Seismic Design of Multistorey Concrete Structures
Lecture 3 Response Spectrum & Ductility
Course Instructor: Dr. Carlos E. Ventura, P.Eng. Department of Civil Engineering The University of British Columbia [email protected]
Short Course for CSCE Calgary 2006 Annual Conference
Instructor: Dr. C.E. Ventura
Response Spectrum Concepts
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SeismicSeismic HazardHazard
soil Site conditions can have significant effect on response.
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Instructor: Dr. C.E. Ventura Linear Response of Structures
• Single-degree-of-freedom oscillators
W
K T
Wg 0.0 0.2 0.4 0.6 0.8 Vibration Period T = 2π time, sec K
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Crustal Eq. vs Subduction Eq.
1000 1000 Max_Acceleration_Kobe = 587 cm/s 2 Max_Acceleration_Llayllay = 605 cm/s 2 500 500
0 0
Acceleration (cm/s/s) Acceleration 500 (cm/s/s) Acceleration 500
1000 1000 0 10203040506070 0 10203040506070 Time (s) Time (s) 100 100 Max_Velocity_Llayllay = 73 cm / s Max_Velocity_Kobe = 74 cm / s 50 50
0 0
Velocity (cm/s) 50 Velocity (cm/s) 50
100 100 0 10203040506070 0 10203040506070 Time (s) Time (s) 20 20 Max_Diplacement_Kobe = 20 cm Max_DiplacementLlayllay = 15 cm 10 10 0 0
Displacement (cm) Displacement 10 (cm) Displacement 10
20 20 0 10203040506070 0 10203040506070 Time (s) Time (s) SI_for_Llayllay = 81 SI_for_Kobe = 72 1995 Kobe eq. 1985 Chile eq. Seismic Design of Multistorey Concrete Structures No. 18
9 Instructor: Dr. C.E. Ventura
Crustal Eq. vs Subduction Eq.
SA spectra
Spectral Acceleration (g) 2.5
2
1.5 SA (g) SA 1
0.5
0 0.5 1 1.5 2 2.5 3 3.5 4 T (sec) KOBE LLAYLLAY
5% damping
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Instructor: Dr. C.E. Ventura
Crustal Eq. vs Subduction Eq.
SV spectra SD spectra
Spectral Velocity (cm/s) Spectral Displacement (cm/s) 300 60
250 50
200 40
150 30 SD (cm) SD SV (cm/s)
100 20
50 10
0 0 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 T (sec) T (sec) KOBE KOBE LLAYLLAY LLAYLLAY
5% damping
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Instructor: Dr. C.E. Ventura Design Spectra
• Acceleration response spectrum S is shown to the right. a
• Displacement response Period, T spectrum can be calculated acceleration response spectrum assuming simple harmonic T motion using expression below: 0 Sd
T 2 S = S g d 4π 2 a Period, T displacement response spectrum
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Instructor: Dr. C.E. Ventura Effect of Period Change
• Rehabilitation design requires an understanding T1 T2 S of how dynamic response a changes when the system is modified. Period, T • Change in period results in acceleration response spectrum change in response S amplitude. d
Period, T displacement response spectrum
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12 Instructor: Dr. C.E. Ventura Effect of Damping Change
• Damping dissipates S energy and reduces a response amplitude
lower damping Period, T higher damping acceleration response spectrum
Sd
Period, T displacement response spectrum
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Instructor: Dr. C.E. Ventura Inelastic Response
• Inelastic response reduces stiffness and W increases energy K dissipation. K K • Reduced stiffness e i equates to increased Shear effective period of vibration. • Energy dissipation equates to increased Displacement damping.
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13 Instructor: Dr. C.E. Ventura Inelastic Response
W lower damping Sa K higher damping
• Base shear is limited by strength. Period, T acceleration response spectrum • Displacements… T1 T2 – tend to increase due to period shift Sd
– tend to decrease due to damping increase – Effects may cancel out (equal Period, T displacement rule) displacement response spectrum
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Instructor: Dr. C.E. Ventura
Ductility Concepts
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14 Instructor: Dr. C.E. Ventura Ductility SDOF response: V m Ve
k
K ∆ 1 m ma
∆e ∆
V m Natural Period of Vibration: T = 2π K
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Instructor: Dr. C.E. Ventura ∆ m V
Ve
V e = R Vy Vy Force Reduction Factor
∆y ∆e ∆u ∆
∆ Displacement u = µ Ductility ∆ ∆y
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Equal displacement rule V (for structures with T>0.5s)
Ve ∆u≈∆e
This is not really a rule but simply an observation, however it forms the basis for much of seismic Vy design.
