En Curs05 Dsis

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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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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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



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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

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