# Hajjar and Gourley 1996

This paper presents an analytical study to derive a polynomial equation to calculate the 3D cross-section strength of square and rectangular CFTs. For this purpose, a fiber based analysis method from the literature was adopted to analyze zero length CFT beam-columns. After ensuring the accuracy of this method with the experimental results, a wide range of CFT cross-sections were analyzed. A polynomial equation was then proposed and verified using the experimental findings

## Analytical Study

In the cross-section analysis program, each CFT cross-section was divided into individual steel and concrete fibers. The stress-strain response of each fiber was monitored throughout the analysis. A constant axial load level was selected and the curvature of CFT cross-section was increased incrementally. The location of the neutral axis was tracked until the equilibrium was satisfied. After reaching the equilibrium condition, the moment capacity was calculated. The whole process was repeated for different axial load levels. The stress-strain curves used in the analysis were taken from the available literature and the effects of concrete confinement and a biaxial stress condition in the steel were accounted for implicitly. This method of analysis was applied for short CFT column specimens tested in past experimental studies from the literature, and the experimental results were predicted accurately. In the case of tensile loading, the concrete was neglected and the strength of the cross-section was calculated as the steel tube yield strength.

The analysis method presented above was executed for a wide range of square cross-section types to provide data for verification of a polynomial cross section strength equation. The D/t ratio was varied from 24 to 96. The yield strength of the steel was taken as 46 ksi, while the range of the concrete strength was varied from 3.5 to 15.1 ksi. A total of sixteen square cross-sections were analyzed and 100 load points were obtained to form their 3D strength surfaces. In addition, rectangular cross-sections with an aspect ratio of up to 1 to 2 were also analyzed. The yield strength of the steel tube for the rectangular cross sections was also ranged up to 70 ksi. A three-dimensional polynomial cross-section strength equation was then determined in terms of normalized axial load and normalized biaxial bending moments. As the axial load-moment interaction curve of the CFT cross-sections was approximately symmetric about the moment axes, the origin of the surfaces was shifted such that the shifted moment axes corresponded approximately to the peak moment achieved in the cross section in the presence of axial compression). The axial load was then redefined and normalized with respect to the new origin to make use of the approximate symmetry of the interaction diagrams. By performing a least squares analysis of the analysis results for the sixteen cross-section types, a high-order polynomial formula was proposed in terms of four coefficients that vary with D/t and f’_{c}/f_{y}. For the rectangular cross sections, the four coefficients are calculated for both major axis and minor axis D/t ratios and then averaged for use in the equation. Moreover, nominal axial strength and nominal moment strength values were also needed in the cross-section strength equation to normalize the axial load and moment values. The nominal axial load was calculated by adding the full strengths of concrete and steel components. A rectangular stress block was utilized to calculate the nominal moment strength, with the tensile strength of the concrete also being taken into account. These nominal strengths were compared with experimental results to insure accuracy. Finally, the cross-section strength formula was compared with both fiber-based cross section analysis results and experimental results for a wide range of square and rectangular CFTs cross sections. In the both cases, strong correlation was achieved.

## 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.