If ∆u=∆e then R=µ∆
∆ ∆y ∆e = ∆u
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Instructor: Dr. C.E. Ventura
V For T<0.5s equal displacement rule does not hold Ve ∆u>∆e
Equal energy rule is one often used Vy R=√(2 µ∆-1)
but this is not good as the period gets very small and ∆y ∆e ∆u ∆ as the R value increases
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16 Instructor: Dr. C.E. Ventura Effect of Yielding on Maximum Displacement
C1 is the ratio between inelastic and elastic responses
(after FEMA 440)
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(after FEMA 440)
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17 Instructor: Dr. C.E. Ventura Idealized Force-Deformation Relationship
(after FEMA 440)
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Instructor: Dr. C.E. Ventura Strength, Strength Degradation & Stability
(after FEMA 440)
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18 Instructor: Dr. C.E. Ventura Measures of ductility:
1. Displacement ductility – entire structure 2. Rotational ductility – member 3. Curvature ductility – section beam yields-forms ∆y ∆u plastic hinges
Elastic Deformation Elastic+Plastic Deformation
θp-plastic rotation
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Instructor: Dr. C.E. Ventura What’s ∆ ?
SDOF ∆ majority of mass m
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MDOF T<1s, uniform buildings
Frame Building Shear Wall Building Linear y~ 0.64 H y ~ 0.77 H y ~ 0.67 H
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Instructor: Dr. C.E. Ventura
θy My My
2L M M L θ = M My y y 3EI
My My
Ф
θu= θy+ θp
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Shear Wall ∆u= ∆p + ∆y ∆y Vy θ p µθ =1+ θ y
θp What’s θy for shear wall? ∆ F My M 3 2 θy: θy y FL ML ∆ = = 3EI 3EI LL L θ ∆ ML θ = = L 3EI L Beams: θ = M y y M L 3EI M=F*L θ = y y 3EI
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Instructor: Dr. C.E. Ventura M Curvature ductility My
φu φ p µφ = =1+ φ y φ y
φy-well defined Фy Фu Ф
φu, φp –difficult to estimate Фu=Фy+ Фp
θ p φ p ≈ lp
lp= length of plastic hinge, not well defined
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21 Instructor: Dr. C.E. Ventura Concrete sections
Fy Wall Fy
H θ θy p
lp lp/2 W Фp=θp/lp
lp = .08H + .022dbfy (m & MPa) for column and beam sections ≈ 0.5W
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Instructor: Dr. C.E. Ventura Strains
In terms of performance strains are the most important parameter, especially in concrete. Unconfined concrete can take strains of 0.004 to 0.005 without failing, while confined concrete can go up to a strain of 0.010 or more
µθ and µφ are not good general indicators of damage as the strains are dependent on the size of the member.
θp is a better indicator of strains.
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εs Strains M M
Ф D d'
C
ε ε = ε +ε c c cy cp c = βD .1 < β < .2 c ' ε = ε . +φ .c d = γD 0.6 < γ < 0.9 c sy d ' p l = αD 0.5 <α <1 θ p φ = p p β β l p εc =εsy. +θ p. c c γ α εc = εsy. ' +θ p. d lp
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Instructor: Dr. C.E. Ventura
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 β β ' d = γD 0.6 < γ < 0.9 εc =εsy. +θ p. γ α lp= αD 0.5 < α <1
New concrete code will limit θp for different situations.
θp < 0.02 will be one limit - for average values α, β, γ
εc ≈ .002(.15/.7)+.02(.15/.75)
εc ≈ .0004+.004 = .0044 -would be o.k. even for unconfined concrete.
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So, how do we use all these concepts for seismic design?
Î NBCC 2005
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Instructor: Dr. C.E. Ventura Equivalent Static Force (NBCC)
1995 NBCC 2005 NBCC
V = vSIFW / (R/U) V =Fa,vSaMvIeW/ (RdRo) vS S Base response spectrum v - amplitude a Based on UHRS S - shape F F or F Site Conditions a v Independent of T Depends on T and Sa
Importance of Structure I Ie R/U R R Inelastic Response d o Implied overstrength Explicit Overstrength
MDOF Increase S in long Mv Forces from higher modes periods Calibrated to dyn. analysis
MDOF Ft Ft Distribution of forces Higher force in top story Same as 1995
MDOF J J Overturning forces Seismic Design of Multistorey Concrete StructuresRevised for UHS No. 48
24 Instructor: Dr. C.E. Ventura Equivalent Static Load Procedure Elastic Base Shear, Moment • Elastic base shear is derived from Ve = S(Ta) W MV , where S(Ta) is the acceleration spectrum at the fundamental period Ta W is the weight of the structure contributing to inertia forces, and
MV is a factor to account for higher mode shears
• Base moment is given by Me = Ve* he J, where he is the height of the resultant of the lateral forces, and J is a moment reduction factor
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Instructor: Dr. C.E. Ventura Assume we have a building with T=1.5 sec.
