Regularity in plan
Regularity in plan influences essentially the choice of the structural model. The reasoning Clause 4.2.3.2
behind the provisions of EN 19981 in this respect is that structures that are regular in plan tend to respond to seismic excitation along their main structural directions in an uncoupled manner. Accordingly, for the design of regular structures in plan it is acceptable to analyse them in a simplified way, using planar models in each main structural direction.
4.3.2.1. Criteria for structural regularity in plan
A building can be characterized as regular in plan if it meets all of the following numbered Clause 4.2.3.2(I) conditions, at all storey levels:
(1) The distribution in plan of the lateral stiffness and the mass is approximately Clause 4.2.3.2(2) symmetrical with respect to two orthogonal horizontal axes. Normally, the horizontal components of the seismic action are consequently applied along these two axes. As absolute symmetry is not required, it is up to the designer to judge whether this condition is met or not.
Clauses
(2) The outline of the structure in plan should have a compact configuration, delimited by a convex polygonal line. What counts in this respect is the structure, as defined in plan by its vertical elements, and not the floor (including balconies and any other cantilevering parts). Any single reentrant corner or edge recess of the outline of the structure in plan should not leave an area between it and the convex polygonal line enveloping it which is more than 5% of the area inside the outline. For a rectangular plan with a single reentrant corner or edge recess, this is equivalent to, for example, a recess of 20% of the parallel floor dimension in one direction and of 25% in the other; or, if there are four such reentrant corners or edge recesses, to, for example, a recess of 25% of the parallel floor dimension in both directions. L, C, H, I or Xshaped plans should respect this condition, in order for the structure to be considered as regular in plan.
(3) It should be possible to consider the floors as rigid diaphragms, in the sense that their inplane stiffness is sufficiently large, so that the floor inplan deformation due to the seismic action is negligible compared with the interstorey drifts and has a minor effect on the distribution of seismic shears among the vertical structural elements. Conventionally, a rigid diaphragm is defined as one in which, when it is modelled with its actual inplane flexibility, its horizontal displacements due to the seismic action nowhere exceed those resulting from the rigid diaphragm assumption by more than 10% of the corresponding absolute horizontal displacements. However, it is neither required nor expected that fulfilment of this latter definition is computationally checked. For instance, a solid reinforced concrete slab (or castinplace topping connected to a precast floor or roof through a clean, rough interface or shear connectors) may be considered as a rigid diaphragm, if its thickness and reinforcement (in both horizontal directions) are well above the minimum thickness of 70 mm and the minimum slab reinforcement of Eurocode 2 (which is a Nationally Determined Parameter (NDP) to be specified in the National Annex to Eurocode 2) required in clause 5.10 of EN 19981 for concrete diaphragms (rigid or not). For a diaphragm to be considered rigid, it should also be free of large openings, especially in the vicinity of the main vertical structural elements. If the designer does not feel confident that the rigid diaphragm assumption will be met due to the large size of such openings and/or the small thickness of the concrete slab, then he or she may want to apply the above conventional definition to check the rigidity of the diaphragm.
(4) The aspect ratio of the floor plan, A = LmJLmin, where Lmax and Lmin are respectively the larger and smaller inplan dimensions of the floor measured in any two orthogonal directions, should be not more than 4. This limit is to avoid situations in which, despite the inplane rigidity of the diaphragm, its deformation due to the seismic action as a deep beam on elastic supports affects the distribution of seismic shears among the vertical structural elements.
(5) In each of the two orthogonal horizontal directions,x and_y, of nearsymmetry according to condition 1 above, the 'static' eccentricity, e, between the floor centre of mass and the storey centre of lateral stiffness is not greater than 30% of the corresponding storey torsional radius, r.
