Response Spectrum Concepts

Transcription

1 1 Lecture 3 Response Spectrum & Ductility Course Instructor: Dr. Carlos E. Ventura, P.Eng. Department of Civil Engineering The University of British Columbia ventura@civil.ubc.ca Short Course for CSCE Calgary 2006 Annual Conference Response Spectrum Concepts No. 2

2 2 Seismic Hazard soil Site conditions can have significant effect on response. No. 3 Linear Response of Structures Single-degree-of-freedom oscillators W K T Vibration Period T = 2π W g K time, sec No. 4

3 3 No. 5 No. 6

4 4 No. 7 No. 8

5 5 No. 9 No. 10

6 6 No. 11 No. 12

7 7 No. 13 No. 14

8 8 No. 15 No. 16

9 9 No. 17 Crustal Eq. vs Subduction Eq. Acceleration (cm/s/s) Max_Acceleration_Kobe = 587 cm/s 2 Acceleration (cm/s/s) Max_Acceleration_Llayllay = 605 cm/s Time (s) Time (s) Velocity (cm/s) Max_Velocity_Kobe = 74 cm / s Velocity (cm/s) Max_Velocity_Llayllay = 73 cm / s Time (s) Time (s) Displacement (cm) Max_Diplacement_Kobe = 20 cm Displacement (cm) Max_DiplacementLlayllay = 15 cm Time (s) SI_for_Kobe = Time (s) 1995 Kobe eq Chile eq. SI_for_Llayllay = 81 No. 18

10 10 Crustal Eq. vs Subduction Eq. SA spectra 2.5 Spectral Acceleration (g) 2 SA (g) KOBE LLAYLLAY T (sec) 5% damping No. 19 Crustal Eq. vs Subduction Eq. SV spectra SD spectra 300 Spectral Velocity (cm/s) 60 Spectral Displacement (cm/s) SV (cm/s) 150 SD (cm) T (sec) KOBE LLAYLLAY T (sec) KOBE LLAYLLAY 5% damping No. 20

11 11 No. 21 Design Spectra Acceleration response spectrum is shown to the right. S a Displacement response spectrum can be calculated assuming simple harmonic motion using expression below: Period, T acceleration response spectrum T 0 S d S 2 T = 4π d 2 S a g Period, T displacement response spectrum No. 22

12 12 No. 23 Effect of Period Change Rehabilitation design requires an understanding of how dynamic response changes when the system is modified. Change in period results in change in response amplitude. T 1 T 2 S a Period, T acceleration response spectrum S d Period, T displacement response spectrum No. 24

13 13 Effect of Damping Change Damping dissipates energy and reduces response amplitude lower damping higher damping S a Period, T acceleration response spectrum S d Period, T displacement response spectrum No. 25 Inelastic Response Inelastic response reduces stiffness and increases energy dissipation. Reduced stiffness equates to increased effective period of vibration. Energy dissipation equates to increased damping. Shear W K K e Displacement K i No. 26

14 14 Inelastic Response W K lower damping higher damping S a Base shear is limited by strength. Displacements tend to increase due to period shift tend to decrease due to damping increase Effects may cancel out (equal displacement rule) Period, T acceleration response spectrum T S 1 T d 2 Period, T displacement response spectrum No. 27 Ductility Concepts No. 28

15 15 Ductility SDOF response: V m V e k m ma 1 K e V Natural Period of Vibration: T = 2π m K No. 29 m V V e V V e = y R V y Force Reduction Factor y e u Displacement Ductility u y = µ No. 30

16 16 V Equal displacement rule (for structures with T>0.5s) V e V y u e This is not really a rule but simply an observation, however it forms the basis for much of seismic design. If u = e then R=µ y e = u No. 31 V V e V y For T<0.5s equal displacement rule does not hold u > e Equal energy rule is one often used R= (2 µ -1) y e u but this is not good as the period gets very small and as the R value increases No. 32

17 17 Effect of Yielding on Maximum Displacement C1 is the ratio between inelastic and elastic responses (after FEMA 440) No. 33 (after FEMA 440) No. 34

