Equivalent Static Eccentricities in the Simplified Methods of Seismic Analysis of Buildings

Abstract

The equivalent static eccentricities of seismic forces are usually defined by codes with simple expressions of the static eccentricity. This paper presents certain formulae for the exact calculation of these eccentricities on the basis of the dynamic response of a simplified model. From the parametric analysis of such formulae the determinative role of the torsional and lateral stiffness of the system becomes obvious for the correct evaluation of the equivalent static eccentricities. Finally, a proposal is made for the improvement of the static torsional provisions of the current codes.

Static versus modal analysis: influence on inelastic response of multi-storey asymmetric buildings

“…In some cases it is close to that produced by the application of equivalent seismic forces at the mass centres (e.g., in torsionally rigid systems, particularly for small values of structural eccentricity) while in other cases it is totally different (e.g., in torsionally flexible systems with large structural eccentricity). Owing to the dependence of the elastic response on both e s and θ , it is difficult to define simple analytical equations of the above-mentioned design eccentricities, i.e., of eccentricities able to lead to reliable estimate of the elastic response of both torsionally flexible and stiff systems, (Calderoni et al 2002;Anastassiadis et al 1998). For this reason, when dealing with design procedures based on static analysis, formulations of design eccentricities found in literature are often restricted to torsionally rigid systems ( θ > 1).…”

Seismic nonlinear static new method of spatial asymmetric multi‐storey reinforced concrete buildings

“…According to the documented equivalent linear static seismic design method, two discrete loading cases of the asymmetric single‐storey building are needed for each horizontal seismic component. For example, when the floor's lateral static force has the same orientation with the principal II ‐axis, then the design eccentricities e f,I (near the flexible side of the building) and e r,I (near the stiff side of the building) are calculated by using the ‘equivalent static eccentricities or dynamic eccentricities e d1 and e d2 ’ (see explicit relationships in Anastassiadis et al , 1998) and the accidental eccentricity e a,I according to following equations (Figure 2): where e o,I is the static eccentricity of the building along to principal I ‐axis, β = 0.05 or 0.10, L I is the side length of the building that is perpendicular to the lateral static loading, according to Seismic Codes α = 1.50 and γ = −0.50 (NBCC/95, 1995; EAK/2003, 2003). The envelope E , II of the seismic demand displacements/stresses of the building resulting from the two (Equations (1) and (2) above‐mentioned discrete static analyses represents the seismic demands of the horizontal seismic component along II ‐axis, only. Similarly, for loading along the principal I ‐axis, the design eccentricities e f,II (near the flexible side of the building) and e r,II (near the stiff side of the building) are calculated by using the ‘equivalent static eccentricities e d3 and e d4 ’ and the accidental eccentricity e a,II according to following equations (Figure 3): where e o,II is the static eccentricity of the building along the principal II ‐axis, β = 0.05 or 0.10, L II is the side length of the building that is perpendicular to lateral static loading, according to the Seismic Codes α = 1.50 and γ = −0.50 (NBCC/95, 1995; EAK/2003, 2003), which are adequately documented as regards torsional provisions.The envelope E , I of the seismic demand displacements/stress of the building resulting from the two (Equations (3) and (4) above‐mentioned discrete static analyses represents the seismic demands of the seismic horizontal component along I ‐axis only.Finally, according to the established equivalent linear static seismic design method, the extreme values E , ex of the seismic demand displacements/stresses due to the spatial action of the two horizontal seismic components, arise by the ‘square root of sum of squares’ rule to E , I and E , II : …”

“…According to the documented equivalent linear static seismic design method, two discrete loading cases of the asymmetric single‐storey building are needed for each horizontal seismic component. For example, when the floor's lateral static force has the same orientation with the principal II ‐axis, then the design eccentricities e f,I (near the flexible side of the building) and e r,I (near the stiff side of the building) are calculated by using the ‘equivalent static eccentricities or dynamic eccentricities e d1 and e d2 ’ (see explicit relationships in Anastassiadis et al , 1998) and the accidental eccentricity e a,I according to following equations (Figure 2): where e o,I is the static eccentricity of the building along to principal I ‐axis, β = 0.05 or 0.10, L I is the side length of the building that is perpendicular to the lateral static loading, according to Seismic Codes α = 1.50 and γ = −0.50 (NBCC/95, 1995; EAK/2003, 2003).…”

Demonstration of torsional behaviour using vibration-based single-storey model with double eccentricities

“…A clear and comprehensive study of the single-storey system with single eccentricity was presented by Anastassiadis et al (1998). In that the main difference between the torsional responses of the building systems lies in their torsional stiffness and strength some interest attaches to studies like those of De la Chopra (1994a, 1994b) who proved that the discrepancies between the computed and actual stiffness values of the structural elements imply that a building with nominally symmetric plan is actually asymmetric to some unknown degree and will undergo torsional vibrations, when subjected to purely translational ground motion.…”