Neogi, Sen, and Chapman 1969
The elasto-plastic behavior of pin-ended circular CFTs, loaded either concentrically or eccentrically about one axis, was studied numerically and then compared to experimental results performed by the authors and results published in other article
Experimental Study, Discussion, and Results
The main parameters of the test were the steel type, L/D ratio, and the eccentricity. All of the columns behaved in a ductile manner and no local buckling of the tubes occurred. The shape of the deformed columns remained symmetric throughout the test. The curvature was distributed uniformly over the length of the beam up to the maximum load. Beyond this point, the increase in curvature was concentrated at the midpoint of the section. As the load decreased, the moment at the center of the section continued to increase, probably due to the strength gain from the triaxial compression of the concrete. These results were important in comparing the analytical method.
Theoretical Discussion
As a CFT undergoes loading, the structural action changes as Poisson's ratio for concrete exceeds that of steel. Initially, for low strains, the steel expands at a faster rate (Poisson's ratio of 0.283 versus 0.15 to 0.25 for the concrete), thus providing no restraining effect on the concrete. Then at higher loads, the concrete expansion exceeds the steel expansion. The radial pressure induces hoop tension in the tube, changing the stress state from uniaxial to biaxial in the steel and from uniaxial to triaxial in the concrete. The steel transfers some of its longitudinal load to the concrete at this point since it cannot sustain as much load in the presence of a hoop stress. In this state of stress, the member provides strengths well in excess of the sum of the individual components, although shear failure may occur in the concrete before the load is completely transferred. The increase in failure load provided by the confinement depends on, among other things, the magnitude of strain at failure. This strength gain therefore varies inversely with length and eccentricity. All slender columns and shorter columns with high eccentricities will fail by buckling before large enough strains to transfer the load are reached. The authors suggested, then, that for practical columns which generally fall into this category, it is unnecessary to take triaxial effects into account.
Analytical Study
Two methods were used to obtain the load-deflection curve of axially loaded sections. In the analytical formulation it was assumed that complete interaction between the materials took place, each material was subjected to a uniaxial state of stress (concrete in tension was ignored), the stress-strain curves were fully reversible (i.e., immediate elastic unloading was not assumed), and no local buckling or shear failure occurred.
Axially Loaded Columns. The axially loaded CFTs were analyzed numerically using the tangent-modulus approach. The computer program the authors developed used an iterative technique to find a value of strain to equilibrate (within a set tolerance) the following equations:
The first equation is the Euler buckling equation with the respective tangent moduli of the steel and concrete. The stress-strain curve for the steel was a trilinear curve with elastic, elasto-plastic, and perfectly-plastic regions. For some calculations, in order to avoid a sudden drop in stiffness at yield (intersection of the elastic and elasto-plastic lines on the curve) which will cause certain columns to buckle, the elasto-plastic region was modeled as a parabola rather than a straight line. The concrete stress-strain curve was based on a variation of Hognestad's equation.
Beam-Columns. The beam-columns were analyzed by two methods: by calculating the 'exact' deflected shape by Newmark's iterative method, and by assuming a deflected shape in the form of a partial-cosine wave. The first method involved subdividing the beam-column into a number of sections along the length and performing an incremental solution for the moments and curvatures until a preset tolerance was reached. By integrating the curvature over the length, deflections could be obtained and the load-deflection curve could be plotted. The latter method assumed that the deflected shape is part of a cosine wave which satisfies equilibrium only at mid-height. A similar iterative procedure was used to select values of deflection and neutral axis distance until convergence of the internal and external moments was achieved. Since the methods of the computational approach incremented central deformations and not axial load, they had the advantage of being able to trace the load-deflection curve into the post-buckling region.
Comparison of Results
The experimental and analytical results agreed for columns with L/D ratios of greater than 15, suggesting triaxial effects are negligible for such columns, since buckling occurs before the strains may reach a large enough value for confinement to occur. For columns with smaller L/D ratios, some strength gain was realized due to triaxial effects, rendering the calculated loads conservative. This effect diminishes, however, as the eccentricity increases. The partial-cosine wave estimate of the deflected shape was always conservative but never more than 5% below the exact shape calculation.
Reference
Neogi, P. K., Sen, H. K., and Chapman, J. C. (1969). “Concrete-Filled Tubular Steel Columns Under Eccentric Loading,” The Structural Engineer, Vol. 47, No. 5, pp. 187-195.