Structural Analysis for PerformanceBased Earthquake Engineering
• Basic modeling concepts • Nonlinear static pushover analysis • Nonlinear dynamic response history analysis • Incremental nonlinear analysis • Probabilistic approaches
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 1
Nonlinear Static Pushover Analysis
• Why pushover analysis? • Basic overview of method • Details of various steps • Discussion of assumptions • Improved methods
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Advanced Analysis 15-5b - 2
Why Pushover Analysis?
• Performance-based methods require reasonable estimates of inelastic deformation or damage in structures.
• Elastic Analysis is not capable of providing this information.
• Nonlinear dynamic response history analysis is capable of providing the required information, but may be very time-consuming. Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 3
Why Pushover Analysis?
• Nonlinear static pushover analysis may provide reasonable estimates of location of inelastic behavior.
• Pushover analysis alone is not capable of providing estimates of maximum deformation. Additional analysis must be performed for this purpose. The fundamental issue is… How Far to Push? Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 4
Why Pushover Analysis?
• It is important to recognize that the purpose of pushover analysis is not to predict the actual response of a structure to an earthquake. (It is unlikely that nonlinear dynamic analysis can predict the response.)
• The minimum requirement for any method of analysis, including pushover, is that it must be “good enough for design”. Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 5
Basic Overview of Method
• Development of Capacity Curve • Prediction of “Target Displacement” Capacity-Spectrum Approach (ATC 40) Simplified Approach (FEMA 273, NEHRP) Uncoupled Modal Response History Modal Pushover
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 6
Development of the Capacity Curve (ATC 40 Approach) 1. Develop Analytical Model of Structure Including: Gravity loads Known sources of inelastic behavior P-Delta Effects 2. Compute Modal Properties: Periods and Mode Shapes Modal Participation Factors Effective Modal Mass 3. Assume Lateral Inertial Force Distribution 4. Construct Pushover Curve 5. Transform Pushover Curve to 1st Mode Capacity Curve 6. Simplify Capacity Curve (Use bilinear approximation) Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 7
Development of the Capacity Curve Pushover Curve
Capacity Curve Modal Acceleration
Base Shear
Roof Displacement
Modal Displacement
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 8
Development of the Demand Curve
1. 2. 3. 4. 5.
Assume Seismic Hazard Level (e.g 2% in 50 years) Develop 5% Damped ELASTIC Response Spectrum Modify for Site Effects Modify for Expected Performance and Equivalent Damping Convert to Displacement-Acceleration Format
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 9
Elastic Spectrum Based Target Displacement Base Shear/Weight or Pseudoacceleration (g) Elastic Spectrum based demand curve for X% equivalent viscous damping
Point on capacity curve representing X% equivalent viscous damping.
Target Displacement
Spectral Displacement
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 10
Review of MDOF Dynamics (1) Original Equations of Motion:
Mu&& + Cu& + Ku = − MRu&&g
KΦ = MΦΩ 2
Transformation to Modal Coordinates:
⎧1⎫ ⎪1⎪ ⎪⎪ R=⎨ ⎬ ⎪. ⎪ ⎪⎩1⎪⎭
u = Φy
⎧ y1 ⎫ ⎪y ⎪ ⎪ 2⎪ y=⎨ ⎬ Φ = [φ1 φ2 φ3 ... φ n ] ⎪.⎪ ⎪⎩ yn ⎪⎭ MΦ&y& + CΦy& + KΦy = − MRu&&g Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 11
Review of MDOF Dynamics (2) Use of Orthogonality Relationships: T T T & & & Φ MΦy + Φ CΦy + Φ KΦy = −Φ MRu&&g T
Φ MΦ = M T
*
φi Mφi = mi T
*
φi Cφi = ci
Φ T KΦ = K *
φ i Kφ i = k i
Φ CΦ = C T
T
T
*
*
*
SDOF equation in Mode i :
m &y& + c y& + k y = −φi MRu&&g * i i
* i i
* i i
T
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 12
Review of MDOF Dynamics (3) *
Simplify by dividing through by mi and noting
ci* = 2ξ iω i * mi
ki* 2 = ωi * mi
φi MR &y&i + 2ξ iω i y& i + ω yi = − T u&&g = −Γi u&&g φi Mφi T