ey<0.3ry
The torsional radius rx in equation (D4.1) is defined as the square root of the ratio of (a) the torsional stiffness of the storey with respect to the centre of lateral stiffness to (b) the storey lateral stiffness in the (orthogonal to x) y direction; for ry, the storey lateral stiffness in the (orthogonal toy) x direction is used in the denominator. (6) The torsional radius of the storey in each of the two orthogonal horizontal directions, x andy, of nearsymmetry according to condition 1 above is not greater than the radius of gyration of the floor mass:
The radius of gyration of the floor mass in plan, ls, is defined as the square root of the ratio of (a) the polar moment of inertia of the floor mass in plan with respect to the centre of mass of the floor to (b) the floor mass. If the mass is uniformly distributed over a rectangular floor area with dimensions I and b (that include the floor area outside of the outline of the vertical elements of the structural system), ls is equal to V[(/2 + b2)l 12],
Condition 6 ensures that the period of the fundamental (primarily) translational mode in each of the two horizontal directions, x and y, is not shorter than the lower (primarily) torsional mode about the vertical axis z, and prevents strong coupling of the torsional and translational response, which is considered uncontrollable and potentially very dangerous. In fact, as ls is defined with respect to the centre of mass of the floor in plan, the torsional radii rx and ry that should be used in equation (D4.2) for this ranking of the three abovementioned modes to be ensured are those defined with respect to the storey centre of mass, rnu. and r , which are related to the torsional radii rx, ry defined with respect to the storey centre of lateral stiffness as r^ = al(rx2 + ex2) and rmy = ~*l(r2 + e2). The greater the 'static' eccentricities ex, ey between the centres of mass and stiffness, the larger the margin provided by equation (D4.2) against a torsional mode becoming predominant. It is worth remembering that if the elements of the lateralloadresisting system are distributed in plan as uniformly as the mass, then the condition of equation (D4.2) is satisfied (be it marginally) and does need to be checked explicitly, whereas if the main lateralloadresisting elements, such as strong walls or bracings, are concentrated near the plan centre, this condition may not be met, and equation (D4.2) needs to be checked.
It is worth noting that, if the lower few eigenvalues are determined within the context of a modal responsespectrum analysis, they may be used directly to determine whether equation (D4.2) is satisfied for the building as a whole: if the period of a predominantly torsional mode of vibration is shorter than those of the primarily translational ones in the two horizontal directions x and y, then equation (D4.2) may be considered as satisfied.
An exhaustive review of the available literature on the seismic response of torsionally unbalanced structures41 has shown that conditions 5 and 6 provide a margin against excessive torsional response. In Fig. 4.1, solid black symbols represent good or satisfactory behaviour, while open and grey symbols correspond to poor behaviour, according to nonlinear dynamic analyses of various degrees of sophistication and reliability. In Fig. 4.1 the regularity region of EN 19981 is that to the left of the rightmost inclined line and above the horizontal line at rib = 0.35 (the ratio r/b ranges from 0.3r/ls to 0.4r//s, depending on the aspect ratio of the floor plan, l/b).
The centre of lateral stiffness is defined as the point in plan with the following property: Clause 4.2.3.2(8) any set of horizontal forces applied at floor levels through that point produces only translation of the individual storeys, without any rotation with respect to the vertical axis (twist). Conversely, any set of storey torques (i.e. of moments with respect to the vertical axis, z) produces only rotation of the floors about the vertical axis that passes through the centre of lateral stiffness, without horizontal displacement of that point inx and y at any storey. If such a point exists, the torsional radius, r, defined as the square root of the ratio of torsional stiffness with respect to the centre of lateral stiffness to the lateral stiffness in one horizontal direction, is unique and well defined. Unfortunately, the centre of lateral stiffness, as defined above, and with it the torsional radius, r, are unique and independent of the lateral loading only in singlestorey buildings. In buildings of two storeys or more, such a definition is not unique and depends on the distribution of lateral loading with height. This is especially so if the structural system consists of subsystems which develop different patterns of storey horizontal displacements under the same set of storey forces (e.g. moment frames exhibit a shearbeam type of horizontal displacement, while walls and frames with bracings concentric or eccentric  behave more like vertical cantilevers). For the general case of such systems, Section 4 of EN 19981 refers to the National Annex for an appropriate approximate definition of the centre of lateral stiffness and of the torsional radius, r.
