# Liu and Goel 1988

The cyclic load behavior of rectangular CFT bracing members was investigated. The main parameters of the study were: the presence of concrete, strength of concrete, effective slenderness ratio, and width-thickness ratio. A simplified method of computing the first cycle buckling load was presented and compared to the tangent modulus approach given by Knowles and others. Both of these methods were then compared to the experimental results. The paper gave detailed descriptions of the buckling failure modes in hollow and concrete-filled bracing members.

## Experimental Study, Discussion, and Results

Each tested specimen was placed diagonally inside a four-hinge testing frame. The tubes were welded to gusset plates which were in turn welded to the frame. An actuator connected to a concrete reaction wall loaded the frame, simulating the movement of a building under cyclic loads. The movement of the frames alternately stressed the brace in tension and compression. Failure occurred in all the specimens by overall buckling in the in-plane direction (bending about the minor axis). The presence of concrete may increase the number of cycles to failure and dissipate more energy, provided the tube's width-thickness ratio was not too small. Concrete may change the local buckling mode, reduce its severity, and delay the occurrence of cracking in the steel which is very beneficial in cyclic applications. The concrete forces the buckling outward which has two advantages. First, the distance between the top and bottom flanges of the tube increases rather than decreases in the case of hollow tubes. This effectively prevents the section modulus from decreasing significantly. Also, the cracking at local buckling is spread over a larger area than the hollow counterpart, alleviating severe strain concentrations, which extends the cyclic life of the specimen. The strength of concrete (4 to 8 ksi in this study) did not show a significant influence on the behavior of the CFTs since the failure was governed primarily by the buckling failure of the steel tubes. On the other hand, severe local buckling in the plastic hinge zones rapidly deteriorates the peak forces in the member. In summary, the concrete in the braces played a different role than the concrete in short columns. Rather than provide added strength, it served to augment the steel tube's buckling strength.

## Theoretical Discussion

In previous papers (Knowles, 1969 et al.), the tangent modulus formula was presented to calculate buckling loads. This approach uses the tangent moduli of concrete and steel along with the Euler buckling equation to give the following form:

Assuming strain compatibility between the steel and concrete combined with the equilibrium condition

the buckling load may be obtained. Although this procedure has shown very accurate results, it requires the use of a computer to perform the calculations, namely to obtain the tangent moduli from the complicated stress-strain functions. Therefore, the authors wanted to develop a simpler process, hence the approximate method presented in this paper. The method relied on the following assumptions: the steel tube and concrete acted independently, thus allowing the AISC formula to be used for the tube; the stress-strain relationship was linear up to the buckling load; and the longitudinal strain in the concrete was the same as the steel. The first step was to calculate the buckling load of the steel without the presence of concrete using the AISC formula. Second, the strain and the stress in the concrete were computed using the latter two assumptions stated above. Knowing the stresses in both materials led to a simple calculation of the first cycle buckling load:

P_o=A_sf_y+A_cf'_c

This approach resulted in values in good agreement with those obtained by the more complex tangent modulus method and the experimental results were also quite close to these values.

## References

Liu, Z. and Goel, S. C. (1988). “Cyclic Load Behavior of Concrete-Filled Tubular Braces,” Journal of Structural Engineering, ASCE, Vol. 114, No. 7, pp. 1488-1506.