# Knowles and Park 1970

Design equations to compute the ultimate strength of CFT columns subjected to axial compression were presented in this paper. The design formulation was based on the tangent modulus approach as discussed in Knowles and Park (1969). The authors also derived a method of calculating the slenderness ratio below which some increase in concrete strength due to confinement is likely. The results of previous tests by other investigators as well as the authors' own tests were compared with the proposed design equations. The authors also discussed, at length, many of the proposed design formulations and pointed out some of the false assumptions embedded in these equations.

## Theoretical Discussion

The tangent modulus theory proposed in the paper was justified by experimental evidence, namely the results of Furlong (1968) and the authors' own work from 1969. Both papers concluded that little bond exists between the concrete and steel. In the formulation of the design equations as discussed in the next section, the two materials were assumed to act independently. Therefore they will not have the same longitudinal strains. The existence of bond would instill error into this assumption.

## Design Formulation

The design equations presented were based upon summing the separate tangent modulus buckling loads of the concrete core and the steel tube, which has been shown by the authors in a previous investigation to be an accurate prediction of ultimate strength. The tangent modulus of the respective materials may be obtained from the stress-strain curves determined experimentally. The authors derived elaborate formulas for the ultimate buckling strength of concrete and steel. The concrete equation was based on a stress/strain relationship expressed by the Hognestad parabola. The AISC formulation was used to determine the ultimate strength of a hollow tube. Summing the two equations resulted in the following expressions: 1) for steel tubes with a slenderness ratio,

and 2) for slender tubes governed exclusively by a buckling mode of failure,

The c and s subscripts refer to concrete and steel, respectively, and

where w is the weight of the concrete. The first term in both equations reflects the ultimate axial strength of the concrete and the second term reflects the strength of the steel. The authors also constructed an equation to determine the slenderness ratio of the concrete core above which an increase in concrete strength due to steel confinement is not likely to occur. From concrete cylinder tests they performed, the authors found that concrete showed a rapid volumetric expansion at an average longitudinal strain of 0.002 and an average stress of 0.954 of the maximum. These results agreed with those of other investigators. Based on the buckling formulas for each material, two equations were derived to express minimum slenderness ratios above which buckling will occur before the concrete becomes confined. Expressed in terms of concrete strains, an increase in the concrete strength due to confinement will not occur if:

where εvol is the strain when volumetric expansion occurs and εo is the strain at f'c. In terms of steel stress-strain properties, an increase in the concrete strength will not occur if :

where fsu is the ultimate steel strength and Est is the tangent modulus of the steel.

## Comparison of Results

The design equation for the lower range of slenderness ratios was conservative for most experimental tests (17 conducted previously by the author and 100 from other investigators). The equation for the higher range was conservative as well, although it was only checked against two results. For square tubes, the equations were up to 12% unconservative suggesting the possible need of a reduction factor to account for the square shape of the concrete core. All of these results were summarized in a tabular format. A comparison of the equations proposed by others revealed that they do not give a better prediction of ultimate loads than the authors' design equations. The authors conducted a thorough discussion of these previous papers and presented all of their formulas.

## Further Research

Other problems requiring further investigation were mentioned by the authors. These include examining the effects of various methods of load application at the ends of the specimen, beam-column and slab-column connections, creep at high loads, the effects of corrosion and fire, and the properties of low cost spiral welded steel pipe columns.