Direct Displacement-Based Design Using Inelastic Design Spectrum

Transcription

1 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 March 27,

2 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 March 27,

3 Equivalent Linear System: Period ò For bilinear systems T eq = T n µ 1+ αµ α f y f y ( ) 1+ αµ α 1 α k ò For elasto-plastic systems T eq = T n µ Force 1 k u y 1 ksec Deformation u m March 27,

4 Equivalent Linear System: Damping ò For bilinear systems 2 ( µ 1) ( 1 α ) ζ eq = π µ ( 1 + αµ α ) ò For elasto-plastic systems ζ eq ( µ 1) 2 = π µ f y E D fforce ( 1+ αµ α ) y u y um Deformation E S March 27,

5 Equivalent Linear System March 27,

6 Force Substitute Damping f y k E S E D k µ 0.5 u y Deformation um Gulkan & Sozen, Shibata & Sozen (Takeda model for R/C structures) March 27,

7 Elastic Design Spectrum March 27,

8 Elastic Design Spectrum Pseudo-Acceleration Deformation March 27,

9 DDBD Using Elastic Spectra: Step-by by-step Procedure 1. Estimate the yield deformation for the system 2. Establish acceptable plastic rotation, θ p 3. Determine design displacement and ductility factor: u m = u y + h θ p and µ = u m / u y March 27,

10 DDBD Using Elastic Spectra: Step-by by-step Procedure 4. Estimate the total equivalent viscous damping: 2 ( µ 1) ( 1 α ) ζ eq = π µ 1 + αµ α ˆ ζ ζ + ζ eq ( ) = eq March 27,

11 DDBD Using Elastic Spectra: Step-by by-step Procedure 5. Enter deformation design spectrum and read T eq. Í Determine the secant stiffness k sec = 2π T 2 2 eq m March 27,

12 DDBD Using Elastic Spectra: Step-by by-step Procedure 6. Determined the required yield strength: f y ksec um = 1+ αµ α Force f y f y ( ) 1+ αµ α 1 k 1 1 ksec α k u y Deformation u m March 27,

13 DDBD Using Elastic Spectra: Step-by by-step Procedure 7. Estimate member size and detail (reinforcement in R/C structures, connections in steel structures) to provide f y. Í Calculate initial elastic stiffness k. Í Calculate yield deformation: u y = f y / k 8. Repeat steps 3 to 7 until a satisfactory solution is obtained. March 27,

14 Example ò R/C viaduct ò Superstructure weight = 190 kn/m ò Bent spacing = 39.6 m W = 7517 kn 9 m 1.5 m H k = 3EI H 3 (a) (b) March 27,

15 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 = kn/cm ÍLateral yield strength = kn ÍYield displacement = 8.82 cm, Design displacement = 26.8 cm ÍElastic period = 1.78 sec, Secant period = 3.14 sec March 27,

16 Inelastic Design Spectrum March 27,

17 Conversion of Elastic to Inelastic Design Spectrum ò Use available R y -µ-t n relationships ÍNewmark and hall ÍKrawinkler et al ÍFajfar et al. ÍMiranda & Bertero ò Relationships based on nonlinear analysis of SDF systems March 27,

18 Constant Ductility Inelastic Design Spectrum Newmark & Hall Krawinkler et al. March 27,

19 Inelastic Design Spectrum Pseudo-Acceleration Deformation March 27,

20 DDBD Using Inelastic Spectra: Step-by by-step Procedure 1. Estimate the yield deformation for the system 2. Establish acceptable plastic rotation, θ p 3. Determine design displacement and ductility factor: u m = u y + h θ p and µ = u m / u y March 27,

21 DDBD Using Elastic Spectra: Step-by by-step Procedure 4. Enter deformation design spectrum and read T n. Í Determine the elastic stiffness k = 2 2π T 2 n m March 27,

22 DDBD Using Inelastic Spectra: Step-by by-step Procedure 5. Determine the required yield strength Í f y = ku y 6. Estimate member size and detail (reinforcement in R/C structures, connections in steel structures) to provide f y. Í Calculate initial elastic stiffness k. Í Calculate yield deformation: u y = f y / k 7. Repeat steps 3 to 6 until a satisfactory solution is obtained. March 27,

23 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 = kn/cm ÍLateral yield strength = 1907 kn ÍYield displacement = 7.99 cm, Design displacement = 26.0 cm ÍElastic period = 1.16 sec March 27,

24 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: f o = ma 3. From known f y, calculate: R y = f o / f y 4. Determine ductility demand µ from R y -µ-t n relationships 5. Calculate displacement and plastic rotation Í u m =(µ/ R y )(T n / 2π) 2 A Í θ p = (u m -u y )/h March 27,

25 Evaluation of Example Design: DDBD Using Elastic Spectra ò Demands from inelastic analysis of the design structure Íu m = 39.7 cm. ͵ = 4.52 Íθ p = rad. ò Design using elastic design spectrum Íu m = 26.8 cm (32.6% underestimation) ͵ = 3.04 (32.6% underestimation) Íθ p = 0.02 rad (Demand exceeds acceptable value by > 72%) March 27,

26 Evaluation of Example Design: DDBD Using Inelastic Spectra ò Demands from inelastic analysis of the design structure Íu m = 25.9 cm. ͵ = 3.25 Íθ p = rad. ò Design using elastic design spectrum Íu m = 26.0 cm ͵ = 3.06 Íθ p = 0.02 rad ÍPredictions are nearly the same as the inelastic demands March 27,

27 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 March 27,

28 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 March 27,