Windows Application For Computing Spectral Shapes
The Spectral Shapes Windows Application
Single Degree Of Freedom System
Frequencies of Spectral ordinates
Earthquake response spectrum has its origin in earthquake engineering,
and determines the influence of the earthquake vibrations on an engineering
structure. Under the influence of earthquake vibrations the base of a structure,
partially embedded in the ground, tends to move with the ground. The rapid
ground motion produces stresses and deformations in the structure. These
deformations vary in different parts of the structure. A rigid structure tends
to move with its base, so that the dynamic forces on the base and elsewhere in
the structure are nearly equal. To prevent damage from differential movements in
the structure, it must be designed to withstand the forces of deformation. It
should be able to absorb energy and dissipate it through vibrations, damping or
inelastic deformations. Response spectra are the tools used to describe the
effect of the structure to the earthquake vibrations. These are defined in terms
of the maximum responses of a set of single degree of freedom (SDOF) systems
characterized by their natural frequencies and damping ratios, when each
oscillator is subjected, independently, to the vibration at its base. (The
number of independent coordinates required for specifying the configuration or
position of a system is referred to as the number of degrees of freedom. A
single degree of freedom system can, thus, be completely defined using
only one spatial coordinate.) With
suitable modifications to the response spectra to account for the effects of
local geology in modifying the earthquake vibration these are used for
earthquake resistant (or aseismic) design of structures . For understanding
earthquake response spectrum the response of the SDOF to the earthquake
ground motion must be understood.
The
figure given below shows a very simple structure. Here mass less elastic columns
CE and DF support a slab ABCD of mass m. The
elastic columns are characterized by a spring constant k.
When subjected the earthquake ground motion the structure deviates from
its original position, as shown in the figure, and dissipates energy due to
internal damping proportional to velocity of the mass. The constant c represents
the damping in the structure. In the treatment given below the displacement of
base of the structure in the horizontal direction is represented by ug(t)
and the horizontal displacement of the slab by u(t) .The slab is in equilibrium
under three different forces, namely:
(a)
The inertial force fI = m (d2u/dt2 +d2ug/dt2),
(b)
The spring force fS =k u, and
(c)
The damping force fD = c (d2u/dt2).
In the equilibrium state fI + fD+ fS=0.
This condition gives the equation of motion during the earthquake vibration.
m (d2u/dt2)
+c (du/dt) +k u =-m(d2ug/dt2)
or
d2u/dt2 +2wx
(du/dt). +w2u
= -d2ug/dt2… (1)
Here
w=
Ö(k/m) and x
= c/2Ö(km)
This
structure is a single degree of freedom (SDOF)
system of natural frequency w
and damping c. For any given earthquake displacement time history ug(t)
the equation of motion can be solved for u(t) at any specified frequency and
damping. For each time history, and for a specified frequency and damping value,
a maximum value umax of u(t) can be determined, where umax
represents the value of the maximum displacement of the slab with respect to its
original position. A plot of umax against frequency is called displacement response spectrum of the earthquake. The ordinates of
this plot are called spectral
displacement, and are represented by SD(w,x).
Thus for any frequency and damping value
SD
=max |u(t)| … (3)
The
maximum spring force is the given by:
fS
= k umax … (4)
The
spring force can also be visualized as a pseudo inertial force on the mass m, so
that
fS
= ma … (5)
Here
`a’ is pseudo absolute acceleration. From equations (4) and (5) one gets a = umax.
At high frequencies the maximum value of `a’ during the vibration may be
obtained from Equation (1). This turns out to be w2u.
The pseudo absolute acceleration (PAA) is represented by the symbol SA.
SA=
uamax =w2u
… (6)
SA
is called the spectral acceleration. The maximum strain energy in the
spring equal to (1/2) k SD2. This can be associated with a
velocity SV, such that the kinetic energy associated with it is equal
to the maximum strain energy, i.e.
(1/2)
m SV2 = (1/2)k SD2…(7)
Thus
SV = w
SD … (8)
SV
is also a pseudo velocity because it is not related to the actual velocity
of the structure. Thus the three different spectral ordinates are related with
each other as follows:
SA
= w
SV = w2SD….
