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Seismic Design of Multistorey Concrete Structures
Course Instructor:
Dr. Carlos E. Ventura, P.Eng.Department of Civil EngineeringThe University of British [email protected]
Short Course for CSCE Calgary 2006 Annual Conference
Lecture 3Response Spectrum & Ductility
Instructor: Dr. C.E. Ventura
No. 2Seismic Design of Multistorey Concrete Structures
Response Spectrum Concepts
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Instructor: Dr. C.E. Ventura
No. 3Seismic Design of Multistorey Concrete Structures
soil
Seismic Hazard
Site conditions can have significant effect on response.
Seismic Hazard
Instructor: Dr. C.E. Ventura
No. 4Seismic Design of Multistorey Concrete Structures
time, sec0.0 0.2 0.4 0.6 0.8
Vibration Period TW g
K= 2π
Linear Response of Structures
• Single-degree-of-freedom oscillators
W
KT
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Instructor: Dr. C.E. Ventura
No. 5Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 6Seismic Design of Multistorey Concrete Structures
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Instructor: Dr. C.E. Ventura
No. 7Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 8Seismic Design of Multistorey Concrete Structures
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Instructor: Dr. C.E. Ventura
No. 9Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 10Seismic Design of Multistorey Concrete Structures
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Instructor: Dr. C.E. Ventura
No. 11Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 12Seismic Design of Multistorey Concrete Structures
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Instructor: Dr. C.E. Ventura
No. 13Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 14Seismic Design of Multistorey Concrete Structures
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Instructor: Dr. C.E. Ventura
No. 15Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 16Seismic Design of Multistorey Concrete Structures
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Instructor: Dr. C.E. Ventura
No. 17Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 18Seismic Design of Multistorey Concrete Structures
Crustal Eq. vs Subduction Eq.
1995 Kobe eq. 1985 Chile eq.SI_for_Kobe 72= SI_for_Llayllay 81=
cm Max_DiplacementLlayllay 15=cm Max_Diplacement_Kobe 20=
0 10 20 30 40 50 60 7020
10
0
10
20
Time (s)
Dis
plac
emen
t (cm
)
0 10 20 30 40 50 60 7020
10
0
10
20
Time (s)
Dis
plac
emen
t (cm
)
cm / sMax_Velocity_Kobe 74=cm / sMax_Velocity_Llayllay 73=
0 10 20 30 40 50 60 70100
50
0
50
100
Time (s)
Vel
ocity
(cm
/s)
0 10 20 30 40 50 60 70100
50
0
50
100
Time (s)
Vel
ocity
(cm
/s)
cm/s 2 Max_Acceleration_Llayllay 605=cm/s 2 Max_Acceleration_Kobe 587=
0 10 20 30 40 50 60 701000
500
0
500
1000
Time (s)
Acc
eler
atio
n (c
m/s
/s)
0 10 20 30 40 50 60 701000
500
0
500
1000
Time (s)
Acc
eler
atio
n (c
m/s
/s)
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Instructor: Dr. C.E. Ventura
No. 19Seismic Design of Multistorey Concrete Structures
Crustal Eq. vs Subduction Eq.
SA spectra
5% damping
0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
KOBELLAYLLAY
Spectral Acceleration (g)
T (sec)
SA (g
)
Instructor: Dr. C.E. Ventura
No. 20Seismic Design of Multistorey Concrete Structures
Crustal Eq. vs Subduction Eq.
SV spectra
5% damping
0.5 1 1.5 2 2.5 3 3.5 40
50
100
150
200
250
300
KOBELLAYLLAY
Spectral Velocity (cm/s)
T (sec)
SV (c
m/s
)
0.5 1 1.5 2 2.5 3 3.5 40
10
20
30
40
50
60
KOBELLAYLLAY
Spectral Displacement (cm/s)
T (sec)
SD (c
m)
SD spectra
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Instructor: Dr. C.E. Ventura
No. 21Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 22Seismic Design of Multistorey Concrete Structures
• Acceleration response spectrum is shown to the right.
Design Spectra
Period, T
Sa
acceleration response spectrum
Period, T
Sd
displacement response spectrum
T0
gSTS ad 2
2
4π=
• Displacement response spectrum can be calculated assuming simple harmonic motion using expression below:
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Instructor: Dr. C.E. Ventura
No. 23Seismic Design of Multistorey Concrete Structures
Instructor: Dr. C.E. Ventura
No. 24Seismic Design of Multistorey Concrete Structures
Effect of Period Change
• Rehabilitation design requires an understanding of how dynamic response changes when the system is modified.
