Keywords: structural engineering, semi-rigid behaviour, steel structures, component method, component behaviour, beam-to-column joints, cyclic behaviour, seismic behaviour.

Steel joints subjected to cyclic loading whereby the amplitude of the applied forces exceeds the yield strain of one or more of the relevant connection components usually exhibit degradation of its moment-rotation response that eventually leads to failure of the joint. This phenomenon has been the object of extensive experimental investigations in what is commonly called "low-cycle fatigue", the most recent impetus following the Kobe and Northridge earthquakes. As a result of these two earthquakes, unexpected failures of steel joints were observed, reflecting the poor behaviour of some less ductile components that could not dissipate the high frequency content of those two seismic events.

Concentrating on end-plate beam-to-column bolted joints, and depending on its actual design, the cyclic response may present slippage. To explore the influence of slippage on the behaviour of steel frames subjected to cyclic loading two cyclic models experimentally calibrated were chosen and numerically implemented. The Richard-Abbott model is based on a formula developed in 1975 [1] to reproduce the elastic-plastic behaviour of several materials and was initially used to simulate the static monotonic response of joints and later applied to cyclic situations. The second model, the model proposed by Mazzolani [2], based on the Ramberg-Osgood model, allows the mathematical simulation of hysteretic behaviour with slippage. As originally proposed, each complete cycle was divided in four branches (I, II, III and IV), the definition of branches I and II being similar to branches III and IV. However, in unsymmetrical joints, all parameters must be defined separately for the positive (branches I and II) and negative (branches III and IV) zones. Given that, in joints with slippage, the corresponding branch may start in the unloading zone, thus preventing the application of the model, a modified version was proposed in Simões et al [3]. It consists of the definition of each cycle with two single branches, ascending and descending, thus eliminating the limitation of slippage not being able to occur in the unloading branch. The two cyclic models described above were numerically implemented using the Delphi programming environment and Object Pascal. For the Richard-Abbott model, the controlling variable is rotation , while for the modified Mazzolani model the controlling variable is bending moment M. They were calibrated using two composite joints tested experimentally at the University of Coimbra [3].

A typical low-rise office building was selected, that basically consists of a two- storey building, with a inner service area (lifts, staircases, WC and storage and ducts), surrounded by a flexible office area, free from structural members. The structural layout consists of an orthogonal grid with five alignments with 3 spans of 7.5-5-7.5m in the transverse direction and 4 alignments with 4 equal spans of 7.5 m in the longitudinal direction. The total height of the steel frames is 7 m (3.5 m in each floor). The structure is composed of HEA 220 columns and composite beams supporting a concrete slab. Given the symmetry of the structure and to simplify the analysis, a major axis internal frame was selected, deemed to represent the structural response of the building.

To loading strategies were adopted: (i) a single cyclic horizontal force applied at the second floor level and, (ii) the same cyclic load, superimposed with the gravity loading described earlier, corresponding to a serviceability level. The cyclic load follows the ECCS load application strategy. The structural analysis was carried out using the finite element code Seismosoft, the joints being modelled using hysteretic bi-linear spring elements.

The results for both loading cases have shown increased horizontal displacements for the second case, together with increased negative moments. It is worth noting that slippage only occurs for rotations above 30 mrad. In this case (modified Mazzolani model), 15% and 35% degradation of resistance are observed for the ascending and descending branches, respectively. Analogously, 60% and 15% stiffness degradation is also noted for the ascending and descending branches, respectively. Globally, a decrease of bending moment of approximately 20% is clearly observed, both for the external nodes and the internal nodes of the first floor. A redistribution of internal forces to the internal nodes of the second floor is also noted. Similar results are obtained for load case 2. These results stress the importance of a correct assessment of this effect.

1

Richard, R.M. and Abbott, B.J. (1975) Versatile Elasto-Plastic Stress-Strain Formula. Journal of the Engineering Mechanics Division, ASCE, 101, EM4, 511-515.

2

Mazzolani, F.M. (1988) Mathematical Model for Semi-Rigid Joints Under Cyclic Loads, in R. Bjorhovde et al. (eds) Connections in Steel Structures: Behaviour, Strength and Design, Elsevier Applied Science Publishers, London, 112-120.

3

Simões, R., Simões da Silva, L. and Cruz P., "Behaviour of End-Plate Beam-to-Column Composite joints under cyclic Loading", International Journal of Steel and Composite Structures, 1(3), 355-376 (2001).

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