Clauses
For singlestorey buildings Section 4 of EN 19981 allows the determination of the centre of lateral stiffness and the torsional radius on the basis of the moments of inertia of the crosssections of the vertical elements, neglecting the effect of beams, as
In equations (D4.3) and (D4.4), EIX and EIy denote the section rigidities for bending within a vertical plane parallel to the horizontal directions x or y, respectively (i.e. about an axis parallel to axis y or x, respectively). Moreover, Section 4 of EN 19981 allows the use of equations (D4.3) and (D4.4) to determine the centre of lateral stiffness and the torsional radius in multistorey buildings also, provided that their structural system consists of subsystems which develop similar patterns of storey horizontal displacements under storey horizontal forces proportional to m,2„ namely only moment frames (exhibiting a shearbeam type of horizontal displacement pattern), or only walls (deflecting like vertical cantilevers). For wall systems, in which shear deformations are also significant in addition to the flexural ones, an equivalent rigidity of the section should be used in equations (D4.3) and (D4.4). It is noted that, unlike the general and more accurate but tedious method outlined above, which yields a single pair of radii rx and ry for the entire building, to be used to check if equations (D4.1) and (D4.2) are satisfied at all storeys, if the crosssection of vertical elements changes from one storey to another, the approximate procedure of equations (D4.3) and (D4.4) gives different pairs of rx and ry, and possibly different locations of the centre of stiffness in different storeys (which affects, in turn, the static eccentricities ex andey).
4.3.2.2. Design implications of regularity in plan
Implications for the analysis: the 2D (plane) versus 3D (spatial) structural model
If a building is characterized as regular in plan, the analysis for the two horizontal components Clauses of the seismic action may use an independent 2D model in each of the two horizontal 4.2.3.1(2), directions of (near) symmetry,x andy. In such a model, the structure is considered to consist 4.2.3.1 (3), of a number of plane frames (moment frames, or frames with concentric or eccentric 4.3.1 (5)
bracings) and/or walls (some of which may actually belong in a plane frame together with coplanar beams and columns), all of them constrained to have the same horizontal displacement at floor levels.
Each 2D model will be analysed for the horizontal component of the seismic action parallel to it (possibly with consideration of the vertical component as well, if required), and will yield internal forces and other seismic action effects only within vertical planes parallel to that of the analysis. This means that the analysis will give no internal forces for beams, bracings or even walls which are in vertical planes orthogonal to the horizontal component of the seismic action considered. Bending in columns and walls will also be uniaxial, with axial force only due to the horizontal component of the seismic action which is parallel to the plane of the analysis.
Given the proliferation of commercial computer programs for linear elastic seismic response analysis  static or dynamic  in 3D, there is little sense today in pursuing analyses with two independent 2D models instead of a spatial 3D model. This is particularly so if the analysis is done for the purposes of seismic design, as in that case the software normally has capabilities to postprocess the results, in order to serve the specific needs of design. Such postprocessing is greatly facilitated if a single (3D) model is used for the entire structure. However, if two independent analyses are done using two different 2D models, the results of these analyses may have to be processed by a special postprocessing module that reads and interprets topology data and internal force results from two different sources. Alternatively, the combination of the internal force results can be done manually. It should be noted that internal force results from the two different 2D analyses need to be combined primarily in columns, due to the requirement to consider that the two horizontal components of the seismic action act simultaneously and to combine their action effects (either via the 1:0.3 rule or through the square root of the sum of the squares (SRSS)). It is true that the facility provided in Section 5 of EN 19981 for the biaxial bending of columns (namely to dimension the column for a uniaxial bending moment equal to that from the analysis divided by 0.7, neglecting the simultaneously acting orthogonal component of bending moment) is quite convenient in this respect. However, the need to combine the column axial forces due to the two horizontal components of the seismic action (via the 1:0.3 or the SRSS rule) remains, even within the framework of uniaxial bending mentioned in the last sentence for concrete columns. A possible way out for such columns might be to: (1) dimension the vertical reinforcement of the two opposite sides of the crosssection considering uniaxial bending (with the 1/0.7 magnification on the moment) with axial force due to the horizontal component of the seismic action which is orthogonal to the two opposite sides considered; (2) repeat the exercise for the two other sides and the corresponding horizontal component of the action; and (3) add the resulting vertical reinforcement requirements on the section, neglecting any positive contribution of any one of them to the flexural resistance in the orthogonal direction of bending.
All things considered, it is not worthwhile using linear analysis with two independent 2D models in building structures which are regular in plan. In that regard, the characterization of a structure as regular or nonregular in plan is important only for the default value of the part of the behaviour factor q which is due to the redundancy of the structural system, as explained below.