18 18 Idealized Force-Deformation Relationship (after FEMA 440) No. 35 Strength, Strength Degradation & Stability (after FEMA 440) No. 36

19 19 Measures of ductility: 1. Displacement ductility entire structure 2. Rotational ductility member 3. Curvature ductility section y u beam yields-forms plastic hinges Elastic Deformation Elastic+Plastic Deformation θ p -plastic rotation No. 37 What s? SDOF majority of mass m No. 38

20 20 MDOF T<1s, uniform buildings Frame Building y~ 0.64 H Shear Wall Building y ~ 0.77 H Linear y ~ 0.67 H No. 39 M y θ y M y M y θ 2L y = M y 3 L EI M y M M M y Ф θ u = θ y + θ p No. 40

21 21 Shear Wall V y y u = p + y µ θ =1+ θ θ p y θ p What s θ y for shear wall? θ y : M y Beams: θ y L L θ y = M y 3 L EI M y L F M=F*L θ 3 2 FL ML = = 3EI 3EI ML θ = = L 3EI M y L θ y = 3EI No. 41 Curvature ductility φ φ u µ φ = = 1+ y φ φ p y M M y φ y -well defined φ u, φ p difficult to estimate Ф y Ф u =Ф y + Ф p Ф u Ф θ p φ p l p l p = length of plastic hinge, not well defined No. 42

22 22 F y Concrete sections Wall F y H θ y θ p W l p l p /2 Ф p =θ p /l p lp =.08H +.022d b f y (m & MPa) 0.5W for column and beam sections No. 43 Strains In terms of performance strains are the most important parameter, especially in concrete. Unconfined concrete can take strains of to without failing, while confined concrete can go up to a strain of or more µ θ and µ φ are not good general indicators of damage as the strains are dependent on the size of the member. θ p is a better indicator of strains. No. 44

23 23 Strains M ε s M D Ф d' C ε = ε + ε c cy cp c ε c = εsy. + φ p. c ' d θ p φ p = l p c c ε c = εsy. + θ p. ' d l p ε c c = βd.1 < β <.2 d ' = γd 0.6 < γ < 0.9 l p = αd 0.5 <α <1 β β ε ε c = sy + θ p. γ α No. 45 α, β and γ vary - dependent mainly on the amount of reinforcement and the axial load on the member but they don t vary by that much c = βd.1 < β <.2 d ' = γd 0.6 < γ < 0.9 l p = αd 0.5 < α <1 β β ε ε. c = sy + θ p. γ α New concrete code will limit θ p for different situations. θ p < 0.02 will be one limit - for average values α, β, γ ε c.002(.15/.7)+.02(.15/.75) ε c = would be o.k. even for unconfined concrete. No. 46

24 24 So, how do we use all these concepts for seismic design? NBCC 2005 No. 47 Equivalent Static Force (NBCC) Base response spectrum Site Conditions Importance of Structure Inelastic Response MDOF Forces from higher modes MDOF Distribution of forces 1995 NBCC V = vsifw / (R/U) vs v - amplitude S - shape F Independent of T I R/U Implied overstrength Increase S in long periods F t Higher force in top story 2005 NBCC V =F a,v S a M v I e W/ (R d R o ) S a Based on UHRS F a or F v Depends on T and S a I e R d R o Explicit Overstrength M v Calibrated to dyn. analysis F t Same as 1995 MDOF Overturning forces J J Revised for UHS No. 48

25 25 Equivalent Static Load Procedure Elastic Base Shear, Moment Elastic base shear is derived from V e = S(T a ) W M V, where S(T a ) is the acceleration spectrum at the fundamental period T a W is the weight of the structure contributing to inertia forces, and M V is a factor to account for higher mode shears Base moment is given by M e = V e * h e J, where h e is the height of the resultant of the lateral forces, and J is a moment reduction factor No. 49 Assume we have a building with T=1.5 sec. Sa, g Vancouver Montreal st mode Period T, seconds