(9)
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Damping is Specified by the user.
Response spectrum is computed at the following frequencies (Hrtz)
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0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.60
0.70
0.80
0.90
1.00
1.10 1.20
1.30 |
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
|
3.15
3.30
3.45
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00 5.25
5.50
5.75
6.00
6.25
6.50
|
6.75
7.00
7.25
7.50
7.75
8.00
8.50
9.00
9.50 10.00
10.50
11.00
11.50
12.00
12.50
13.00
13.50
|
14.00
14.50
15.00
16.00
17.00
18.00
20.00
22.00
25.00
28.00
31.00
34.00
37.00
40.00
43.50
45.50
50.00 |
The Spectral Shapes Windows Application
The application may be used for generating mean and mean + standard deviation design response spectral shapes. The values of mean and the mean squares are stored in the applications directory, and are used to update the spectral shapes when more ground motion time histories become available. The application has the provision of generating spectrum compatible accelerograms, which are basically useful for understanding the application. The opening window of the application has a self explanatory menu.
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Clicking on the 'Generate time history' menu item opens the following three dialog boxes in succession. The time histories, which are generated, use a trapezium window for strong motion defined by a rise time, a duration of constant signal amplitude followed by a signal decay time. The generated time histories are compatible with a specified spectral shape given in an ASCII file ( extension .spc). The time history file to be used by the application for generating or updating spectral shapes must have the extension .ths (it is again an ASCII file, and the extension .ths has been chosen only to indicate that it is a time history file).
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After the menu item 'Fresh spectral shapes is clicked on the windows menu bar the dialog box for initializing the parameters of the spectral shapes appears as shown on the left hand side. of the figure given below: The generated mean and mean + standard deviation shapes then appear as shown on the right hand side of this figure. The generated mean and mean + standard deviation shapes are listed in ASCII files SPShapeMean.SPC and SPShapeMeanPlusSigma.SPC in the applications' directory. The application makes use of a file named Shape_0.spf, which contains the spectral frequencies and the sums and the sums of the squares for each frequency. After the spectral shapes are generated the Shape_0.spf file is modified. In subsequent updates of the spectral shapes fresh .spf files are created.
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Updating The Spectral Shapes
On clicking on the menu item Update shapes update initialization dialog box appears as shown on the right hand side of the following figure. In the update Input file section the .spf file corresponding to the the last update is selected, and in the time history file name the file which contains the new set of the time histories is selected. A number is added before the extension .spf in the spectrum shape output file to produce a new update file (which will be used in the next update). As the various update files (with .spf extension) are stored (until deleted manually) the updates can be effected from any stage. On clicking OK in this dialog box the updated mean and mean + standard deviation spectral shapes appear as shown on the right hand side of this figure with the details as shown.
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Design
Response Spectra (DRS) specify the upper limit within which the response of the
structure should be restricted during an earthquake in order to protect it from
damage. DRS are, generally, specified in terms of a spectral shape and a
normalization factor. Peak ground
acceleration (PGA) has been widely used as the normalization factor.
In the early days of aseismic design use of standard response spectra was
proposed [1].
These were derived from eight accelerograms - two horizontal components each
from four earthquakes recorded at sites within 8 to 56 km distance from the
earthquake source. These earthquakes are listed below:
Table-
: Earthquakes used for deriving Standard Response Spectra.
|
S.No. |
EARTHQUAKE |
DATE |
MAGNI-TUDE |
|
1. |
Imperial valley, California |
December 12, 1934 |
6.5 |
|
2 |
El Centro, Imperial Valley, California |
May19,1940 |
7.1 |
|
3. |
Olympia, Puget Sound, Wash. |
April 13, 1949 |
7.1 |
|
4. |
Taft, Kern County, California |
July 21, 1952 |
7.7 |
Spectral
shapes computed from the accelerograms of these earthquakes were normalized to
have equal area under the zero damped spectral intensity curve for a
predetermined frequency range to produce a standard spectral shape. The spectral
intensity, SI, was defined as follows:
2.5
SI(b)
= ň
SV(b,t)dt
0.1
Where
SV represents the velocity spectrum and b
is the coefficient of damping. Spectral intensity is not, any more, considered a
good descriptor of earthquake damage, and hence is not used for normalizing
response spectra. Current practices are based on normalization of accelerograms
with respect to PGA before computing spectral shapes. Though, it is not
considered ideal, PGA has found wider acceptance because:
§
Acceleration, being a
measure of force, is easy to comprehend.
