b) the “modal response spectrum analysis", which is applicable to all types of buildings (see 4.3.3.3).
(4) As an alternative to a linear method, a non-linear method may also be used, such as:
c) non-linear static (pushover) analysis;
d) non-linear time history (dynamic) analysis,
provided that the conditions specified in (5) and (6) of this subclause and in 4.3.3.4 are satisfied.
NOTE For base isolated buildings the conditions under which the linear methods a) and b) or the nonlinear ones c) and d), may be used are given in Section 10. For non-base-isolated buildings, the linear methods of 4.3.3.1(3) may always be used, as specified in 4.3.3.2.1. The choice of whether the nonlinear methods of 4.3.3.1(4) may also be applied to non-base-isolated buildings in a particular country , will be found in its National Annex. The National Annex may also include reference to complementary information about member deformation capacities and the associated partial factors to be used in the Ultimate Limit State verifications in accordance with 4.4.2.2(5).
(5) Non-linear analyses should be properly substantiated with respect to the seismic input, the constitutive model used, the method of interpreting the results of the analysis and the requirements to be met.
(6) Non-base-isolated structures designed on the basis of non-linear pushover analysis without using the behaviour factor q (see 4.3.3.4.2.1(1)d), should satisfy 4.4.2.2(5), as well as the rules of Sections 5 to 9 for dissipative structures.
(7) Linear-elastic analysis may be performed using two planar models, one for each main horizontal direction, if the criteria for regularity in plan are satisfied (see 4.2.3.2).
(8) Depending on the importance class of the building, linear-elastic analysis may be performed using two planar models, one for each main horizontal direction, even if the criteria for regularity in plan in 4.2.3.2 are not satisfied, provided that all of the following special regularity conditions are met:
a) the building shall have well-distributed and relatively rigid cladding and partitions;
b) the building height shall not exceed 10 m;
c) the in-plane stiffness of the floors shall be large enough in comparison with the lateral stiffness of the vertical structural elements, so that a rigid diaphragm behaviour may be assumed.
d) the centres of lateral stiffness and mass shall be each approximately on a vertical line and, in the two horizontal directions of analysis, satisfy the conditions: rx2 > ls2 + eox2, ry2 > ls2 + eoy2, where the radius of gyration ls, the torsional radii rx and ry and the natural eccentricities eox and eoy are defined as in 4.2.3.2(6).
NOTE The value of the importance factor, γI, below which the simplification of the analysis in accordance with 4.3.3.1(8) is allowed in a country, may be found in its National Annex.
(9) In buildings satisfying all the conditions of (8) of this subclause with the exception of d), linear-elastic analysis using two planar models, one for each main horizontal direction, may also be performed, but in such cases all seismic action effects resulting from the analysis should be multiplied by 1,25.
(10)P Buildings not conforming to the criteria in (7) to (9) of this clause shall be analysed using a spatial model.
(11)P Whenever a spatial model is used, the design seismic action shall be applied along all relevant horizontal directions (with regard to the structural layout of the building) and their orthogonal horizontal directions. For buildings with resisting elements in two perpendicular directions these two directions shall be considered as the relevant directions.
4.3.3.2 Lateral force method of analysis 4.3.3.2.1 General
(1)P This type of analysis may be applied to buildings whose response is not significantly affected by contributions from modes of vibration higher than the fundamental mode in each principal direction.
(2) The requirement in (1)P of this subclause is deemed to be satisfied in buildings which fulfil both of the two following conditions.
a) they have fundamental periods of vibration T1 in the two main directions which are smaller than the following values
≤ ⋅ s 0 , 2 4 C
1
T T (4.4)
where TC is defined in 3.2.2.2;
b) they meet the criteria for regularity in elevation given in 4.2.3.3.
4.3.3.2.2 Base shear force
(1)P The seismic base shear force Fb, for each horizontal direction in which the building is analysed, shall be determined using the following expression:
--`,`,,,`,``,,,```````,,`,,`,,-`-`,,`,,`,`,,`---
( )⋅ ⋅λ
=S T m
Fb d 1 (4.5)
where
Sd (T1) is the ordinate of the design spectrum (see 3.2.2.5) at period T1;
T1 is the fundamental period of vibration of the building for lateral motion in the direction considered;
m is the total mass of the building, above the foundation or above the top of a rigid basement, computed in accordance with 3.2.4(2);
λ is the correction factor, the value of which is equal to: λ = 0,85 if T1 < 2 TC and the building has more than two storeys, or λ = 1,0 otherwise.