Period, T
Sa
acceleration response spectrum
Period, T
Sd
displacement response spectrum
T1
• Change in period results in change in response amplitude.
T2
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Instructor: Dr. C.E. Ventura
No. 25Seismic Design of Multistorey Concrete Structures
Effect of Damping Change
• Damping dissipates energy and reduces response amplitude
Period, T
Sa
acceleration response spectrum
Period, T
Sd
displacement response spectrum
higher dampinglower damping
Instructor: Dr. C.E. Ventura
No. 26Seismic Design of Multistorey Concrete Structures
Inelastic Response
Shear
Displacement
Ke Ki
W
K
• Inelastic response reduces stiffness and increases energy dissipation.
• Reduced stiffness equates to increased effective period of vibration.
• Energy dissipation equates to increased damping.
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Instructor: Dr. C.E. Ventura
No. 27Seismic Design of Multistorey Concrete Structures
Inelastic Response
• Base shear is limited by strength.
W
K higher dampinglower damping
– tend to decrease due to damping increase
• Displacements…– tend to increase due to period shift T2
Sa
acceleration response spectrum
Period, T
Sd
displacement response spectrum
T1
Period, T
– Effects may cancel out (equal displacement rule)
Instructor: Dr. C.E. Ventura
No. 28Seismic Design of Multistorey Concrete Structures
Ductility Concepts
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Instructor: Dr. C.E. Ventura
No. 29Seismic Design of Multistorey Concrete Structures
∆ ∆e
K1
Ve
V
m∆
V
ma
m
k
KmT π2=
SDOF response:Ductility
Natural Period of Vibration:
Instructor: Dr. C.E. Ventura
No. 30Seismic Design of Multistorey Concrete Structures
∆ ∆e ∆u∆y
Vy
Ve
V m
∆
∆=∆∆ µ
y
uDisplacementDuctility
RVV
y
e =
Force ReductionFactor
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Instructor: Dr. C.E. Ventura
No. 31Seismic Design of Multistorey Concrete Structures
∆ ∆e ∆u
Vy
Ve
=
V
∆y
Equal displacement rule(for structures with T>0.5s)
∆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∆
Instructor: Dr. C.E. Ventura
No. 32Seismic Design of Multistorey Concrete Structures
∆ ∆e ∆u∆y
Vy
Ve
V For T<0.5s equal displacementrule does not hold
∆u>∆e
Equal energy rule is one often used
R=√(2 µ∆-1)
but this is not good as the period gets very small and as the R value increases
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Instructor: Dr. C.E. Ventura
No. 33Seismic Design of Multistorey Concrete Structures
Effect of Yielding on Maximum Displacement
C1 is the ratio between inelastic and elastic responses
(after FEMA 440)
Instructor: Dr. C.E. Ventura
No. 34Seismic Design of Multistorey Concrete Structures
(after FEMA 440)
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Instructor: Dr. C.E. Ventura
No. 35Seismic Design of Multistorey Concrete Structures
(after FEMA 440)
Idealized Force-Deformation Relationship
Instructor: Dr. C.E. Ventura
No. 36Seismic Design of Multistorey Concrete Structures
Strength, Strength Degradation & Stability
(after FEMA 440)
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Instructor: Dr. C.E. Ventura
No. 37Seismic Design of Multistorey Concrete Structures
Measures of ductility:
1. Displacement ductility – entire structure2. Rotational ductility – member3. Curvature ductility – section
Elastic Deformation
∆y ∆u
Elastic+Plastic Deformation
beam yields-forms plastic hinges
θp-plastic rotation
Instructor: Dr. C.E. Ventura
No. 38Seismic Design of Multistorey Concrete Structures
majority of mass m∆ SDOF
What’s ∆ ?
