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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.
2
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
3
Seismic action North-south component of the El Centro, California
record during Imperial Valley earthquake from 18.05.1940
4
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
5
Seismic response
6
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
7
8
Elastic displacement response spectrum: Du0
8
Elastic displacement response spectrum: Du0
8
Elastic displacement response spectrum: Du0
8
Elastic displacement response spectrum: Du0
8
Elastic displacement response spectrum: Du0
8
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
9
|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
10
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
11
Combined D-V-A spectrum
12
Characteristics of elastic response spectra
13
Characteristics of elastic response spectra
14
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
15
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
16
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
20
Elastic design spectra
TB TC TDT
PSA
20
Elastic design spectra
TB TC TDT
PSA
20
Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
20
Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
20
Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
TB TC TDT
SD
20
Elastic design spectra
TB TC TDT
PSA
TB TC TDT
PSV
TB TC TDT
SD
20
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
21
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
22
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
23
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
24
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
26
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
27
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
29
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
30
http://steel.fsv.cvut.cz/suscos