Equivalent Lateral Forces (ELF)

Introduction

Seismic ELF analysis is the most well known method for the seismic analysis of structures. Although it is quite conservative, its simplicity makes it a very popular method 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.

Defining an ELF seismic load case

Pre-requisites - before creating an ELF seismic load case

enable dynamics and seismic analysis in the project settings
define masses and a combination of mass groups (same as for any dynamic load case)
define storeys
enable the reduced model

Then create an ELF seismic load case

create a new load case
select Action type = Variable
select Load type = Dynamic
select Specification = Seismicity

Then, in order to set the seismic load case to ELF, go to the settings group Equivalent lateral forces and select the desired ELF method (disabled by default)

A this point, most of the settings of a standard seismic load case will disappear from the dialogue, as many of them are relevant only for a dynamic response spectrum analysis. For an ELF load case, the following remain:

Direction X (resp. Y, Z)

Tickbox that enables the seismic action in the X (resp. Y, Z) direction

Response spectrum X (resp. Y, Z)

Selection of the seismic response spectrum for direction X (resp. Y, Z)

Factor X (resp. Y, Z)

Multiplying factor for the seismic action in direction X (resp. Y, Z)

Acceleration factor

Multiplying factor for the entire seismic action

Overturning reference level

Reference level for the calculation of overturning moments. Also used as reference level when using a linear or polynomial distribution of accelerations

Accidental eccentricity

Accidental eccentricity settings act the same as for dynamic seismic analysis

Mass combi

Mass combination selected for the calculation of the seismic action

Specific settings for an ELF seismic load case

Equivalent lateral forces

ELF method

Defines how the distribution of accelerations will be calculated in the building. The various methods are detailed in the next section. The available choices are:

Disabled

ELF calculation of the seismic load case is disabled, i.e. the load case will use dynamic, multi-modal response spectrum analysis

Linear distribution of accelerations

The acceleration is increasing linearly with the global Z coordinate, starting with zero at the overturning reference level (see above)

Polynomial distribution of accelerations (ASCE 7-10 12.8.3)

The acceleration is increasing according to a polynomial function, according to ASCE 7-10 standard, starting with zero at the overturning reference level (see above)

Distribution of accelerations from eigenshape

The acceleration is distributed proportionally to the mode shape of an eigenmode selected by the user

Seismic force from

Defines how the total seismic force (base shear) is calculated. The available options depend on the selected ELF method.

Max acceleration of spectrum

The maximum acceleration of the selected response spectrum is used

Input fundamental period

The acceleration corresponding to a user input value of fundamental period is used

Selected eigenmode

The acceleration corresponding to the period of the user selected eigenmode is used; please note, that this implies a modal analysis of the structure

Approximate fundamental period (ASCE 7-10 12.8-7)

The acceleration corresponding to approximate fundamental period according ASCE 7-10 eq. 12.8-7. This is available only when ELF method = Polynomial distribution of accelerations (ASCE 7-10 12.8.3)

Fundamental period

User input fundamental period to be used for the calculation of the seismic force when Seismic force from = Input fundamental period

Mode shape

Eigenmode to be used for the distribution of the accelerations and/or the calculation of the seismic force

ASCE additional parameters

Structure type

This item is for values of approximate period parameters, see ASCE 7-10 table 12.8-2

Steel moment frame

Concrete moment frame

Steel eccentrically braced frames

Steel buckling-restrained braced frames

All other structural systems

Structural height from model

When this option is ticked, the structural height is obtained from the model as the maximal z-coordinate of top storey in project minus theOverturning reference level (Reference level for the calculation of overturning moments). When it is not ticked, the structural height must be defined manually.

h_n (structural height)

Structural height of building, used for the calculation of the approximate fundamental period according to ASCE 7-10 eq. 12.8-7.

C_t

Coefficient for ASCE 7-10 eq. 12.8-7

Exponent for ASCE 7-10 eq. 12.8-7

T_a [s]

Approximate fundamental period, see ASCE 7-10 section 12.8.2.1 and equation 12.8-7

Limit fundamental period

When this option is ticked, the fundamental period used for the calculation of the seismic force is limited by the upper boundary value defined in ASCE 7-10 section 12.8.2 .

C_u

Coefficient for upper limit on calculated period, see ASCE 7-10 table 12.8-1

Maximum fundamental period [s]

Maximum fundamental period according to ASCE 7-10 section 12.8.2, defined as T_a multiplied by C_u.

Calculation of the Equivalent Lateral Forces

Equivalent Lateral Forces are applied as one concentrated force at the mass center of each storey of the building.

First, the total seismic force (base shear) is calculated. It is then distributed to the storeys according to the selected method (vertical distribution of accelerations). The same procedure is applied for each direction (X,Y,Z).

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

Approximate fundamental period (ASCE 7-10 12.8-7) : the value of acceleration corresponding to the calculated approximate fundamental period

In the 2^nd and 3^rd cases above, the used fundamental period may be reduced by the calculated upper limit, if Limit fundamental period is ticked

Additionally, when a seismic response spectrum according to ASCE 7-10 is used (spectrum generator for IBC standard), the reference acceleration a_ref cannot be less than the specified minimum acceleration according to section 12.8.1.1:

where

additionally, if S₁≥0.6 the following applies:

The parameters S_DS, S₁, I_e and R are defined in the settings of the IBC response spectrum generator.

g is the acceleration of gravity, defined in the project settings.

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 selected ELF method setting

Linear distribution of accelerations

where

z_j is the level of the mass center of storey j

z_overturn is the Overturning reference level, 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 section 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 allow to "smear" the concentrated storey forces in such a way that the resultant of each storey force is applied at the mass center 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 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 to display the storey forces applied to the structure.