26 26 Design shears in a building of two different structural types located in Vancouver and Montreal first mode. Structure type Period T 1 Modal weight S a (T 1 ) g Base shear Vancouver Shear cantilever Flexural cantilever W.616W W.158W Montreal Shear cantilever Flexural cantilever W.616W W.045W How do we understand the meaning of the R factors? R d = Force Reduction Factor based on ductility of system R o = Force Reduction Factor based on reliable overstrengths R d and R o are now in a table with building height limits based on the severity of the regions seismicity and on the ductility of the structural system. Basically, the lower the seismicity and the higher the ductility the higher the allowed building. No. 52

27 27 R d, R o Factors No. 53 Inelastic Response in NBCC, Ductility factor - R d Ductility factor, R d Related to the amount of ductility capacity the structure is believed to possess. Varies from 1.0 (e.g. unreinforced masonry) to 5.0 (e.g. ductile steel moment frame). Don t get reduction in force for nothing!! For higher R values, material codes (e.g. A23.3) have stricter detailing requirements. Needed to achieve higher ductility capacity. No. 54

28 28 R d (Ductility) Factor APPLIED LOAD (kn) DISPLACEMENT, (mm) Load Deformation. R o (Over strength) Factor R R R R R R o = size φ yield sh mech R size = rounding of sizes and dimensions R φ = difference between nominal and factored resistance, 1 / φ R yield = ratio of actual yield to minimum specified yield R sh = overstrength due to strain hardening R mech = overstrength arising from mobilising full capacity of structure (collapse mechanism) No. 56

29 29 R o (Over strength) Factor R R R R R R o = size φ yield sh mech V V e / R d V= / R R V e d V yi o R size R φ R yield V R size R f = rounding of sizes and dimensions = difference between nominal and factored resistance R yield = ratio of actual yield to minimum specified yield No. 57 R o (Over strength) Factor R R R R R R o = size φ yield sh mech σ R sh - Steel F y APPLIED LOAD (kn) ε DISPLACEMENT, (mm) R sh = overstrength due to strain hardening No. 58

30 30 R o (Over strength) Factor R R R R R R o = size φ yield sh mech V Last Hinge M pb M pb M pb M pb M pb M pb M pc M pc M pc M pc V Capacity design: M pc = α M pb V yi First Hinge R mech = overstrength arising from mobilising full capacity of structure (collapse mechanism) No. 59 R o (Over strength) Factor R mech amount of strength that can be developed before a collapse mechanism forms. For a frame this is related to the ratio of column strength to the beam strength, β. N R mech = + β N + 1 No. 60

31 31 R o (Over strength) Factor R o depends on the type of lateral force resisting system No. 61 Derivation of R o for Concrete Structures No. 62

32 32 R factors for Concrete Structures System Cat. R d R o R d R o Moment Resisting Frames D MD Coupled walls D 4.0 D (1) Shear walls D MD Conventional constr. (2) (1) Ductile partially coupled wall (2) Structures designed in accordance with CSA-A23.3 Cl No. 63 Notes on R d, R o, and General Restrictions Rd is a ductility based force reduction virtually the same as the 1990 and 1995 NBCC. Ro is a force reduction factor based on reliable overstrengths typically found in structures. Previous codes had a few height restrictions, such as systems over 60m in high seismic zones had to be ductile; Buildings greater than 3 storeys in higher zones had to have reasonably ductile systems; Buildings less than 3 storeys had few constraints. The above led to some anomalies, i.e. 2 storey brittle post disaster buildings. The 2001 CSA SI6.1 Steel Code introduced a more extensive set of height limits based on system, ductility, and seismic zone. This is reflected in this table. The approach taken by CSA SI6.1 had several advantages, addressed several concerns, and was extended to the other structural materials. No. 64

33 33 Useful Reference Special issue of Canadian Journal of Civil Engineering on 2005 NBCC Earthquake Provisions (April 2003) No. 65 Note: Several of the slides were kindly provided by Dr. Mete Zosen and Dr. Luis Garcia of Purdue University No. 66