1 Vancouver 0.8 Montreal 0.6
Sa, g 0.4 1.256st mode 0.2 .073 0 01234 Period T, seconds
Seismic Design of Multistorey Concrete Structures
25 Instructor: Dr. C.E. Ventura Design shears in a building of two different structural types located in Vancouver and Montreal – first mode.
Structure Period Modal Sa(T1) Base type T1 weight g shear Vancouver Shear cantilever 1.50 .811W .256 .208W Flexural cantilever 1.50 .616W .256 .158W
Montreal Shear cantilever 1.50 .811W .073 .059W Flexural cantilever 1.50 .616W .073 .045W
Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura How do we understand the meaning of the R factors?
Rd = Force Reduction Factor based on “ductility” of system
Ro = Force Reduction Factor based on “reliable” overstrengths
•Rd and Ro 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.
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Rd, Ro Factors
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Inelastic Response in NBCC, Ductility factor - Rd
• Ductility factor, Rd – 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.
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Rd (Ductility) Factor
200 1.5
150 1.0 100 0.5 50
0 0.0 .
-50 Load -0.5 -100 APPLIED LOAD (kN) LOAD APPLIED -1.0 -150
-200 -1.5 -200 -150 -100 -50 0 50 100 150 200 -6-4-20246 DISPLACEMENT, (mm) Deformation
Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
Ro (Over strength) Factor
RRRo= RRRsizeφ yield sh mech
Rsize = rounding of sizes and dimensions
Rφ = difference between nominal and factored resistance, 1 / φ
Ryield= ratio of actual “yield” to minimum specified “yield”
Rsh = overstrength due to strain hardening
Rmech= overstrength arising from mobilising full capacity of structure (collapse mechanism)
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28 Instructor: Dr. C.E. Ventura Ro (Over strength) Factor
RRRo= RRRsizeφ yield sh mech V
Ve / Rd
Vyi
V= Ve / RRdo RsizeRVφ Ryield
∆
Rsize = rounding of sizes and dimensions Rf = difference between nominal and factored resistance Ryield= ratio of actual “yield” to minimum specified “yield”
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Instructor: Dr. C.E. Ventura Ro (Over strength) Factor
RRRo= RRRsizeφ yield sh mech
σ 200 Rsh - Steel 150
100
50
0 Fy -50
-100 APPLIED LOAD (kN) APPLIED LOAD -150
-200 -200 -150 -100 -50 0 50 100 150 200 ε DISPLACEMENT, (mm)
Rsh = overstrength due to strain hardening
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Ro (Over strength) Factor
RRRo= RRRsizeφ yield sh mech
V Last Hinge
Mpb Mpb Mpb Mpb Mpb Mpb
Mpc Mpc Mpc Mpc Vyi V First Hinge Capacity design:
M pc = α Mpb ∆
Rmech= overstrength arising from mobilising full capacity of structure (collapse mechanism)
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Instructor: Dr. C.E. Ventura
Ro (Over strength) Factor
•Rmech 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 + β R = mech N + 1
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Ro (Over strength) Factor
•Ro depends on the type of lateral force resisting system
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Derivation of Ro for Concrete Structures
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R factors for Concrete Structures
System Cat. Rd Ro RdRo
Moment Resisting Frames D 4.0 1.7 6.8 MD 2.5 1.4 3.5
Coupled walls D 4.0 1.7 6.8 D(1) 3.5 1.7 6.0
Shear walls D 3.5 1.6 5.6 MD 2.0 1.4 2.8
Conventional constr.(2) - 1.5 1.3 2.0
(1) Ductile partially coupled wall (2) Structures designed in accordance with CSA-A23.3 Cl. 1-20
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Instructor: Dr. C.E. Ventura
Notes on Rd, Ro, 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.
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Useful Reference
Special issue of Canadian Journal of Civil Engineering on 2005 NBCC Earthquake Provisions (April 2003)
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Instructor: Dr. C.E. Ventura
Note: Several of the slides were kindly provided by Dr. Mete Zosen and Dr. Luis Garcia of Purdue University
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