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Elastic and design response spectrum - EN 1998-1:2004
Explanation

### Elastic response spectrum definition

Within the scope of EN 1998 the earthquake motion at a given point on the surface is represented by an elastic ground acceleration response spectrum, henceforth called an “elastic response spectrum”.

The following image illustrates the shape of the elastic response spectrum.

The horizontal seismic action is described by two orthogonal components assumed as being independent and represented by the same response spectrum.

For the horizontal components of the seismic action, the elastic response spectrum Se(T) is defined by the following expressions:

`0 ≤ T ≤ TB: Se(T) = ag * S * [1 + (T / TB) * (η * 2.5 - 1)]` (3.2)
`TB ≤ T ≤ TC: Se(T) = ag * S * η * 2.5` (3.3)
`TC ≤ T ≤ TD: Se(T) = ag * S * η * 2.5 * (TC / T)` (3.4)
`TD ≤ T ≤ 4s: Se(T) = ag * S * η * 2.5 * (TC * TD / T2)` (3.5)

where

Se(T) is the elastic response spectrum;
T is the vibration period of a linear single-degree-of-freedom system;
ag is the design ground acceleration on type A ground (ag = γ * agR);
TB is the lower limit of the period of the constant spectral acceleration branch;
TC is the upper limit of the period of the constant spectral acceleration branch;
TD is the value defining the beginning of the constant displacement response range of the spectrum;
S is the soil factor;
The values in fonction of the type of soil can be found on the Parameters for Type 1 and 2 Elastic response spectra
η is the damping correction factor with a reference value of η = 1 for 5% viscous damping, provided below.

The value of the damping correction factor η may be determined by the expression:

η = √(10 / (5 + ξ)) ≥ 0.55      (3.6)

where ξ is the viscous damping ratio of the structure, expressed as a percentage.

Expressions above should normally be applied for vibration periods not exceeding 4,0 s. For structures with vibration periods longer than 4,0 s, a more complete definition of the elastic displacement spectrum is possible.

Methodolody: Expand explanation
Calculation process
Input data
Design ground acceleration on type A ground (ag):
(m/s²)
TB (see note 2 below):
s
TC (see note 3 below):
s
TD (see note 4 below):
s
ξ (viscous damping ratio of the structure):
%
Soil factor (S) (see note 5 below):
Behaviour factor (q) (see note 6 below):
Lower bound factor for the horizontal design spectrum (β):
Notes
1. The values of accelerations in Spain can be found on this interactive map.
2. TB is the lower limit of the period of the constant spectral acceleration branch. The values in fonction of the type of soil can be found on the Parameters for Type 1 and 2 Elastic response spectra.
3. TC is the upper limit of the period of the constant spectral acceleration branch. The values in fonction of the type of soil can be found on the Parameters for Type 1 and 2 Elastic response spectra.
4. TD is the value defining the beginning of the constant displacement response range of the spectrum. The values in fonction of the type of soil can be found on the Parameters for Type 1 and 2 Elastic response spectra.
5. S is the soil factor. The values in fonction of the type of soil can be found on the Parameters for Type 1 and 2 Elastic response spectra.
6. q is the behaviour factor. Its value has to be extracted from the different sections of the EN 1998-1
7. β is the lower bound factor for the horizontal design spectrum. The value to be ascribed to β for use in a country can be found in its National Annex. The recommended value for β is 0,2.
Calculate
Graph - Elastic response spectrum
Explanation
Tabulated results

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