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Instructor: Dr. C.E. Ventura
No. 39Seismic Design of Multistorey Concrete Structures
MDOF T<1s, uniform buildings
Linear y ~ 0.67 H
Shear Wall Building y ~ 0.77 H
Frame Building y~ 0.64 H
Instructor: Dr. C.E. Ventura
No. 40Seismic Design of Multistorey Concrete Structures
θy My
M
Ф
M
θu= θy+ θp
My
My My
My
EILM yy 3
=θ
2L
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Instructor: Dr. C.E. Ventura
No. 41Seismic Design of Multistorey Concrete Structures
Shear Wall
θy My My
∆y Vy ∆u= ∆p + ∆y
θp
θy:
L L
y
p
θθ
µθ +=1
What’s θy for shear wall?
EILM yy 3
=θBeams:
EIML
EIFL
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23
==∆
EIML
L 3=
∆=θ
EILM y
y 3=θ
∆ F
θ
L
M=F*L
Instructor: Dr. C.E. Ventura
No. 42Seismic Design of Multistorey Concrete Structures
Curvature ductility
y
p
y
u
φφ
φφµφ +== 1
M
My
ФФy Фu
Фu=Фy+ Фp
φy-well defined
φu, φp –difficult to estimate
p
pp l
θφ ≈
lp= length of plastic hinge, not well defined
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Instructor: Dr. C.E. Ventura
No. 43Seismic Design of Multistorey Concrete Structures
Concrete sections
lp = .08H + .022dbfy (m & MPa) for column and beam sections≈ 0.5W
θp H
W
Wall
Фp=θp/lp
lp/2 lp
θy
Fy Fy
Instructor: Dr. C.E. Ventura
No. 44Seismic Design of Multistorey Concrete Structures
StrainsIn 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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Instructor: Dr. C.E. Ventura
No. 45Seismic Design of Multistorey Concrete Structures
D d'
C
Ф
εs
εc cpcyc εεε +=
cdc
psyc .. ' φεε +=
p
pp l
θφ =
ppsyc l
cdc .. ' θεε +=
αβθ
γβεε .. psyc +=
0.90.6 d ' <<= γγD
MM
lp= αD 0.5 <α <1
c = βD .1 < β < .2
Strains
Instructor: Dr. C.E. Ventura
No. 46Seismic Design of Multistorey Concrete Structures
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
0.90.6 d ' <<= γγD
New concrete code will limit θp for different situations.
θp < 0.02 will be one limit - for average values α, β, γ
-would be o.k. even for unconfined concrete.
αβθ
γβεε .. psyc +=
lp= αD 0.5 < α <1
εc ≈ .002(.15/.7)+.02(.15/.75)
εc ≈ .0004+.004 = .0044
c = βD .1 < β < .2
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Instructor: Dr. C.E. Ventura
No. 47Seismic Design of Multistorey Concrete Structures
So, how do we use all these concepts for seismic design?
NBCC 2005
Instructor: Dr. C.E. Ventura
No. 48Seismic Design of Multistorey Concrete Structures
Equivalent Static Force (NBCC)
IeIImportance of Structure
Fa or Fv
Depends on T and Sa
FIndependent of T
Site Conditions
Sa
Based on UHRS
vSv - amplitude
S - shapeBase response spectrum
2005 NBCC V =Fa,vSaMvIeW / (RdRo)
1995 NBCCV = vSIFW / (R/U)
JRevised for UHSJMDOF
Overturning forces
FtSame as 1995
FtHigher force in top story
MDOFDistribution of forces
MvCalibrated to dyn. analysis
Increase S in long periods
MDOF Forces from higher modes
RdRo
Explicit OverstrengthR/U
Implied overstrengthInelastic Response
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Instructor: Dr. C.E. Ventura
No. 49Seismic Design of Multistorey Concrete Structures
Equivalent Static Load ProcedureElastic Base Shear, Moment
• Elastic base shear is derived fromVe = S(Ta) W MV , where
S(Ta) is the acceleration spectrum at the fundamental period TaW 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
Instructor: Dr. C.E. Ventura
Seismic Design of Multistorey Concrete Structures
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4Period T, seconds
Sa, g
Vancouver
Montreal
1st mode.256
.073
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4Period T, seconds
Sa, g
Vancouver
Montreal
1st mode.256
.073
Assume we have a building with T=1.5 sec.
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Instructor: Dr. C.E. Ventura
Seismic Design of Multistorey Concrete Structures
.059W
.045W.073.073
.811W
.616W1.501.50
MontrealShear cantileverFlexural cantilever
.208W
.158W.256.256
.811W
.616W1.501.50
VancouverShear cantileverFlexural cantilever
Baseshear
Sa(T1)g
Modalweight
PeriodT1
Structuretype
Design shears in a building of two different structural types located in Vancouver and Montreal – first mode.
