Keywords: RC frame, earthquake loading, anchorage slip, pinching effect, Bauschinger effect, moment-curvature relationship.

The nonlinear dynamic responses of RC frame structures under earthquake excitations are usually developed at certain critical regions, and these regions are often located at points of maximum internal forces such as the beam-column joints. This means that an accurate numerical model, which can simulate the hysteretic behaviour of RC columns and beams, is necessary in order to exactly predict the nonlinear response of the frame structures. Since earthquake-induced energy is dissipated through the formation of plastic hinges in the beams and columns, especially, the determination of influence factors which affect to the nonlinear response at a joint is essential step in the construction of a numerical model. Typically, initial stiffness, bond-slip, anchorage slip, and axial force effect are some of the influence factors which must be included in the numerical model because the major sources of deformation in RC frame structures are flexural rotation, shear deformation including shear sliding, and bond-slip. The hysteretic load-deformation behaviour of frame member arises from a combination of this deformation mechanism.

Many analytical models have been proposed to date for the nonlinear analysis of RC frame structures, and these range from very refined and complex local models to simplified global models. Of these models, a numerical model based on the moment-curvature relation is popular used in the case of frame structures. Since the first introduction of a bilinear moment-curvature relation by Clough and Johnson [3] the various mechanical models for the hysteretic moment-curvature relation have been introduced to analyze the behaviour of RC beams subject to cyclic loadings. Such models include cyclic stiffness degradation and further modifications to take into consideration the pinching effect due to the shear force and the strength degradation after yielding of steel. In addition, by using the bilinear hysteretic curve instead of trilinear, a more simplified model has been proposed. Nevertheless, these models still have limitations in simulating exact structural behaviour of RC columns because of the exclusion of the bond-slip, fixed-end rotation effect, axial force and Bauschinger effect.

The present study concentrates on the introduction of a moment-curvature relation of a RC section that can simulate the cyclic behaviour of RC columns and RC frame. Unlike most mathematical or mechanical models found in the literature, the proposed model has taken into account the bond-slip effect, Bauschinger effect of the steel, axial force effect and fixed-end rotation effect. In advance, a modification of the hysteretic moment curvature relation to consider an increase of the ultimate resisting capacity and the pinching phenomenon in the axially loaded RC columns is also introduced on the basis of the energy conservation. However, differently from many numerical and mechanical models considering the strength degradation [1,4,6], this effect is still not included in the proposed model. An additional concern for the strength degradation under cyclic loading beyond the yield strength may be required to estimate the exact damage level undergone by a section after a certain number of cycles. Through correlation studies between analytical results and experimental values [2,5] from typical RC column tests, the following conclusions are obtained: (1) The inclusion of pinching effect is important in structures dominantly affected by an axial force; (2) to accurately predict the structural behaviour of the beam to column subassemblage where the nonlinear response is concentrated, a modification of the moment-curvature relation to consider the fixed end rotation is strongly required; (3) to effectively simulate axially loaded RC columns, the axial load effect must be considered; and finally (4) the proposed model can be effectively used to predict the structural response of RC columns under cyclic loadings, and its application can be extended to the dynamic analysis of a frame structure.

1

Chung Y.S., Meyer C., Shinozuka M., "Modeling of concrete damage", ACI Structural Journal, 86(3), 259-271, 1998.

2

Clough, R.W., Gidwani, J., "Reinforced Concrete Frame 2: Seismic Testing and Analytical Correlation", Earthquake Engrg. Research Center Report No. EERC 76-15, Univ. of California, Berkeley, USA, 1976.

3

Clough, W., Johnson, S.B., "Effect of stiffness degradation on earthquake ductility requirements", Proceedings of Japan Earthquake Engineering Symposium, October, 1966.

4

Dowell R.K., Seible F., Wilson E.L., "Pivot Hysteresis model for reinforced concrete members", ACI Structural Journal, 95(5), 607-617, 1998.

5

Low, S.S., Moehle, J.P., "Experimental study of reinforced concrete columns subjected to multi-axial cyclic loading", Earthquake Engrg. Research Center Report No. EERC 87-14, Univ. of California, Berkeley, USA, 1987.

6

Takeda T., Sozen M.A., Nielsen N.N., "Reinforced concrete response to simulated earthquake", Journal of the Structural Division, ASCE, 96(ST-12), 1970.

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