Direct Displacement Based Design Using Inelastic Design Spectrum
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Direct Displacement-Based Design Using Inelastic Design Spectrum Rakesh K. Goel California Polytechnic State University, San Luis Obispo
Anil K. Chopra University of California, Berkeley
R. K. Goel
March 27, 2002
1
Objectives òDemonstrate application of inelastic design spectra to direct displacementbased design (DDBD) òDemonstrate potential limitations of current DDBD that use elastic design spectra and equivalent linear systems
R. K. Goel
March 27, 2002
2
Equivalent Linear System: Period ò For bilinear systems µ T eq = T n 1 + αµ − α
fy
1
Force
ò For elasto-plastic systems
f y (1 + αµ − α )
αk
ksec k
1
1
T eq = T n µ
uy
um
Deformation
R. K. Goel
March 27, 2002
3
ò For bilinear systems 2 (µ − 1)(1 − α ) ζ eq = π µ (1 + αµ − α )
R. K. Goel
ES
ED
2 (µ − 1)
π
f y (1 + αµ − α )
fy
ò For elasto-plastic systems ζ eq =
Force
Equivalent Linear System: Damping
uy
µ
um
Deformation
March 27, 2002
4
Equivalent Linear System
R. K. Goel
March 27, 2002
5
Force
Substitute Damping
fy k ES ED
k µ0.5
uy
um
Deformation
Gulkan & Sozen, Shibata & Sozen (Takeda model for R/C structures) R. K. Goel
March 27, 2002
6
Elastic Design Spectrum
R. K. Goel
March 27, 2002
7
Elastic Design Spectrum
Pseudo-Acceleration R. K. Goel
Deformation March 27, 2002
8
DDBD Using Elastic Spectra: Step-by-Step 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
R. K. Goel
March 27, 2002
9
DDBD Using Elastic Spectra: Step-by-Step Procedure 4. Estimate the total equivalent viscous damping: 2 (µ − 1)(1 − α ) ζ eq = π µ (1 + αµ − α )
ζˆ eq = ζ +ζ eq
R. K. Goel
March 27, 2002
10
DDBD Using Elastic Spectra: Step-by-Step Procedure 5. Enter deformation design spectrum and read Teq. Í Determine the secant stiffness
k sec =
R. K. Goel
2π T
2
2 eq
m
March 27, 2002
11
DDBD Using Elastic Spectra: Step-by-Step Procedure
ksecum fy= 1 + αµ − α
f y (1 + αµ − α ) fy
1
Force
6. Determined the required yield strength:
αk
ksec k
1
1 uy
um
Deformation
R. K. Goel
March 27, 2002
12
DDBD Using Elastic Spectra: Step-by-Step 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.
R. K. Goel
March 27, 2002
13
Example ò R/C viaduct ò Superstructure weight = 190 kN/m ò Bent spacing = 39.6 m W = 7517 kN
9m
1.5 m
k=
3EI H3
(b)
(a) R. K. Goel
H
March 27, 2002
14
Design Summary: DDBD Using Elastic Spectra ò Starting yield displacement = 4.5 cm. ò Convergence achieved after three iterations ò The final design has: ÍLongitudinal column reinforcement = 1.3% ÍInitial stiffness = 95.17 kN/cm ÍLateral yield strength = 839.7 kN ÍYield displacement = 8.82 cm, Design displacement = 26.8 cm ÍElastic period = 1.78 sec, Secant period = 3.14 sec R. K. Goel
March 27, 2002
15
Inelastic Design Spectrum
R. K. Goel
March 27, 2002
16
Conversion of Elastic to Inelastic Design Spectrum ò Use available Ry-µ-Tn relationships ÍNewmark and hall ÍKrawinkler et al ÍFajfar et al. ÍMiranda & Bertero
ò Relationships based on nonlinear analysis of SDF systems
R. K. Goel
March 27, 2002
17
Constant Ductility Inelastic Design Spectrum
Newmark & Hall R. K. Goel
Krawinkler et al. March 27, 2002
18
Inelastic Design Spectrum
Pseudo-Acceleration R. K. Goel
Deformation March 27, 2002
19
DDBD Using Inelastic Spectra: Step-by-Step 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
R. K. Goel
March 27, 2002
20
DDBD Using Elastic Spectra: Step-by-Step Procedure 4. Enter deformation design spectrum and read Tn. Í Determine the elastic stiffness
k=
R. K. Goel
2π T
2
2 n
m
March 27, 2002
21
DDBD Using Inelastic Spectra: Step-by-Step 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. R. K. Goel
March 27, 2002
22
Design Summary: DDBD Using Inelastic Spectra ò Starting yield displacement = 4.5 cm. ò Convergence achieved after five iterations ò The final design has: ÍLongitudinal column reinforcement = 5.5% ÍInitial stiffness = 238.6 kN/cm ÍLateral yield strength = 1907 kN ÍYield displacement = 7.99 cm, Design displacement = 26.0 cm ÍElastic period = 1.16 sec R. K. Goel
March 27, 2002
23
Evaluation of Design: Inelastic Analysis 1. Calculate initial elastic period from m and k 2. Determine A from elastic design spectrum Í Elastic Design Force: fo = mA
3. From known fy, calculate: Ry = fo / fy 4. Determine ductility demand µ from Ry-µ-Tn relationships 5. Calculate displacement and plastic rotation Í um =(µ/ Ry)(Tn / 2π)2A Í θp = (um-uy)/h R. K. Goel
March 27, 2002
24
Evaluation of Example Design: DDBD Using Elastic Spectra ò Demands from inelastic analysis of the design structure
ò Design using elastic design spectrum
Íum = 39.7 cm. ͵ = 4.52 Íθp = 0.0343 rad.