§
The procedures of
computing spectral shapes are better defined and understood.
§
The computations deliver,
directly, the ordinates of the response spectral shape (called the dynamic
amplification factors - DAF, because when multiplied by the design value of PGA
these provide the ordinates of the design response spectrum).
In
1969 Newmark and Hall proposed that spectral acceleration at different
frequencies are related to the PGA through amplification factors[2].
These amplification factors are he dynamic amplification factors (DAF). They
determined the DAFs using statistical analysis of the observed response spectra
at certain control points, which were chosen to define the response spectra.
These DAF values are given below in Table 9.
Here, the response is defined at four discrete points, called control
points. Between the controls points the spectra have been assumed to be smooth,
and are defined by straight-line segments. These spectral shapes have been in
use as standard or site-independent spectra shapes. Standard response spectral
shapes from 33 accelerograms from twelve earthquakes recorded under varying site
conditions were derived by Blume and his associates [3]
. In derivation of these
spectra the accelerograms were grouped according to several different criteria
(e.g. PGA values, site soil characteristics, epicentral distance, geographical
locations etc.) to determine the influence of various factors on the shape of
the response spectral shapes. On the basis of statistical analysis of various
groups, response spectral shapes corresponding to three different levels of
exceedance probability were estimated. These three levels were called the `large’,
‘small’ and `negligible’ exceedance probability levels, and correspond to
the mean, M (50% probability of exceedance), M+s
(15% probability of exceedance) and M+2s
(2.3% probability of exceedance) s
being the standard deviation . It was suggested that risk variability due to
factors like regional seismicity, geotectonics etc. be considered before
choosing the spectral shapes for a site. A
comparison of these shapes with the corresponding shapes of Housners spectra
showed that the Housners shapes were lower below the mean shapes for periods
shorter than 0.4 seconds, and higher for longer periods.
TABLE-1 : DAF VALUES FOR RESPONSE SPECTRUM FOR HORIZONTAL DESIGN
RESPONSE SPECTRA: RELATIVE VALUES OF AMPLIFICATION FACTORS AT CONTROL POINTS.
|
Percent
of critical damping |
Acceleration |
Displacement
D (.25Hz
) |
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|
A (33
Hz) |
B (9
Hz) |
C (2.5
Hz) |
||
|
0.5 |
1.0 |
4.96 |
5.95 |
3.20 |
|
2.0 |
1.0 |
3.54 |
4.25 |
2.50 |
|
5.0 |
1.0 |
2.61 |
3.13 |
2.05 |
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7.0 |
1.0 |
2.27 |
2.72 |
1.88 |
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10.0 |
1.0 |
2.28 |
2.28 |
1.70 |
[1] Housener,G.W.(1959). Behavior of structures during Earthquakes. Proc. Am. Soc. Engrs. Vol. 85.
[2] Newmark, N.M. and Hall, W.J.(1969). Seismic Design Criteria For Nuclear Power Facilities. 4 WCEE, Santiago, Chile, Proc. pp 37-50.
[3] Blume, J.A. and Associates (1973). Recommendations for Shapes of Earthquake Response Spectra. WASH-1254, Government Printing Press, Washington D.C.
[4] IAEA(1979). Earthquakes and Associated Topics in Relation to Nuclear Power Plant Siting. Safety Guide 50-SG-S1, International Atomic Energy Agency, Vienna.
[5] USAEC(1973). Design Response Spectra for Seismic Design of Nuclear Power Plants. Regulatory Guide 1.60
