# Cai 1988

This article highlighted the results of seven phases of tests conducted and documented in earlier papers written in Chinese. Therefore, this paper gave only limited details regarding the nature of the experiments. However, a significant number of tests were performed, lending credence to the conclusions the author draws. He also presented detailed descriptions of failure mechanisms for short and long CFT columns and derives formulas for the ultimate strength of these columns. A method to convert the strength of what the author referred to as a “nonstandard” column (unequal end eccentricities) to a “standard” column (equal end eccentricities causing single curvature), and formulas to predict the equivalent length of standard columns, are also proposed. This latter procedure is described below under Cai (1991).

## Contents

## Experimental Study, Discussion, and Results

The seven phases of tests conducted encompassed a wide range of loading techniques including the following: concentrically loaded short and long columns, pure bending, eccentrically loaded columns bent in single curvature and double curvature, and cantilever columns. The three parameters of greatest interest in the author's tests were the slenderness ratio, the end eccentricity ratio (for the beam-columns), and the ratio of the end moments.

Short Column Behavior. The behavior of the short columns depended on the D/t ratio. For thin-walled tubes ( ), the load-strain curve was linear until the first yield lines were observed in the steel. The ultimate load was reached when the slope of the load-strain curve became zero, at which point the entire steel cross-section had yielded and the concrete reached its ultimate compressive strength. For very short, thick-walled tubes (D/t = 10), the steel confined the concrete as described in previous work by many authors. As the concrete expanded outward against the tube wall, hoop stresses were induced in the steel, effecting a decrease in the amount of longitudinal capacity of the tube and a transfer of axial load to the concrete. The confinement of the concrete allowed the section to realize a net gain in strength. Both ranges of D/t underwent failure when the resultant compressive force carried by the two materials reached ultimate.

Long and Intermediate Column Behavior. Axially loaded columns with a large slenderness ratio (long columns) fail by elastic buckling. A failure of this type may be described accurately using Euler's formula. However, columns between the range of short and long columns (intermediate columns) buckle inelastically. A tangent modulus may be used in place of the elastic modulus in the Euler formula to model the behavior of intermediate columns, but this requires a detailed knowledge of the CFT's stress-strain behavior. Also, columns of long and intermediate length are significantly affected by initial out-of-straightness, eccentric loading, and other imperfections, which further complicates the column's behavior.

## Theoretical Discussion

*Short Culumns* The author derived a formula for predicting the ultimate strength of short columns. The basic assumptions were that at the limit stage, the concrete was in a state of triaxial compression and the steel was in a biaxial state of stress, the steel conformed to the Von Mises yield criterion, and the concrete failure criterion was an empirical formula developed by the author to fit experimental data. After some algebra, the ultimate load may be expressed by the following formula:

where the confinement ratio

*Long and Intermediate Columns*
The author proposed the following formula to determine the ultimate load of a column that fails by overall inelastic or elastic buckling:

P_{o} is the ultimate strength of a axially loaded stub column, ϕ_{l} is an empirical strength reduction factor due to the slenderness ratio, and phi_e is an empirical strength reduction factor due to the eccentricity ratio (the ratio of load eccentricity to the radius of the concrete core). For combined axial load and bending, the author presented an interaction equation to based on the above parameters:

for an eccentricity ratio less than or equal to 1.55 and

for eccentricity ratios greater than 1.55.

## Comparison of Results

*Short Columns* The experimental and theoretical results showed good agreement. The ratio of the tested to computed ultimate axial load for the 44 specimens varied between 0.704 and 1.283 with an arithmetic mean of 0.985 and a standard deviation of 0.139.

*Long and Intermediate Columns* The ratio of the tested to computed strength reduction factor for the 48 tests ranged from 0.877 to 1.352 with an arithmetic mean of 1.049 and a standard deviation of 0.107.

## References

Cai, S.-H. (1988). “Ultimate Strength of Concrete-Filled Tube Columns,” 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. 702-727.

Cai, S.-H. (1991). “Influence of Moment Distribution Diagram on Load-Carrying Capacity of Concrete-Filled Steel Tubular Columns,” Proceedings of the Third International Conference on Steel-Concrete Composite Structures, Wakabayashi, M. (ed.), Fukuoka, Japan, September 26-29, 1991, Association for International Cooperation and Research in Steel-Concrete Composite Structures, pp. 113-118.