# Knowles and Park 1969

This paper investigated axially loaded CFTs and hollow tubes over a wide range of slenderness ratios, with particular attention paid to the effect of the slenderness ratio on the lateral pressure exerted by the tube on the concrete. The authors also looked at the effect of loading the materials together and individually (i.e., load the concrete and not the steel and vice versa). They examined concentrically loaded columns theoretically by the tangent modulus approach and they constructed a straight line interaction formula to estimate the behavior of eccentrically loaded CFTs.

## Experimental Study, Discussion, and Results

*Concentrically Loaded Columns*. All of the hollow tubes tested under axial loads failed by inelastic flexural buckling; no local buckling was observed before the ultimate load was reached. Since local buckling is often sudden and catastrophic, the authors suggested that the ratio of the wall thickness to the diameter of the tube should be limited, although no specific values were given. The concrete-filled tubes failed in the same manner as the hollow tubes, with the region of plasticity always located at midheight. Tests of concrete cylinders showed a sudden and large increase in volume at a strain of around 0.002. In the CFT this increase in volume will cause the steel to exert a confining pressure and increase the strength of the concrete. The authors found that for concrete core slenderness ratios (K*L/rc) less than 44.3, confinement of the concrete will occur. For most of the author's tests, though, overall column buckling preceded strains of sufficient magnitude to cause volumetric expansion and confinement. It was noted that the square tube columns with small slenderness ratios did not gain additional strength due to confinement. Although it has been shown by other investigators that square tubes provide less confinement than circular tubes, square ties in reinforced concrete have produced good confinement results. The authors stated that the issue of square tube confinement has yet to be resolved.

*Eccentrically Loaded Columns*. Both the hollow and concrete-filled tubes failed by overall column buckling at midheight, where the moment was largest.

## Theoretical Discussion

*Axially Loaded Columns*. A CFT can be loaded axially in three distinct ways: load the steel but not the concrete, load the concrete but not the steel, and load the steel and the concrete such that the longitudinal strain is the same in both materials. Each method produces a different section behavior. The first method essentially mimics the behavior of a steel tube alone, with the concrete failing at the maximum load the steel tube can carry. Tests by Gardner and Jacobson (1967) showed that loading the steel tube alone does not increase the failure load above that of a hollow tube. Under eccentric loading, the concrete may tend to delay the onset of local buckling and increase the bending resistance. Loading the concrete alone idealizes the use of the steel tube which provides a confining stress and does not resist axial load. Steel used in this way is approximately two times as effective as steel resisting longitudinal stresses. But in reality, some bond will exist between the two materials, inducing some longitudinal stress in the steel tube. This creates a biaxial state of stress which decreases the circumferential capacity and the amount of confining pressure that the steel is able to exert on the concrete. Gardner and Jacobson found that this type of loading does not increase the failure load above that obtained by loading both materials simultaneously. Loading both materials so that longitudinal strains are equal is the probable situation in actual structures and was the type of loading used in the experiments. The theoretical ultimate buckling stress of the axially loaded columns was obtained using the tangent-modulus formula:

where K*L/r is the slenderness ratio and Et is the tangent modulus obtained by summing the respective tangent moduli of the steel and concrete. Hollow tube stub columns and short unconfined concrete cylinders were tested to obtain the tangent modulus-stress relationships used in the formulation. The authors presented a number of graphs illustrating their stress-strain results. The tangent modulus method produced generally accurate and conservative estimates of ultimate loads. The formula underestimated the strength for short columns, as expected. At small slenderness ratios, the concrete was able to reach high enough strains to allow expansion of the concrete and cause a corresponding lateral confinement from the steel, which increased the strength of the concrete. Also mentioned were additional methods of calculating the ultimate load. One method involves using the tangent modulus of the steel tube and assuming equal longitudinal strains in the concrete and steel. Using this measured longitudinal strain, the load in the concrete may be obtained from its stress-strain curve. This method tends to overestimate the capacity, however. A second approach is to use the tangent modulus obtained from CFT stub column tests. The authors did not recommend this because end conditions will affect the results. Friction at the ends will cause lateral pressure that is unaccounted for and the two materials may not be loaded equally. Therefore the authors recommended the method used in the paper.

*Beam-Columns*. Theoretical estimates of the bending capacity were determined by using a straight line interaction equation for the beam-columns:

The value of Po was obtained by testing an equivalent length column with no moment and Mo was obtained by a flexural test of a simply-supported CFT beam. Pu was the measured load at failure of the beam-column and Mu was calculated by multiplying Pu by the actual eccentricity (the sum of the initial applied eccentricity and the measured lateral deflection) at the ultimate load. The straight line interaction formula predicted a theoretical ultimate load equal to or less than the experimental for all of the tests except for the circular columns with large eccentricities (1.0 in. for these tests). A rigorous explanation of the overestimate was not given. The authors cautioned against using this straight line interaction formula for slender columns. They also felt it would be conservative for short columns.

## References

Knowles, R. B. and Park, R. (1969). “Strength of Concrete Filled Steel Tubular Columns,” Journal of the Structural Division, ASCE, Vol. 95, No. ST12, pp. 2565-2587.