NOTE The factor λ accounts for the fact that in buildings with at least three storeys and translational degrees of freedom in each horizontal direction, the effective modal mass of the 1st (fundamental) mode is smaller, on average by 15%, than the total building mass.
(2) For the determination of the fundamental period of vibration period T1 of the building, expressions based on methods of structural dynamics (for example the Rayleigh method) may be used.
(3) For buildings with heights of up to 40 m the value of T1 (in s) may be approximated by the following expression:
4 / 3 t
1 C H
T = ⋅ (4.6)
where
Ct is 0,085 for moment resistant space steel frames, 0,075 for moment resistant space concrete frames and for eccentrically braced steel frames and 0,050 for all other structures;
H is the height of the building, in m, from the foundation or from the top of a rigid basement.
(4) Alternatively, for structures with concrete or masonry shear walls the value Ct in expression (4.6) may be taken as being
c t 0,075/ A
C = (4.7)
where
( )
( )
[ i wi 2]
c A 0,2 l /H
A =Σ ⋅ + (4.8)
and
Ac is the total effective area of the shear walls in the first storey of the building, in m2;
Ai is the effective cross-sectional area of shear wall i in the direction considered in the first storey of the building, in m2;
--`,`,,,`,``,,,```````,,`,,`,,-`-`,,`,,`,`,,`---
H is as in (3) of this subclause;
lwi is the length of the shear wall i in the first storey in the direction parallel to the applied forces, in m, with the restriction that lwi/H should not exceed 0,9.
(5) Alternatively, the estimation of T1 (in s) may be made by using the following expression:
d
T1=2⋅ (4.9)
where
d is the lateral elastic displacement of the top of the building, in m, due to the gravity loads applied in the horizontal direction.
4.3.3.2.3 Distribution of the horizontal seismic forces
(1) The fundamental mode shapes in the horizontal directions of analysis of the building may be calculated using methods of structural dynamics or may be approximated by horizontal displacements increasing linearly along the height of the building.
(2)P The seismic action effects shall be determined by applying, to the two planar models, horizontal forces Fi to all storeys.
j j
i b i
i s m
m F s
F ⋅
⋅ ⋅
= Σ (4.10)
where
Fi is the horizontal force acting on storey i;
Fb is the seismic base shear in accordance with expression (4.5);
si, sj are thedisplacements of masses mi, mj in the fundamental mode shape;
mi,mj are the storey masses computed in accordance with 3.2.4(2).
(3) When the fundamental mode shape is approximated by horizontal displacements increasing linearly along the height, the horizontal forces Fi should be taken as being given by:
j j
i b i
i z m
m F z
F ⋅
⋅ ⋅
= Σ (4.11)
where
zi, zj are the heights of the masses mi mj above the level of application of the seismic action (foundation or top of a rigid basement).
(4)P The horizontal forces Fi determined in accordance with this clause shall be distributed to the lateral load resisting system assuming the floors are rigid in their plane.
--`,`,,,`,``,,,```````,,`,,`,,-`-`,,`,,`,`,,`---
4.3.3.2.4 Torsional effects
(1) If the lateral stiffness and mass are symmetrically distributed in plan and unless the accidental eccentricity of 4.3.2(1)P is taken into account by a more exact method (e.g. that of 4.3.3.3.3(1)), the accidental torsional effects may be accounted for by multiplying the action effects in the individual load resisting elements resulting from the application of 4.3.3.2.3(4) by a factor δ given by
e
6 , 0
1 L
⋅ x +
δ = (4.12)
where
x is the distance of the element under consideration from the centre of mass of the building in plan, measured perpendicularly to the direction of the seismic action considered;
Le is the distance between the two outermost lateral load resisting elements, measured perpendicularly to the direction of the seismic action considered.
(2) If the analysis is performed using two planar models, one for each main horizontal direction, torsional effects may be determined by doubling the accidental eccentricity eai of expression (4.3) and applying (1) of this subclause with factor 0,6 in expression (4.12) increased to 1,2.