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8/10/2019 En Curs05 Dsis
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Structural Dynamics and EarthquakeEngineering
Course 5
Single degree of freedom systems
Seismic response of linear elastic systems
Elastic response spectra
Inelastic response
Course notes are available for download athttp://www.ct.upt.ro/users/AurelStratan/
Seismic action
Ground acceleration: accelerogram
Properties of a SDOF system (m, c, k) +
Relative displacement, velocity and acceleration of aSDOF system
( )gu t
gmu cu ku mu+ + =
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Seismic action
North-south component of the El Centro, Californiarecord during Imperial Valley earthquake from 18.05.1940
Determination of seismic response
Equation of motion:
/m:
Numerical methods
central difference method
Newmark method ...
Response depends on:
natural circular frequency n (or natural period Tn)
critical damping ratio
ground motion
22n n g
u u u u + + =
gmu cu ku mu+ + =
( ), ,nu u t T
gu
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Seismic response
Elastic response spectra
Response spectrum: representation of peak values of
seismic response (displacement, velocity, acceleration)of a SDOF system versus natural period of vibration, for agiven critical damping ratio
( ) ( )0 , max , ,n nt
u T u t T =
( ) ( )0 , max , ,n ntu T u t T =
( ) ( )0 , max , ,t t
n nt
u T u t T =
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Elastic displacement response spectrum: Du0
Pseudo-velocity and pseudo-acceleration
Spectral pseudo-velocity:
units of velocity
different from peak velocity
Strain energy
2n
n
V D DT
= =
( )22 2 2
00
2 2 2 2
n
S
k Vku kD mV E
= = = =
Spectral pseudo-acceleration:
units of acceleration
different from peak acceleration
2
0 0 0S nf ku m u mA= = =
2
2 2
0
2n n
n
u D DT
= = =
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2n
n
V D DT
= =
2
2 2n
n
A D DT
= =
D
Combined D-V-A spectrum
Displacement, pseudo-velocity and pseudo-acceleration
spectra:
same information
different physical meaning
A line inclined at +45 for lgA - lg2= const. spectralpseudo-acceleration: an axis inclined to -45
Similarly, spectral displacement: an axis inclined to +45
2
2
n
n
n n
TAV D or A V D
T
= = = =
( )2nT A V = lg lg lg 2 lgnT A V+ =
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Combined D-V-A spectrum
Characteristics of elastic response spectra
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Characteristics of elastic response spectra
Characteristics of elastic response spectra
ForTnTf spectral displacement Dis close to
spectral pseudo-acceleration A is small
BetweenTaand Tc A > BetweenTband Tc A can be considered constant
BetweenTdand Tf D>
BetweenTdand Te Dcan be considered constant
BetweenTcand Td V>
BetweenTcand Td Vcan be considered constant
0gu
0gu
0gu
0gu
0gu
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Characteristics of elastic response spectra
Tn>Td response region sensible to displacements Tn
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Elastic design spectra
idealized "smooth" spectra based on statistical interpretation (median; median plus
standard deviation)of severalrecordscharacteristicfor a given site
Elastic design spectra
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Elastic design spectra
TB T
C T
DT
PSA
TB T
C T
DT
PSV
TB T
C T
DT
SD
Inelastic response of SDOF systems
Most structures designed for seismic forces lower than
the ones assuring an elastic response during the designearthquake
design of structures in the elastic range for rare seismic eventsconsidered uneconomical
in the past, structures designed for a fraction of the forcesnecessary for an elastic response, survived major earthquakes
f S f S f S
u u u um
a b c d
f S
u
a
b
c
d
f S
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Inelastic response of SDOF systems
Elasto-plastic system: stiffness k
yield force fy yield displacement uy
Elasto-plastic idealization:equal area under the actualand idealised curves up to
the maximum displacement um
Cyclic response of the elasto-
plastic system
Corresponding elastic system
Corresponding elastic system:
same stiffness
same mass
same damping
Inelastic response:
yield force reduction factor Ry
ductility factor
the same periodof vibration (atsmall def.)
0 0
y
y y
f uR
f u= =
m
y
u
u =
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Equation of motion
Equation of motion:
/m
Seismic response of an inelastic SDOF system dependson:
natural circular frequency of vibration n
critical damping ratio yield displacement uy force-displacement shape
( ),S gmu cu f u u mu+ + =
( )22 ,n n y S gu u u f u u u + + =
( ),Sf u u
( ) ( ), ,S S yf u u f u u f=
Effects of inelastic force-displacement relationship
4 SDOF systems
(El Centro):
Tn = 0.5 sec
= 5%
Ry = 1, 2, 4, 8
Elastic system:
vibr. about theinitial positionof equilibrium
up=0
Inelastic syst.:
vibr. about anew position ofequilibrium
up0
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Elastic inelastic
Design of a structure responding in the elastic range:f0fRd
Design of a structure responding in the inelastic range:umuRd Rd
ductility demand ductility capacity
um/u0ratio
El Centro
ground motion
= 5%
Ry = 1, 2, 4, 8
Tn>Tf um independent
of Ry umu0
Tn>Tc umdepends on
Ry umu0
Tn u0
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Ry-relationship: idealisation
Tnin the displacement- and velocity-sensitive region: "equal displacement" rule um/u0=1 Ry=
Tnin the acceleration-sensitive region:
"equal energy" rule um/u0>1
Tn