However, the facility of two independent 2D models is very important for nonlinear analysis, static (pushover) or dynamic (timehistory). Reliable, widely accepted and numerically stable nonlinear constitutive models (including the associated failure criteria) are available only for members in uniaxial bending with (littlevarying) axial force; their extension to biaxial bending for wide use in 3D analysis belongs to the future. So, for the use of nonlinear analysis the characterization of a building structure as regular or nonregular in plan is very important.
Clause 4.3.3.1 (8) Within the framework of the lateral force procedure of analysis, two independent 2D models may also be used for buildings which have:
(1) a height less than 10 m, or 40% of the plan dimensions
(2) storey centres of mass and stiffness approximately on (two) vertical lines
(3) partitions and claddings well distributed vertically and horizontally, so that any potential interaction with the structural system does not affect its regularity
(4) torsional radii in the two horizontal directions at least equal to rx = V(/s2 + e/) and
If conditions 1 to 3 are fulfilled, but not condition 4, then two separate 2D models may still be used, provided that all seismic action effects from the 2D analyses are increased by 25%.
The above relaxation of the regularity conditions for using two independent 2D models instead of a full 3D model is meant to make it easier for the designer (and hence the owner) of small buildings to apply Eurocode 8. For this reason, the extent of the application of this facility will be determined nationally, and a note in Eurocode 8 states, without giving any recommendations for the selection, that the importance class(es) to which this relaxation will apply should be listed in the National Annex.
Implications for the behaviour factor q
As we will see in more detail in Chapters 5 to 7 of this guide, in most types of structural systems system overstrength due to redundancy is explicitly factored into the value of q, as a ratio ajav This is the ratio of the seismic action that causes development of a full plastic mechanism (au) to the seismic action at the first plastification in the system (a,). The value of al may be computed as the lower value over all member ends in the structure of the ratio (SRd  SV)/SE, where ,SRd is the design value of the action effect capacity at the location of first plastification, and >S'E and Sv are the values of the action effect there from the elastic analysis for the design seismic action and for the gravity loads included in the load combination of the 'seismic design situation'. The value of au may be found as the ratio of the base shear on development of a full plastic mechanism according to a pushover analysis to the base shear due to the design seismic action (e.g. see Fig. 5.2). As the designer may not consider it worth performing iterations of pushover analyses and design based on elastic analysis just to compute the ratio ajaj for the determination of the q factor, Sections 57 of EN 19981 give default values for this ratio. For buildings which are regular in plan, the default values range
from ajai  1.0, in buildings with very little structural redundancy, to ajax = 1.3 in multistorey multibay frames, with a default value of ajax = 1.2 used in the fairly common concrete dual systems (frame or wall equivalent), concrete coupledwall systems and steel or composite frames with eccentric bracings.
In buildings which are not regular in plan, the default value of aja¡ is the average of (1) 1.0 and (2) the default values given for buildings regular in plan. For the values of ajal = 1.21.3 specified as the default for the most common structural systems in the case of regularity in plan, the reduction in the default q factor is around 10%. If the designer considers such a reduction unacceptable, he or she may resort to iterations of pushover analyses and design based on elastic analysis, to quantify a possibly higher value of ajal for the (nonregular) structural system.
Fulfilment or not of equation (D4.2) has very important implications for the value of the behaviour factor q of concrete buildings. If at any floor, one or both conditions of equation (D4.2) are not met (i.e. if the radius of gyration of the floor mass exceeds the torsional radius in one or both of the two main directions of the building in plan), then the structural system is characterized as torsionally flexible, and the basic value of the behaviour factor q (i.e. prior to any reduction due to potential nonregularity in elevation (see Section 4.3.3.1)) is reduced to a value of q0 = 2 for Ductility Class Medium (DCM) or qQ = 3 for Ductility Class High (DCH). As nonfulfilment of equation (D4.2) is most commonly due to the presence of stiff concrete elements, such as walls or cores, near the centre of the building in plan, Section 6 of EN 19981 adopts the same reduction of the basic value of the q factor in steel buildings which employ such walls or cores for (part of) their earthquake resistance.
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