## The lateral force method of analysis

4.5.2.1. Introduction: the lateral force method versus modal response spectrum analysis

In the lateral force method a linear static analysis of the structure is performed under a set of lateral forces applied separately in two orthogonal horizontal directions, X and Y. The intent is to simulate through these forces the peak inertia loads induced by the horizontal component of the seismic action in the two directions, X or Y. Owing to the familiarity and experience of structural engineers with elastic analysis for static loads (due to gravity, wind or other static actions), this method has long been - and still is - the workhorse for practical seismic design. The version of the method in Eurocode 8 has been tuned to give similar results for storey shears - considered as the fundamental seismic action effects - as those from modal response spectrum analysis (which is the reference method), at least for the type of structures to which the lateral force method is considered applicable.

For the type of structures where both the lateral force method and modal response spectrum analysis are applicable, the latter gives, on average, a slightly more even distribution of peak internal forces in different critical sections, such as the two ends of the same beam or column. These effects are translated to some savings in materials. Despite such savings, the overall inelastic performance of a structure is normally better if its members are dimensioned for the results of a modal response spectrum analysis, instead of the lateral force method. The better performance is attributed to closer agreement of the distribution of peak inelastic deformations in the non-linear response to the predictions of the elastic modal response spectrum analysis than to those of the lateral force approach.

As the use of modal response spectrum analysis is not subject to any constraints of applicability, it can be adopted by a designer who wishes to master the method as the single analysis tool for seismic design in 3D. In addition to this convenience, modal response spectrum analysis is more rigorous (e.g. unlike the lateral force method, it gives results independent of the choice of the two orthogonal directions, X and Y, of application of the horizontal components of the seismic action), and offers a better overall balance of economy and safety. So, with today's availability of reliable and efficient computer programs for modal response spectrum analysis of structures in 3D, and with the gradual establishment of structural dynamics as a core subject in structural engineering curricula and continuing education programmes in seismic regions of the world, it is expected that modal response spectrum analysis will grow in application and prevail in the long run. Even then, though, the lateral force method of analysis will still be relevant, due to its intuitive appeal and conceptual simplicity.

4.5.2.2. Applicability conditions

The fundamental assumptions underlying the lateral force procedure are that: Clause 4.3.3.2.1

(1) the response is governed by the first translational mode in the horizontal direction in which the analysis is performed

(2) a simple approximation of the shape of that mode is possible, without any calculations.

Section 4 of EN 1998-1 allows the use of the lateral force procedure only when both of the following conditions are met:

(a) The fundamental period of the building is shorter than 2 s and four times the transition period Tc between the constant spectral acceleration and the constant spectral pseudo-velocity regions of the elastic response spectrum.

(b) The building structure is characterized as regular in elevation, according to the criteria set out in Section 4.3.3.1.

If the condition (a) is not met, the second and/or third modes may contribute significantly to the response in comparison to the fundamental one, despite their normally lower participation factors and participating masses: at periods longer than 2 s or 4Tc, spectral values are low, while, when the fundamental period is that long, the second and/or third mode periods may fall within or close to the constant spectral acceleration plateau where spectral values are highest. Under these circumstances, accounting for higher modes through a modal response spectrum analysis is essential.

In structures that are not regular in elevation the effects of higher modes may be significant locally, i.e. near elevations of discontinuity or abrupt changes, although they may not be important for the global response, as this is determined by the base shear and overturning moment. A more important reason for this condition, though, is that the common and simple approximation of the first mode shape may not be applicable when there are irregularities in elevation.

Only condition (a) above is explicitly required to be met in both horizontal directions for the lateral force procedure to be applicable. In principle, a structure that is characterized as regular in elevation in only one of the two directions may be subjected to lateral force analysis in that direction and to modal response spectrum analysis in the other, especially if the structure is analysed with a separate 2D model in each of these two directions. However, it is very unlikely that this is a practical design option. So, in practice, both conditions have to be met in both horizontal directions for the lateral force procedure to be applicable.

Clause The base shear is derived separately in the two horizontal directions in which the structure is

4.3.3.2.2(I) analysed, on the basis of the first translational mode in that horizontal direction:

where Sd(7\) is the value of the design spectrum at the fundamental period T, in the horizontal direction considered and Am is the effective modal mass of the first (fundamental) mode, expressed as a fraction A of the total mass, m, of the building above the foundation or above the top of a rigid basement. If the building has more than two storeys and a fundamental period TY shorter than 2TC (with Tc denoting again the transition period between the constant spectral acceleration and the constant spectral pseudo-velocity ranges), A = 0.85. In buildings with just two storeys, practically the full mass participates in the first mode, and A = 1.0; the same A value is used if T1 > 2Tc, to account for the increased importance of the second (and of higher) modes. The aim of the introduction of the A factor is to emulate the modal response spectrum analysis method, at least as far as the global seismic action effects are concerned (base shear and overturning moment).

