The radius of gyration, l, is the square root of the ratio of the polar moment of inertia to the mass, the polar moment of inertia being calculated about the centre of mass. For a rectangular building of side lengths l and b, and a uniform mass distribution, Equation (4.3) applies.
The requirement for torsional radius r to exceed 3.33 times the mass-stiffness eccentricity e (item 5 on the list at the beginning of this section) relates the torsional resistance to the driving lateral-torsional excitation, correctly favouring configurations with stiff perimeter elements and penalising those relying on central elements for lateral resistance. It is very similar to a requirement that has appeared for many years in the Japanese code.
The requirement for r to exceed radius of gyration, l (item 6 on the list at the beginning of this section), ensures that the first torsional mode of vibration does not occur at a higher period than the first translational mode in either direction, and demonstrating that this applies is an alternative way of showing that 'torsional flexibility' is avoided (EC8 Manual, (ISE/AFPS 2009)).
4.5.3 Regularity in elevation
A building must satisfy all the following requirements to be classified as regular in elevation.
1 All the vertical load resisting elements must continue uninterrupted fro foundation level to the top of the building or (here setbacks are present - see 4 below) to the top of the setback.
2 Mass and stiffness must either remain constant with height or reduce only gradually, ithout abrupt changes. uantification is not provided in EC8; the EC8 Manual (ISE/AFPS 2009) recommends that buildings where the mass or stiffness of any storey is less than 70 per cent of that of the storey above or less than 80 per cent of the average of the three storeys above should be classified as irregular in elevation.
3 In buildings with moment-resisting frames, the lateral resistance of each storey (i.e. the seismic shear initiating failure within that storey, for the code-specified distribution of seismic loads) should not vary 'disproportionately' between storeys. Generally, no quantified limits are stated by EC8, although special rules are given where the variation in lateral resistance is due to masonry infill within the frames. The EC8 Manual (ISE/AFPS 2009) recommends that buildings where the strength of any storey is less than 80 per cent of that of the storey above should be classified as irregular in elevation.
4 Buildings with setbacks (i.e. where the plan area suddenly reduces bettveen successive storeys) are generally irregular, but may be classified as regular if less than limits defined in the code. The limits broadly speaking are a total reduction in idth fro top to botto on any face not exceeding 30 per cent, ith not ore than 10 per cent at any level compared to the level below. However, an overall reduction in width of up to half is permissible within the lowest 15 per cent of the height of the building.
EC8 Part 1 Section 2.2.4 contains some specific design measures for ensuring that structures meet the performance requirements of the code. These apply to all structures, not just buildings, and a crucial requirement concerns capacity design, which determines much of the content of the material-
Figure 4.5 Capacity design - ensuring that ductile links are weaker than brittle ones
Figure 4.5 Capacity design - ensuring that ductile links are weaker than brittle ones specific rules for concrete, steel and composite buildings in Sections 5, 6 and 7 of EC8 Part 1.
Clause 2(P) of Section 188.8.131.52 states:
In order to ensure an overall dissipative and ductile behaviour, brittle failure or the premature formation of unstable mechanisms shall be avoided. To this end, where required in the relevant Parts of EN 1998, resort shall be ade to the capacity design procedure, hich is used to obtain the hierarchy of resistance of the various structural components and failure odes necessary for ensuring a suitable plastic echanis and for avoiding brittle failure odes.
Professor Paulay's 'ductile chain' illustrates the principle of capacity design - see Figure 4.5. The idea is that the ductile link yields at a load that is ell belo the failure load of the brittle links. lthough ost building structures are somewhat less straightforward than the chain used in Tom Paulay's example, one of the great strengths of the capacity design principle is that it relies on simple static analysis to ensure good performance, and is not dependent on the vagaries of a complex dynamic calculation.
Ensuring that columns are stronger than beams in moment frames, concrete beams are stronger in shear than in flexure, and steel braces buckle before columns, are three important examples of capacity design. A general rule for all types of frame building given in EC8 Part 1 Section 184.108.40.206 is that the moment strength of columns connected to a particular node should be 30 per cent greater than the moment strength of the beams:
he rule ust be satisfied for concrete buildings, but the alternative capacity design rules given in EC8 Section 6.6.3 may apply to steel columns (see hapter 6 of this book).
ne feature of capacity design is that it ensures that designers identify clearly hich parts of the structure ill yield in a severe earthquake (the 'critical' regions) and which will remain elastic. An important related clause is given by Clause 3(P) of Section 220.127.116.11 of EC8.
Since the seismic performance of a structure is largely dependent on the behaviour of its critical regions or elements, the detailing of the structure in general and of these regions or elements in particular, shall be such as to aintain the capacity to transit the necessary forces and to dissipate energy under cyclic conditions. To this end, the detailing of connections bettveen structural elements and of regions where non-linear behaviour is foreseeable should receive special care in design.
4.7 Other basic issues for building design 4.7.1 Load combinations
Basic load combinations are given in EN 1990: Basis for design, and for seismic load combinations are as follows:
Design action effect Permanent Earthquake Reduced variable load is the factor defined in EN 1990, which reduces the variable (or live) load from its characteristic (upper bound) value to its 'quasi-permanent' value, expected to be present for most of the time. It is typically in the range 0.0 to 0.8, depending on the variability of the loading type.
4.7.2 'Seismic' mass
The mass taken when calculating the earthquake loads should comprise the full permanent (or dead) load plus the variable (or live) load multiplied by a factor yEi. EC8 Part 1 Section 4.2.4 quantifies this as the factor y2i defined in Section 4.7.1 above multiplied by a further reduction factor j that allows for the incomplete coupling between the structure and its live load:
Typical values of j are in the range 0.5 to 1, depending on the loading type.
Four importance classes are recognised, as shown in Table 4.1, which also shows the recommended y¡ factor; this is, however, a 'Nationally Determined Parameter' (NDP), which may be varied in the National Annex.
Note that whereas in US practice the importance factors are applied to the seismic loads, in EC8 they are applied to the input motions. This makes an important difference when non-linear analysis is employed, since increasing the ground otions by per cent ay cause an increase of less than per
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