2 i
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 13
Review of MDOF Dynamics (4)
φi MR &y&i + 2ξ iω i y& i + ω yi = − T u&&g = −Γi u&&g φi Mφi T
2
Modal Participation Factor:
φi MR Γi = T φi Mφi T
Important Note: Γi depends on mode shape scaling Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 14
Variation of First Mode Participation Factor with First Mode Shape Γ1=1.0
Γ1=1.4 1.0
Γ1=1.6 1.0
Instructional Material Complementing FEMA 451, Design Examples
1.0
Advanced Analysis 15-5b - 15
Review of MDOF Dynamics (5) Any Mode of MDOF system
&y&i + 2ξ iω i y& i + ω i yi = −Γi u&&g 2
SDOF system
&D& + 2ξ ω D& + ω 2 D = −u&& i i i i i i g If we obtain the displacement Di(t) from the response of a SDOF we must multiply by Γ1 to obtain the modal amplitude response yi(t). history
y1 (t ) = Γ1 Di (t ) Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 16
Review of MDOF Dynamics (6) If we run a SDOF Response history analysis:
yi (t ) = Γi Di (t ) If we use a response spectrum:
yi ,max = Γi Di ,max
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 17
Review of MDOF Dynamics (7) In general
yi (t ) = Γi Di (t ) Recalling
ui (t ) = φi yi (t ) Substituting
ui (t ) = Γiφi Di (t ) Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 18
Review of MDOF Dynamics (8) Applied “static” forces required to produce ui(t):
Fi (t ) = Kui (t ) = Γi Kφi Di (t ) Recall
Kφi = ω Mφi 2 i
Fi (t ) = Γi Mφiω Di (t ) = Γi Mφi ai (t ) 2 i
Fi ( t ) = Si ai ( t )
where
Si = Γi Mφi
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 19
Review of MDOF Dynamics (9) Total shear in mode:
Vi = Fi R T
Vi (t ) = Γi (Mφi ) Rai (t ) = Γ φ MRai (t ) T
T i i
ˆ a (t ) Vi ( t ) = M i i Effective Modal Mass:
[ ] φ MR i Mˆ i = T φi Mφi T
2
Mˆ i
Important Note: does NOT depend on mode shape scaling
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 20
Variation of First Mode Effective Mass with First Mode Shape Mˆ 1 = 1.0 M Total
1.0
Mˆ 1 = 0 .9 M Total
1.0
Instructional Material Complementing FEMA 451, Design Examples
Mˆ 1 = 0 .8 M Total 1.0
Advanced Analysis 15-5b - 21
Review of MDOF Dynamics (10) S1 + S 2 + ... + S n = MR n
ˆ S = M ∑ i ,k i k =1
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 22
Simple Numerical Example 0 ⎤ ⎡ 50 − 50 K = ⎢⎢− 50 110 − 60⎥⎥ ⎢⎣ 0 − 60 130 ⎥⎦ ⎧1.267 ⎫ ⎪ ⎪ S1 = ⎨1.060 ⎬ ⎪0.600⎪ ⎩ ⎭ 3
∑S k =1
1, k
= 2.927
0⎤ ⎡1.0 0 M = ⎢⎢ 0 1.1 0 ⎥⎥ ⎢⎣ 0 0 1.2⎥⎦
⎧− 0.338⎫ ⎪ ⎪ S 2 = ⎨ 0.223 ⎬ ⎪ 0.428 ⎪ ⎩ ⎭
3
3
∑S k =1
⎧ 0.071 ⎫ ⎪ ⎪ S 3 = ⎨− 0.183⎬ ⎪ 0.172 ⎪ ⎩ ⎭
2,k
= 0.313
∑S k =1
3, k
= 0.060
⎧1.0⎫ ⎪ ⎪ S1 + S 2 + S 3 = ⎨1.1⎬ ⎪1.2⎪ ⎩ ⎭ Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 23
Review of MDOF Dynamics (11) Displacement Response in single mode:
ui (t ) = Γiφi Di (t ) From Response-History or Response Spectrum Analysis
Total shear in single mode:
ˆ a (t ) Vi ( t ) = M i i Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 24
First Mode Response as Function of System Response Modal Displacement:
Modal Acceleration:
D1 (t ) =
u1,roof (t ) Γ1φ1,roof
V1 (t ) a1 (t ) = Mˆ 1
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 25
Converting Pushover Curve to Capacity Curve
u1,roof (t ) = Γ1φ1,roof D1 (t ) D1(t)
V First Mode SDOF System (modal coords)
V First Mode System (natural coords) Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 26
Converting Pushover Curve to Capacity Curve V1( t ) Modal a1( t ) = ˆ Acceleration M 1
Base Shear
Pushover Curve Roof Displacement
Capacity Curve
Modal Displacement u1 (t ) D1 (t ) = Γ1φ1,roof
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 27
Development of Pushover Curve Potential Plastic Hinge Location (Must be predicted and possibly corrected)
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 28
Event-to-Event Pushover Analysis Create Mathematical Model Apply gravity load to determine initial nodal displacements and member forces Apply lateral load sufficient to produce single yield event Update nodal displacements and member forces Modify structural stiffness to represent yielding
Instructional Material Complementing FEMA 451, Design Examples
Continue Until Sufficient Load or Displacement is obtained.
