Seismic Response of SDOF Systems

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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[email protected]

http://steel.fsv.cvut.cz/suscos