The mechanical effects of short-circuit currents in - Montefiore

8. ANNEX
8.1. THE EQUIVALENT STATIC LOAD AND
EQUIVALENT STATIC LOAD FACTOR
8.1.1. Definition of ESL (Equivalent Static Load)
and ESL factor
The equivalent static load is a static load which can
be taken into account for the design of structures
which are stressed by dynamic loading. We only
consider here concentrated loads (not distributed).
The ESL is a static load with the same application
point as the actual load, which would induce in the
structure the same maximum force (generally in
connection with the bending moment) as the dynamic
load.
ESL is not equal to the peak value of the dynamic
load because inertial forces (distributed along the
structures) exist during dynamic loadings (Figure
8.1).
ESL can be lower, equal or higher than peak dynamic
load, depending on the vibration frequencies (so
called eigenfrequencies) of the structures to which the
load is applied
The dynamic load can be characterised by its PSD
(power spectral density) which point out the range of
frequencies in which the loading will have some true
dynamic effects.
The ESL factor is the ratio between ESL and the
relative peak instantaneous load (compared to its
initial value, if any). The ESL factor can change from
location to another in a structure. For example on
(Figure 8.1) ESL = 0 for two specific location in the
112
span and changing from 0 to approximately 1 for
other positions. It could have been larger than 1 at
mid-span if the frequency of load application would
have been similar to the first or the third
eigenfrequency of the beam. And it could also be
smaller than one in case of non resonant effect, in
some particular location in the structure. In fact
inertial forces can have compensation effect or
elimination effect, depending on the modal shape of
the structure, which can be very different from the
static deformation shape. But these effects are
difficult to be predicted and thus can be neglected in a
simplified method. A simplified method will then be
restricted to particular cases with special loading. In
other cases a dynamic simulation is then required.
ESL factor can be estimated by analytical treatment
of the load. Its value is determined from the linear
theory of the beam based on modal analysis which
includes the modal shape of the structure. Due to the
typical problems in substations and the necessary
simplifications the formula limited to the first
eigenmode only are used. Due to this simplification,
the structural factor is equal to one, only the load
factor has to be evaluated. The ESL factor is given by
the folowing equation :
(8.1)
ESL
factor
t ⎡ −ξ1ω1
( t−τ
)
= ω
− ⎤
1 ⎢⎣ ∫ f ( τ).
e . sin( ω1(
t τ)).
dτ
0
⎥⎦
:where ξ is the modal damping, f(t) is the time
function of the load, ω (rad/s) is the first
eigenfrequency of the structure, t is time duration of
the load application.
If the damping is assumed at 3% and if ESL factor is
depending on the frequency of the structure (f
=ω/2π ), the ESL factor will only depend of the load
shape. For this reason, in chapter 3, we have splitted
Figure 8.1 taken from "Earthquake response of structures", R.W. Clough, Berkeley, chapter 12. Influence of the dynamic loading compared to
static loading. P(t) maximum is equal to P static.