Instructor: Dr. C.E. Ventura
No. 52Seismic Design of Multistorey Concrete Structures
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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Instructor: Dr. C.E. Ventura
No. 53Seismic Design of Multistorey Concrete Structures
Rd, Ro Factors
Instructor: Dr. C.E. Ventura
No. 54Seismic Design of Multistorey Concrete Structures
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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Instructor: Dr. C.E. Ventura
Seismic Design of Multistorey Concrete Structures
Rd (Ductility) Factor
-200 -150 -100 -50 0 50 100 150 200
DISPLACEMENT, (mm)
-200
-150
-100
-50
0
50
100
150
200
APP
LIED
LO
AD
(kN
)
.
-6 -4 -2 0 2 4 6
Deformation
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Load
Instructor: Dr. C.E. Ventura
No. 56Seismic Design of Multistorey Concrete Structures
Ro (Over strength) Factor
Rsize = rounding of sizes and dimensionsRφ = difference between nominal and factored resistance, 1 / φRyield= ratio of actual “yield” to minimum specified “yield”Rsh = overstrength due to strain hardeningRmech= overstrength arising from mobilising full capacity of
structure (collapse mechanism)
R R RR R Ro yieldsize sh mech= φ
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Instructor: Dr. C.E. Ventura
No. 57Seismic Design of Multistorey Concrete Structures
Ro (Over strength) Factor
R R RR R Ro yieldsize sh mech= φ
V
V = / R R d oVe
Ve / Rd
∆
Vyi
RsizeR Vφ Ryield
Rsize = rounding of sizes and dimensionsRf = difference between nominal and factored resistanceRyield= ratio of actual “yield” to minimum specified “yield”
Instructor: Dr. C.E. Ventura
No. 58Seismic Design of Multistorey Concrete Structures
Ro (Over strength) Factor
R R RR R Ro yieldsize sh mech= φ
σ
ε
Fy
Rsh - Steel
-200 -150 -100 -50 0 50 100 150 200
DISPLACEMENT, (mm)
-200
-150
-100
-50
0
50
100
150
200
APP
LIED
LO
AD
(kN
)
Rsh = overstrength due to strain hardening
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Instructor: Dr. C.E. Ventura
No. 59Seismic Design of Multistorey Concrete Structures
Ro (Over strength) Factor
Rmech= overstrength arising from mobilising full capacity ofstructure (collapse mechanism)
R R RR R Ro yieldsize sh mech= φ
∆
Vyi
First Hinge
Last HingeVMpb MpbMpb MpbMpb Mpb
Mpc Mpc Mpc Mpc
Capacity design:M = pc α Mpb
V
Instructor: Dr. C.E. Ventura
No. 60Seismic Design of Multistorey Concrete Structures
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.
R NNmech =++β1
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Instructor: Dr. C.E. Ventura
No. 61Seismic Design of Multistorey Concrete Structures
Ro (Over strength) Factor• Ro depends on the type of lateral force resisting system
Instructor: Dr. C.E. Ventura
No. 62Seismic Design of Multistorey Concrete Structures
Derivation of Ro for Concrete Structures
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Instructor: Dr. C.E. Ventura
No. 63Seismic Design of Multistorey Concrete Structures
R factors for Concrete Structures System Cat. Rd Ro RdRo
Moment Resisting Frames
D MD
4.0 2.5
1.7 1.4
6.8 3.5
Coupled walls
D D(1)
4.0 3.5
1.7 1.7
6.8 6.0
Shear walls
D MD
3.5 2.0
1.6 1.4
5.6 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
Instructor: Dr. C.E. Ventura
No. 64Seismic Design of Multistorey Concrete Structures
Notes on Rd, Ro, and General RestrictionsRd 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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Instructor: Dr. C.E. Ventura
No. 65Seismic Design of Multistorey Concrete Structures
Useful Reference
Special issue of Canadian Journal of Civil Engineering on 2005 NBCC Earthquake Provisions (April 2003)
Instructor: Dr. C.E. Ventura
No. 66Seismic Design of Multistorey Concrete Structures
Note: Several of the slides were kindly providedby Dr. Mete Zosen and Dr. Luis Garcia ofPurdue University