# Equivalent Lateral Forces

15.2

SCIA Engineer 15.2 brings new possibility for seismic calculation - Equivalent Lateral Forces (ELF). It is the most well known method for the seismic analysis of structures. Although it is quite conservative, its simplicity makes it very popular for seismic design.

The ELF method is a static analysis method. However, using it in SCIA Engineer requires the input of some data related to dynamic analysis: masses and at least one combination of mass groups must be defined, as the calculation of the seismic equivalent lateral forces is based on the distribution of masses in the structure. The calculation of storey forces is based on the definition of storeys as well as on the reduced system, which must therefore be defined in order to allow using the ELF analysis.

## Calculation of the Equivalent Lateral Forces

Setting of ELF is very simple - it is just an extension of a seismic load case. There is a new group related to this method.

The user is able to select from methods for calculation of equivalent lateral forces. These method are in compliance with standards - EN 1998 and ASCE 7-10. ELF are calculated at the background after a model analysis. Calculated equivalent lateral forces are applied as one concentrated force at the mass centre of each storey of the building and linear analysis is performed.

### Calculation of the seismic force

The total seismic force *F _{base}* is calculated as follows

where

*M _{tot}* is the total mass of the structure, obtained from the selected mass combination

*a _{ref}* is the reference acceleration, obtained from the selected seismic response spectrum

*c _{dir}* is the

*direction factor*defined in the seismic load case settings

*c _{acc}* is the

*acceleration factor*defined in the seismic load case settings

The value of *a _{ref}* is extracted from the response spectrum, according to the

*Seismic force from*setting

*Max acceleration of spectrum*: the maximum value of acceleration defined in the selected spectrum

*Input fundamental period*: the value of acceleration corresponding to the period defined in the *Fundamental period* setting

*Selected eigenmode*: the value of acceleration corresponding to the period of the eigenmode selected in the *Mode shape* setting

### Distribution of the seismic force to the storeys

The storey force for the *j-th* storey j is calculated as follows

where

*F _{j}* is the horizontal force acting on storey

*j*

*F _{base}* is the total horizontal force acting on the structure (see above)

*m _{j}* is the mass of storey

*j*

*α _{j}* is the distribution key of the accelerations, according to the

*ELF method*setting

#### Linear distribution of accelerations

where

*z _{j}* is the level of the mass center of storey

*j*

*z _{overturn}* is defined by the user in the properties of the seismic load case.

This method corresponds to the simplified approach defined in EN1998-1 § 4.3.3.2.3(3) and formula (4.11).

#### Polynomial distribution of accelerations

where

*z _{j}* is the level of the mass center of storey

*j*

*z _{overturn}* is defined by the user in the properties of the seismic load case

*T* is the reference fundamental period, depending on the selected *Seismic force from* setting:

*Max acceleration of spectrum*: *T* is unknown, the conservative value *k*=2 is used

*Input fundamental period*: *T* is the period defined in the *Fundamental period* setting

*Selected eigenmode*: T is the period of the eigenmode selected in the *Mode shape* setting

This method corresponds to the approach defined in ASCE 7-10 12.8.3

#### Distribution of accelerations from eigenshape

where

*U _{G,j}* is the modal displacement of the mass center of storey j in the direction of the seismic action, obtained from modal analysis of the reduced model.

This is the preferred approach in Eurocode 8, defined in EN1998-1 § 4.3.3.2.3(2) and formula (4.10).

### Application of the storey forces to the model

The calculated storey forces are applied to the structure using the reduced system. The transformation matrices of the IRS method make it possible to "smear" the concentrated storey forces in such a way that the resultant of each storey force is applied at the mass centre of the corresponding storey. The loads are, however, applied in a distributed way to the entire storey, hence avoiding any numerical singularity, as would be the case if point loads would be applied in a conventional way.

## Results

As an ELF load case is fundamentally a static load case, all standard result output can be used in SCIA Engineer, without any restriction. Also, because it is a static load case, none of the issues related to the loss of sign due to the modal superposition apply here.

Additionally, the Summary Storey Results service allows for display of the storey forces applied to the structure.