R. K. Goel
March 27, 2002
Íum = 26.8 cm (32.6% underestimation) ͵ = 3.04 (32.6% underestimation) Íθp = 0.02 rad (Demand exceeds acceptable value by > 72%) 25
Evaluation of Example Design: DDBD Using Inelastic Spectra ò Demands from inelastic analysis of the design structure
ò Design using elastic design spectrum
Íum = 25.9 cm. ͵ = 3.25 Íθp = 0.0199 rad.
R. K. Goel
March 27, 2002
Íum = 26.0 cm ͵ = 3.06 Íθp = 0.02 rad ÍPredictions are nearly the same as the inelastic demands
26
Conclusions ò A direct displacement-based design procedure is presented ÍUses well-known inelastic design spectrum ÍProvides displacement estimates consistent with those from inelastic analysis ÍProduces design that satisfies the design criteria of acceptable plastic rotation ÍThe procedure is as simple as the current DDBD procedure using elastic design spectra
R. K. Goel
March 27, 2002
27
Conclusions ò DDBD procedure based on elastic design spectra ÍUses equivalent linear systems ò Secant stiffness and equivalent damping
ÍProvides displacement estimate which can be significantly smaller than that from inelastic analysis ÍPlastic rotation demand may exceed the acceptable value ò Leaves an erroneous impression that the allowable plastic rotation constraint has been satisfied
R. K. Goel
March 27, 2002
28
Anil K. Chopra University of California, Berkeley
R. K. Goel
March 27, 2002
1
Objectives òDemonstrate application of inelastic design spectra to direct displacementbased design (DDBD) òDemonstrate potential limitations of current DDBD that use elastic design spectra and equivalent linear systems
R. K. Goel
March 27, 2002
2
Equivalent Linear System: Period ò For bilinear systems µ T eq = T n 1 + αµ − α
fy
1
Force
ò For elasto-plastic systems
f y (1 + αµ − α )
αk
ksec k
1
1
T eq = T n µ
uy
um
Deformation
R. K. Goel
March 27, 2002
3
ò For bilinear systems 2 (µ − 1)(1 − α ) ζ eq = π µ (1 + αµ − α )
R. K. Goel
ES
ED
2 (µ − 1)
π
f y (1 + αµ − α )
fy
ò For elasto-plastic systems ζ eq =
Force
Equivalent Linear System: Damping
uy
µ
um
Deformation
March 27, 2002
4
Equivalent Linear System
R. K. Goel
March 27, 2002
5
Force
Substitute Damping
fy k ES ED
k µ0.5
uy
um
Deformation
Gulkan & Sozen, Shibata & Sozen (Takeda model for R/C structures) R. K. Goel
March 27, 2002
6
Elastic Design Spectrum
R. K. Goel
March 27, 2002
7
Elastic Design Spectrum
Pseudo-Acceleration R. K. Goel
Deformation March 27, 2002
8
DDBD Using Elastic Spectra: Step-by-Step 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
R. K. Goel
March 27, 2002
9
DDBD Using Elastic Spectra: Step-by-Step Procedure 4. Estimate the total equivalent viscous damping: 2 (µ − 1)(1 − α ) ζ eq = π µ (1 + αµ − α )
ζˆ eq = ζ +ζ eq
R. K. Goel
March 27, 2002
10
DDBD Using Elastic Spectra: Step-by-Step Procedure 5. Enter deformation design spectrum and read Teq. Í Determine the secant stiffness
k sec =
R. K. Goel
2π T
2
2 eq
m
March 27, 2002
11
DDBD Using Elastic Spectra: Step-by-Step Procedure
ksecum fy= 1 + αµ − α
f y (1 + αµ − α ) fy
1
Force
6. Determined the required yield strength:
αk
ksec k
1
1 uy
um
Deformation
R. K. Goel
March 27, 2002
12
DDBD Using Elastic Spectra: Step-by-Step 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.