Advanced Analysis 15-5b - 29
Initial Gravity Load Analysis
Moment
MG
A
B
A,B Rotation
Moments plotted on tension side.
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 30
Analysis 1: Gravity Analysis Base Shear
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 31
Lateral Load Analysis (Acting Alone) ML
Total Load = V
A
Moment
B B
A Rotation
Moments plotted on tension side.
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 32
Combined Load Analysis Including Total Load V MG+ML Moment
B Total Load = V
A B
Instructional Material Complementing FEMA 451, Design Examples
A
Rotation
Advanced Analysis 15-5b - 33
Analysis 2a First Lateral Analysis Base Shear
V
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 34
Combined Load Analysis: Determine amount of Lateral Load Required to Produce First Yield MG+ψ ML Moment
Total Load = ψV Rotation
For all potential hinges (i) find ψ i such that
M G ,i + ψ i M L , i = M P ,i Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 35
Analysis 2b Adjust Load to First Yield Base Shear Old Tangent Stiffness
New Tangent Stiffness
1 1
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 36
Analysis 3a Modify System Stiffness Apply Remainder of Load
Base Shear
VR=V(1-Ďˆ1) 1 1
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 37
Determine amount of Lateral Load Required to Produce Second Yield
Total Load = (1-Ďˆ1)V
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 38
Analysis 3b Adjust Load to Second Yield Base Shear Old Tangent Stiffness New Tangent Stiffness
2 1
2 1
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 39
Analysis 4a Modify System Stiffness Apply Remainder of Load Base Shear
VR 2 1
2 1
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 40
Analysis 4b Adjust Load to Third Yield Base Shear
Old Tangent Stiffness New Tangent Stiffness
3 2 1
2 3
1
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 41
Analysis 5a‌.. Base Shear
VR
3 2 1
2 3
1
Roof Displacement Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 42
Convert Pushover Curve to Capacity Curve V1 ( t ) Modal a1 ( t ) = ˆ Acceleration M 1
Base Shear
Pushover Curve
Capacity Curve ACTUAL SIMPLIFIED
Roof Displacement
Modal Displacement u (t ) D1 ( t ) = 1 Γ1φ1,roof
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 43
Equivalent Viscous Damping Base Shear/Weight or Pseudoacceleration (g) Elastic Spectrum based demand curve for X% equivalent viscous damping
Point on capacity curve representing X% Equivalent Viscous Damping.
Target Displacement
Spectral Displacement
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 44
Computing Damping Ratio from Damping Energy and Strain Energy
F
F Area=ED
FD
Area=ES
FS K
u
u
ED = π FD u = πCu ω 2 2 = 2πξmω u 2
ES = 0.5 Fs u 2 = 0.5 Ku = 0.5mω u 2
2
ED ξ= 4πES Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 45
Computing Damping Ratio from Damping Force and Elastic Force F F FD
FS
Area=ED
Area=ES
u
ED = π FD u
u
1 ES = Fs u 2
ED FD ξ= = 4πES 2 FS Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 46
Computing “True” Viscous Damping Ratio from Damping Energy and Strain Energy
ED FD ξ= = 4πES 2 FS Note: System must be in steady state harmonic RESONANT response for this equation to work.