4.5.2.4. Estimation of the fundamental period T, Clause Eurocode 8 encourages estimation of the fundamental period T1 through methods based on

4.3.3.2.2(2) mechanics. A fairly accurate such estimate of Tt is provided by the Rayleigh quotient:

I mfi

lm where <5; denotes the lateral drift at degree of freedom i from an elastic analysis of the structure under a set of lateral forces Fi applied to the degrees of freedom of the system. Both Fi and <5, are taken in the horizontal direction, X or Y, in which 7", is sought. For a given pattern (i.e. distribution) of the forces Fl over the degrees of freedom z, the drifts Si are proportional to Fp and the outcome of equation (D4.6) is independent of the absolute magnitudes of Ft. As equation (D4.6) is also rather insensitive to the distribution of these forces to the degrees of freedom z, any reasonable distribution of Fi may be used. It is both convenient and most accurate to use as F: the lateral forces corresponding to the distribution of the total base shear of equation (D4.5) to the degrees of freedom, i, postulated in the lateral force method of analysis (see Section 4.5.2.5 and equation (D4.7)). As at this stage the value of the design base shear is still unknown, the magnitude of lateral forces Fl can be such that their resultant base shear is equal to the total weight of the structure, i.e. as if A5d(T1) is equal to l.Og. Then, a single linear static analysis for each horizontal direction,Xor Y, is used both for (1) the estimation of Tx from equation (D4.6), and (2) for the calculation of the effects of the horizontal component of the seismic action in that direction. The seismic action effects from this analysis are multiplied by the value of ASd(T1) determined from the design spectrum for the now known natural period and used as the horizontal seismic action effects,^ or AY.

Eurocode 8 also allows the use in equation (D4.5) of values of 7\ estimated through Clauses empirical expressions - mostly adopted from the SEAOC '99 requirements.40 For Tx in 4.3.3.2.2(3), seconds and all other dimensions in metres: 4.3.3.2.2(4)

• T, = 0.085//3'4, for steel moment frame buildings less than 40 m tall

• J\ = 0.075//3'4, for buildings less than 40 m tall with concrete frames or with steel frames with eccentric bracings

• Tj = 0.05Hm for buildings less than 40 m tall with any other type of structural system (including concrete wall buildings)

• T] = 0.075/[X^w;(0.2 + (IJRr))2]1/2, in buildings with concrete or masonry walls where H denotes the total height of the building from the base or above the top of a rigid basement, and A.m and lwi denote the horizontal cross-sectional area and the length of wall i, with the summation extending over all ground storey walls i parallel to the direction in which Tx is estimated.

Such expressions represent lower (mean minus one standard deviation) bounds to values inferred from measurements on buildings in California in past earthquakes. As such measurements include the effects of non-structural elements on the response, these empirical expressions give lower estimates of the period than equation (D4.6). They are used because they give conservative estimates of Si(T1) - usually in the constant spectral acceleration plateau - for force-based design. Being derived from a high-seismicity region, these expressions are even more conservative for use in moderate- or low-seismicity areas, where structures have lower required earthquake resistance and hence are less stiff. Moreover, as estimation of 7\ from equation (D4.6) is quite accurate and requires limited extra calculations (only application of equation (D4.6) to the results of the linear static analysis anyway performed for the lateral force analysis), there is no real reason to resort to the use of empirical expressions.

The use of a period calculated from mechanics, regardless of how its value compares to the empirical value, as well as the introduction of the A factor in equation (D4.5), show that Eurocode 8 tries to bring the results of the lateral force method closer to those of modal response spectrum analysis, and not the other way around as US codes do.

4.5.2.5. Lateral force pattern

To translate the peak base shear from equation (D4.5) into a set of lateral inertia forces in Clause 4.3.3.2.3

the same direction (i.e. that of the horizontal component of the seismic action) applied to the degrees of freedom, i, of the structure, a distribution with height, z, of the peak lateral drifts in the same direction is assumed, $(z). Then, as in a single mode of vibration the peak lateral inertia force for the degree of freedom i is proportional to $(z,-)m;, where mi is the mass associated with that degree of freedom, and the base shear from equation (D4.5), Fb, is distributed to the degrees of freedom as follows:

j where the summation in the denominator extends over all degrees of freedom.

Within the field of application of the lateral force method (higher modes unimportant, structures regular in elevation) and in the spirit of the simplicity of the approach, the first-mode drift pattern is normally taken as proportional to elevation, z, from the base or above the top of a rigid basement, i.e. = zt. Moreover, although the presentation above is general, for any arrangement of the masses and degrees of freedom in space, for buildings with floors acting as rigid diaphragms the discretization in equation (D4.7) refers to floors or storeys (index i, with i = 1 at the lowest floor and i = rj at the roof) and lateral forces F, are applied at the floor centres of mass.

The result of equation (D4.7) for <5, = z,- is commonly termed the 'inverted triangular' pattern of lateral forces (although in reality it is just the drifts that have an 'inverted triangular' distribution, and the pattern of forces depends also on the distribution of masses,

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