seismic response of sdof systems
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Seismic Analysis of SDOF SystemsTRANSCRIPT
2C09 Design for seismic and climate changes
Lecture 08: Seismic response of SDOF systems
Aurel Stratan, Politehnica University of Timisoara 13/03/2014
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards and Catastrophic Events
520121-1-2011-1-CZ-ERA MUNDUS-EMMC
Lecture outline 8.1 Time-history response of linear SDOF systems. 8.2 Elastic response spectra. 8.3 Time-history response of inelastic SDOF systems.
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Seismic action Ground acceleration: accelerogram
Properties of a SDOF system (m, c, k) +
Relative displacement, velocity and acceleration of a SDOF system
( )gu t
gmu cu ku mu
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Seismic action North-south component of the El Centro, California
record during Imperial Valley earthquake from 18.05.1940
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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
22 n n gu u u u
gmu cu ku mu
, ,nu u t T
gu
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Seismic response
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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 a given critical damping ratio 0 , max , ,n nt
u T u t T
0 , max , ,n ntu T u t T
0 , max , ,t tn nt
u T u t T
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Elastic displacement response spectrum: Du0
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Elastic displacement response spectrum: Du0
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Elastic displacement response spectrum: Du0
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Elastic displacement response spectrum: Du0
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Elastic displacement response spectrum: Du0
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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 20
0 2 2 2 2n
S
k Vku kD mVE
Spectral pseudo-acceleration: – units of acceleration – different from peak acceleration
20 0 0S nf ku m u mA
22 2
02
n nn
A u D DT
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|u(t)|max = Sde(T) |u(t)|max = Sde(T)
Fm
k
m
k
=kSde(T)=mSae(T)
ag(t)
Pseudo-velocity and pseudo-acceleration
2n
n
V D DT
22 2n
n
A D DT
D
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Combined D-V-A spectrum Displacement, pseudo-velocity and pseudo-acceleration
spectra: – same information – different physical meaning
A line inclined at +45º for lgA - lg2 = const. spectral pseudo-acceleration: an axis inclined to -45º
Similarly, spectral displacement: an axis inclined to +45º
22n
nn n
TA V D or A V DT
2nT A V lg lg lg 2 lgnT A V
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Combined D-V-A spectrum
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Characteristics of elastic response spectra
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Characteristics of elastic response spectra
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Characteristics of elastic response spectra For Tn<Ta
– pseudo-acceleration A is close to – spectral displacement D is small
For Tn>Tf – spectral displacement D is close to – spectral pseudo-acceleration A is small
Between Ta and Tc A > Between Tb and Tc A can be considered constant
Between Td and Tf D > Between Td and Te D can be considered constant
Between Tc and Td V > Between Tc and Td V can 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<Tc response region sensible to accelerations Tc<Tn<Td response region sensible to velocity
Larger damping:
– smaller values of displacements, pseudo-velocity and pseudo-acceleration
– more "smooth" spectra
Effect of damping: – insignificant for Tn 0 and Tn , – important for Tb<Tn<Td
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Elastic design spectra Spectra of past ground motions:
– jagged shape – variation of response for different earthquakes – areas where previous data is not available
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Elastic design spectra idealized "smooth" spectra based on statistical interpretation (median; median plus
standard deviation) of several records characteristic for a given site
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Elastic design spectra
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Elastic design spectra
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Elastic design spectra
TB TC TDT
PSA
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Elastic design spectra
TB TC TDT
PSA
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Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
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Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
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Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
TB TC TDT
SD
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Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
TB TC TDT
SD
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Inelastic response of SDOF systems Most structures designed for seismic forces lower than
the ones assuring an elastic response during the design earthquake – design of structures in the elastic range for rare seismic events
considered uneconomical – in the past, structures designed for a fraction of the forces
necessary for an elastic response, survived major earthquakes
f S f S f Su u u um
a b c d
f S
u
a
bc
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 actual and idealised curves up to the maximum displacement um
Cyclic response of the elasto-plastic system
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Corresponding elastic system Corresponding elastic system:
– same stiffness – same mass – same damping
Inelastic response:
– yield force reduction factor Ry
– ductility factor
the same period of vibration (at small def.)
0 0y
y y
f uRf u
m
y
uu
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Equation of motion Equation of motion:
/m
Seismic response of an inelastic SDOF system depends on: – 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
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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 the
initial position of equilibrium
– up=0 Inelastic syst.:
– vibr. about a new position of equilibrium
– up≠0 25
Elastic inelastic Design of a structure responding in the elastic range:
f0 ≤ fRd Design of a structure responding in the inelastic range:
um ≤ uRd ≤ Rd
ductility demand ductility capacity
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um/u0 ratio El Centro
ground motion – = 5% – Ry = 1, 2, 4, 8
Tn>Tf –um independent of Ry –um u0
Tn>Tc –um depends on Ry –um u0
Tn<Tc –um depends on Ry –um > u0
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Ry - relationship: idealisation Tn in the displacement- and velocity-sensitive region:
– "equal displacement" rule um/u0=1 Ry= Tn in the acceleration-sensitive region:
– "equal energy" rule um/u0>1 Tn<Ta:
– small deformations, elastic response Ry=1
2 1yR
'
1
2 1n a
y b n c
n c
T T
R T T TT T
u
f S
f 0
f y
u =umuy u
f S
f 0
f y
uuy um28
Ry - relationship: idealisation
'
1
2 1n a
y b n c
n c
T T
R T T TT T
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References / additional reading Anil Chopra, "Dynamics of Structures: Theory and
Applications to Earthquake Engineering", Prentice-Hall, Upper Saddle River, New Jersey, 2001.
Clough, R.W. and Penzien, J. (2003). "Dynamics of structures", Third edition, Computers & Structures, Inc., Berkeley, USA
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