ASCE, New York, pp. 773-785.