Direct Displacement Direct Displacement - - Based Design Based Design Using Inelastic Design Spectrum Using Inelastic Design Spectrum Rakesh K. Goel California Polytechnic State University, San Luis Obispo Anil K. Chopra University of California, Berkeley March 27, 2002 1 R. K. Goel
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Direct Displacement Based Design Using Inelastic Design Spectrum
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Direct DisplacementDirect Displacement--Based DesignBased Design Using Inelastic Design SpectrumUsing Inelastic Design Spectrum
Rakesh K. Goel California Polytechnic State University, San Luis Obispo
Anil K. Chopra University of California, Berkeley
March 27, 2002 1 R. K. Goel
ObjectivesObjectives òDemonstrate application of inelastic
design spectra to direct displacement-based design (DDBD)
òDemonstrate potential limitations of current DDBD that use elastic design spectra and equivalent linear systems
March 27, 2002 2 R. K. Goel
Equivalent Linear System: PeriodEquivalent Linear System: Period
March 27, 2002 3 R. K. Goel
ò For bilinear systems
ò For elasto-plastic systems
µTT neq =
ααµ µ
−+ =
1TT neq
muyu
yf
Deformation
Forc
e
1 1
1
kα
ksec
k ( )ααµ −+1f y
Equivalent Linear System: DampingEquivalent Linear System: Damping
March 27, 2002 4 R. K. Goel
ò For bilinear systems
ò For elasto-plastic systems
( ) ( ) ( )ααµµ
αµ π
ζ −+ −− =
1
112
eq
( ) µ
µ π
ζ 12 − = eq
DE
SE
Deformation
Forc
e
f y
( )ααµ −+1f y
umuy
Equivalent Linear SystemEquivalent Linear System
March 27, 2002 5 R. K. Goel
Substitute DampingSubstitute Damping
DE
SE
Deformation
Forc
e
f y
umuy
k
k
µ0.5
Gulkan & Sozen, Shibata & Sozen
(Takeda model for R/C structures) R. K. Goel March 27, 2002 6
March 27, 2002 7 R. K. Goel
Elastic Design SpectrumElastic Design Spectrum
Elastic Design SpectrumElastic Design Spectrum
Pseudo-Acceleration Deformation R. K. Goel March 27, 2002 8
DDBD Using Elastic Spectra:DDBD Using Elastic Spectra: StepStep--byby--Step ProcedureStep Procedure
1. Estimate the yield deformation for the system
2. Establish acceptable plastic rotation, θp
3. Determine design displacement and ductility factor: um = uy + h θp and µ = um / uy
March 27, 2002 9 R. K. Goel
DDBD Using Elastic Spectra:DDBD Using Elastic Spectra: StepStep--byby--Step ProcedureStep Procedure
4. Estimate the total equivalent viscous damping:
( ) ( ) ( )ααµµ
αµ π
ζ −+ −− =
1
112
eq
ζζζ += eqeq
ˆ R. K. Goel March 27, 2002 10
DDBD Using Elastic Spectra:DDBD Using Elastic Spectra: StepStep--byby--Step ProcedureStep Procedure
5. Enter deformation design spectrum and read Teq. Í Determine the
secant stiffness
m T
k
eq
2
2
sec
2π =
R. K. Goel March 27, 2002 11
DDBD Using Elastic Spectra:DDBD Using Elastic Spectra: StepStep--byby--Step ProcedureStep Procedure
6. Determined the required yield strength:
muyu
yf
Deformation
Forc
e
1 1
1
kα
ksec
k ( )ααµ −+1f y
ααµ −+ =
1
sec m
y
ukf
March 27, 2002 12 R. K. Goel
DDBD Using Elastic Spectra:DDBD Using Elastic Spectra: StepStep--byby--Step ProcedureStep Procedure
7. Estimate member size and detail (reinforcement in R/C structures, connections in steel structures) to provide fy. Í Calculate initial elastic stiffness k. Í Calculate yield deformation: uy = fy / k
8. Repeat steps 3 to 7 until a satisfactory solution is obtained.
March 27, 2002 13 R. K. Goel
March 27, 2002 14 R. K. Goel
ExampleExample
ò R/C viaduct ò Superstructure weight = 190 kN/m ò Bent spacing = 39.6 m
9 m 1.5 m
(a)
H k = 3EI H3
W = 7517 kN
(b)
March 27, 2002 15 R. K. Goel
Design Summary:Design Summary: DDBD Using Elastic SpectraDDBD Using Elastic Spectra
ò Starting yield displacement = 4.5 cm. ò Convergence achieved after three iterations ò The final design has:
DDBD Using Inelastic Spectra:DDBD Using Inelastic Spectra: StepStep--byby--Step ProcedureStep Procedure
1. Estimate the yield deformation for the system
2. Establish acceptable plastic rotation, θp
3. Determine design displacement and ductility factor: um = uy + h θp and µ = um / uy
March 27, 2002 20 R. K. Goel
DDBD Using Elastic Spectra:DDBD Using Elastic Spectra: StepStep--byby--Step ProcedureStep Procedure
4. Enter deformation design spectrum and read Tn. Í Determine the
elastic stiffness
m T
k
n
2
22π =
R. K. Goel March 27, 2002 21
March 27, 2002 22 R. K. Goel
DDBD Using Inelastic Spectra:DDBD Using Inelastic Spectra: StepStep--byby--Step ProcedureStep Procedure
5. Determine the required yield strength Í fy = kuy
6. Estimate member size and detail (reinforcement in R/C structures, connections in steel structures) to provide fy. Í Calculate initial elastic stiffness k. Í Calculate yield deformation: uy = fy / k
7. Repeat steps 3 to 6 until a satisfactory solution is obtained.
March 27, 2002 23 R. K. Goel
Design Summary:Design Summary: DDBD Using Inelastic SpectraDDBD Using Inelastic Spectra
ò Starting yield displacement = 4.5 cm. ò Convergence achieved after five iterations ò The final design has: