## Introduction the level of discretization

In constructing the structural model of a building for the purposes of its earthquake-resistant design, the designer should keep in mind that his or her objective is the design of an earthquake-resistant structure and not the analysis perse. This ultimate objective is pursued through a long process, an intermediate stage of which is normally a linear elastic analysis of a mathematical model of the structure, as conceived. A subsequent, and at least equally important phase, is that of the detailed design of members, which comprises dimensioning of regions for the internal force results of the analysis and member detailing for the ductility demands of the design seismic action. The only purpose of modelling and analysis is to provide the data for this penultimate phase of detailed design. Rules for practical dimensioning and detailing of members against cyclic inelastic deformations are sufficiently developed mainly - if not only - for prismatic members. Corresponding rules for 2D members are available only for special cases with a specific structural role, e.g. low-shear-ratio coupling concrete beams in antisymmetric bending, seismic link regions in steel frames with eccentric bracings, or interior or exterior beam-column joint panel zones. So, the structural model should employ primarily 3D beam elements.

According to Section 4 of EN 1998-1, the model of the building structure for linear elastic Clause 4.3.2(1) analysis should represent well the distribution of stiffness in structural elements and of the mass throughout the building. This may not be enough for the purposes of design. As emphasized in the above, the idealization and discretization of the structure should correspond closely to its geometric configuration in 3D, so that it is fit for the main purpose of the analysis, i.e. to provide the seismic action effects for the dimensioning and detailing of members and sections. This means, for instance, that a stick-type model, with all members of a storey combined into a single mathematical element connecting adjacent floors and only three degrees of freedom per storey (for analysis in 3D) is not sufficient for the purposes of seismic design. At the other extreme, a very detailed finite-element discretization, providing very 'accurate' predictions of elastic displacements and stresses on a point-by-point basis, is practically useless, as reliable and almost equally accurate predictions of the 'average' seismic action effects which are necessary for member dimensioning, i.e. the stress resultants, can be directly obtained through an appropriate space frame idealization of the structure. Moreover, some fine effects captured by detailed finite-element analyses, such as those of non-planar distributions of strains in the cross-section of deep members, or shear lag in members with composite cross-section, lose their relevance under inelastic response conditions, such as those encountered under the design seismic action and used as the basis of ultimate limit state calculations and member verification. It should also be recalled that the connection between (1) a 2D element or region modelled using 2D finite-element and (2) 3D beam elements in the same plane requires special treatment, as in shell finite-elements the rotation degrees of freedoms about the normal to the shell surface do not have any stiffness and hence cannot be directly connected to 3D beam elements. For all these reasons, the type of structural model appropriate for an analysis for seismic design is a member-by-member type of model, in which every beam, column or bracing and every part of a wall between successive floors is represented as a 3D beam element, with the three translations and the three rotations at each joint of these elements considered as degrees of freedom. Masses may also be lumped at these nodal points and associated in general with all six degrees of freedom there. If the analysis also considers the vertical component of the seismic action, lumped masses at intermediate points of long-span girders or at the ends of cantilevers should also be included. This requires nodes with six degrees of freedom at these points, regardless of whether other elements frame into them there, or not.

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