R. K. Goel
March 27, 2002
13
Example ò R/C viaduct ò Superstructure weight = 190 kN/m ò Bent spacing = 39.6 m W = 7517 kN
9m
1.5 m
k=
3EI H3
(b)
(a) R. K. Goel
H
March 27, 2002
14
Design Summary: DDBD Using Elastic Spectra ò Starting yield displacement = 4.5 cm. ò Convergence achieved after three iterations ò The final design has: ÍLongitudinal column reinforcement = 1.3% ÍInitial stiffness = 95.17 kN/cm ÍLateral yield strength = 839.7 kN ÍYield displacement = 8.82 cm, Design displacement = 26.8 cm ÍElastic period = 1.78 sec, Secant period = 3.14 sec R. K. Goel
March 27, 2002
15
Inelastic Design Spectrum
R. K. Goel
March 27, 2002
16
Conversion of Elastic to Inelastic Design Spectrum ò Use available Ry-µ-Tn relationships ÍNewmark and hall ÍKrawinkler et al ÍFajfar et al. ÍMiranda & Bertero
ò Relationships based on nonlinear analysis of SDF systems
R. K. Goel
March 27, 2002
17
Constant Ductility Inelastic Design Spectrum
Newmark & Hall R. K. Goel
Krawinkler et al. March 27, 2002
18
Inelastic Design Spectrum
Pseudo-Acceleration R. K. Goel
Deformation March 27, 2002
19
DDBD Using Inelastic Spectra: Step-by-Step 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
R. K. Goel
March 27, 2002
20
DDBD Using Elastic Spectra: Step-by-Step Procedure 4. Enter deformation design spectrum and read Tn. Í Determine the elastic stiffness
k=
R. K. Goel
2π T
2
2 n
m
March 27, 2002
21
DDBD Using Inelastic Spectra: Step-by-Step 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. R. K. Goel
March 27, 2002
22
Design Summary: DDBD Using Inelastic Spectra ò Starting yield displacement = 4.5 cm. ò Convergence achieved after five iterations ò The final design has: ÍLongitudinal column reinforcement = 5.5% ÍInitial stiffness = 238.6 kN/cm ÍLateral yield strength = 1907 kN ÍYield displacement = 7.99 cm, Design displacement = 26.0 cm ÍElastic period = 1.16 sec R. K. Goel
March 27, 2002
23
Evaluation of Design: Inelastic Analysis 1. Calculate initial elastic period from m and k 2. Determine A from elastic design spectrum Í Elastic Design Force: fo = mA
3. From known fy, calculate: Ry = fo / fy 4. Determine ductility demand µ from Ry-µ-Tn relationships 5. Calculate displacement and plastic rotation Í um =(µ/ Ry)(Tn / 2π)2A Í θp = (um-uy)/h R. K. Goel
March 27, 2002
24
Evaluation of Example Design: DDBD Using Elastic Spectra ò Demands from inelastic analysis of the design structure
ò Design using elastic design spectrum
Íum = 39.7 cm. ͵ = 4.52 Íθp = 0.0343 rad.
R. K. Goel
March 27, 2002
Íum = 26.8 cm (32.6% underestimation) ͵ = 3.04 (32.6% underestimation) Íθp = 0.02 rad (Demand exceeds acceptable value by > 72%) 25
Evaluation of Example Design: DDBD Using Inelastic Spectra ò Demands from inelastic analysis of the design structure
ò Design using elastic design spectrum
Íum = 25.9 cm. ͵ = 3.25 Íθp = 0.0199 rad.
R. K. Goel
March 27, 2002
Íum = 26.0 cm ͵ = 3.06 Íθp = 0.02 rad ÍPredictions are nearly the same as the inelastic demands
26
Conclusions ò A direct displacement-based design procedure is presented ÍUses well-known inelastic design spectrum ÍProvides displacement estimates consistent with those from inelastic analysis ÍProduces design that satisfies the design criteria of acceptable plastic rotation ÍThe procedure is as simple as the current DDBD procedure using elastic design spectra
R. K. Goel
March 27, 2002
27
Conclusions ò DDBD procedure based on elastic design spectra ÍUses equivalent linear systems ò Secant stiffness and equivalent damping
ÍProvides displacement estimate which can be significantly smaller than that from inelastic analysis ÍPlastic rotation demand may exceed the acceptable value ò Leaves an erroneous impression that the allowable plastic rotation constraint has been satisfied
R. K. Goel
March 27, 2002
28
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