Keywords: finite element, non-linear, reinforced-concrete, earthquake loading, slabs, design codes.

The proper application of the capacity design approach in earthquake resistant design requires a deep understanding of the behavioural characteristics of the different members comprising the structure. As well, an accurate assessment of the strengths of these members is essential. Of particular importance are the relative strengths of the beams and columns framing into the joint. The role of the compression slab cast monolithically with the beams has been established over the years and has been incorporated in design codes. In contrast, most codes tend to neglect the effects of the slab when the section is subjected to negative moments (i.e. slab in tension). In order to ensure the safety of the occupants of a structure in case of a severe earthquake, design codes specify that the design of earthquake-resistant reinforced concrete structures be such that if the structure were to fail, it should been in a predefined mode. In addition, the members should be able to exhibit sufficient deformability in order to provide the required overall ductility. Acceptable failure modes include formation of plastic hinges in beam-ends while prohibiting them in columns and joints of a typical storey. When a structure in subjected to a lateral earthquake-type load, the slab on one side of the joint will be subjected to compression (positive moments) while on the other side it will be subjected to tension (negative moments). Under negative (hogging) moments, the presence of the floor slab has been observed to significantly enhance the strength and stiffness of the supporting beams.

In order to be able to successfully implement such a design principle, an accurate assessment of the relative strengths of the beams and columns is essential. Is has been established experimentally that the presence of the floor slab can significantly increase the beam's capacity and stiffness when subjected to negative moments. For the purpose of assessing slab contribution effects under negative bending, a nonlinear finite element analysis of an exterior slab-beam-column subassembly was performed. The computer program used for the analysis incorporates five different finite elements in its library and is based on an iterative secant stiffness formulation. Three-dimensional brick elements were used to model the concrete and three-dimensional truss elements were used to discretely model the reinforcement. The nonlinear properties of both the concrete and the reinforcement were used. The reinforcement included both flexural and shear reinforcement. Bond slip was assumed in the joint area and in the longitudinal beam's flexural reinforcement lying within the plastic hinge zone. In order to simulate bond slip in the prescribed areas, three- dimensional bond elements were used to model the interface steel-to-concrete stresses. The large finite element mesh used to model the structure comprised 794 brick elements, 2378 truss elements and 438 bond elements.

In assessing the effects of slab contribution on the response characteristics of the subassembly, reinforcement strains were of great importance and in particular, the slab reinforcement strains. Of particular importance were some behavioural characteristics such as: 1) the strain distribution in the slab reinforcement in the transverse direction, 2) the rate of decay in the slab reinforcement strains in the transverse direction, 3) the membrane and bending actions in the slab, 4) the bond slip effects in the joint area and 5) the effects of bond slip in the longitudinal direction. The strains and deformations resulting were plotted at all load stages up to yielding of slab's longitudinal reinforcement.

The analysis revealed that the bulk of the contribution of the slab reinforcement occurred around the joint area in what will be termed hereafter the "Contribution Zone". This contribution was manifested by the large strains experienced by the slab's longitudinal reinforcement within this zone. Beyond this distance, it appears that the slab strains are relatively insignificant. The results were compared with the provisions of both the Canadian Code (CSA CAN3 A23.3 M84) and the American Code (Joint ACI-ASCE Committee 352).

In the CSA CAN3 A23.3 M84 provisions for "seismic design of ductile frame members subjected to flexure and axial load" and in order to achieve the desired beam hinging mechanism, the Standard specifies a "strong column-weak beam" design approach. Account is given for such an approach in clause 21.4.2.2, which specifies limitations for the relative strengths between the factored resistance of the columns and the nominal resistance of the beams framing into the joint. In this case, the nominal resistance of the beams shall include the effects of the slab reinforcement within a distance of 3 times the slab thickness measured from each side of the beam. In its recommendations for design of beam-column joints in monolithic reinforced concrete structures, the ACI-ASCE joint committee 352 gives guidance on the required flexural strength ratio between the nominal resistance of both the columns and the beams framing into the joint. In this case, no tension slab is considered with the beam.

The finite element analysis performed showed significant contribution of the slab reinforcement and yielded valuable results regarding the assessment of slab contribution under negative moments when the structure is subjected to lateral earthquake type loads. The results are compared with the provisions of both the Canadian (CSA) and American (ACI) codes. The nominal resistance was computed for the recommended slab width as well as for the CSA and ACI recommended widths. The moment-curvature responses for the cross-sections using all three widths were also generated. The effects of increased effective slab widths on the strength and stiffness of the beams as well as their effect on the beam's ductility were studied. Recommendations on an effective slab contribution zone for similar situations are also given.

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