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 47
Harmonic Resonant Response from NONLIN 15000
10000
Force, Kips
5000
0
-5000
Spring Damper
-10000
-15000 -6
-4
-2
0
2
4
6
Displacement, Inches
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 48
Harmonic Non-Resonant Response from NONLIN 2000
1500
1000
Force, Kips
500
0
-500
-1000
Spring Damper
-1500
-2000 -1.00
-0.50
0.00
0.50
1.00
Displacement, Inches Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 49
ED FD Results from NONLIN Using: Ξ = = 4πES 2 FS System Period = 0.75 seconds Harmonic Loading Target Damping Ratio 5% Critical Loading Period (sec) 0.50 0.75 1.00
Damping Force (k) 118 984 197
Spring Force (k) 787 9828 2251
Damping Ratio % 7.50 5.00 3.75
Instructional Material Complementing FEMA 451, Design Examples
Resonant
Advanced Analysis 15-5b - 50
ED FD = Results from NONLIN Using: Ξ = 4πES 2 FS System Period = 0.75 seconds Harmonic Loading Target Damping Ratio 20% Critical Loading Period (sec) 0.50 0.75 1.00
Damping Force (k) 430 999 1888
Spring Force (k) 717 2498 5666
Damping Ratio % 30.0 20.0 16.7
Instructional Material Complementing FEMA 451, Design Examples
Resonant
Advanced Analysis 15-5b - 51
Computing Equivalent Viscous Damping Ratio from Yield-Based Hysteretic Energy and Strain Energy ES
ES ED
u
Viscous System
EH
u
Yielding System
EH ξ≡ 4πES Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 52
Actual Yielding System System Stiffness and Energy Dissipation
“Equivalent� Elastic System System Stiffness
Rigid Rigid
System Energy Dissipation
ES EH
u
Instructional Material Complementing FEMA 451, Design Examples
ES ED
u Advanced Analysis 15-5b - 53
Original Yielding System Initial Frequency:
m
Posin(ωInelt) ω =
Initial Stiffness k
k m
Resonant Frequency:
ω Inelastic
ω2 = (θ − 0.5 sin 2θ ) π
θ = cos (1 − 2 −1
F Fy k uy
umax
)
Maximum Steady State Response (loaded at ωInelastic):
ksec umax
uy
u
umax
4u y = Pπ 4− o ku y
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 54
“Equivalent” Elastic System m
Posin(ωt)
Initial Stiffness k
Resonant Frequency:
ω sec
ksec = m
Maximum Steady State Resonant Response: F
u max
Po = 2ξ sec ksec
Fy ksec uy
umax
u
ξ sec
Equivalent Damping: uy 1 = 0.637(1 − ) = 0.637(1 − ) umax μΔ
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 55
“Equivalent” Elastic System when Strain Hardening is Included F
αK Fmax Fy
K u
ksec uy
ξ sec ≡ 0.637
umax
( Fy umax − Fmax u y ) Fmax umax
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 56
“Equivalent” Elastic System when Strain Hardening is Included F
αK Fmax Fy
K u
ksec uy
ξ Equiv
umax
uy ⎤ ⎡ ⎡ Fy 1 1 ⎤ ≡ 0.637 ⎢ − − ⎥ ⎥ = 0.637 ⎢ ⎣ Fmax u max ⎦ ⎣α ( μ Δ − 1) + 1 μ Δ ⎦ Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 57
Effect of Secondary Stiffness On Equivalent Viscous Damping 0.6
Equivalent Viscous Damping Ratio
α =0.0 0.5
0.4
α =0.05
0.3
α =0.10 α =0.15
0.2
α =0.20
0.1
0 1
2
3
4
5
6
7
8
9
Ductility Demand Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 58
Reduction in Ductility Demand with Strain Hardening Ratio (W = 11250 k, K = 918 k/in., T=1.0 sec, El Centro Ground Motion) 8 Yield Strength
7
600 800 1000
Ductility Demand
6 5 4 3 2 1 0 0.00
0.05
0.10
0.15
0.20
0.25
Strain Hardening Ratio Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 59
Total System Damping (% Critical)
ξTotal = 5 + κξ Equiv
Robust
Moderately Robust
Short
κ=1
κ = .7
κ = .7
Long
κ = .7
κ = .33
κ = .33
Shaking Duration
Pinched Or Brittle
See ATC 40 for Exact Values Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 60
Equivalent Viscous Damping Values for EPP System (Values Shown are Percent Critical) 1.2
5 10 20
1.0
30
40
F/Fy
0.8
0.6
0.4
0.2
0.0 0.0
0.5
1.0
1.5
2.0
2.5
U/Uy Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 61
Equivalent Viscous Damping Values for System With 5% Strain Hardening Ratio (Values Shown are Percent Critical) 1.2
5 10 20
1.0
40
30
F/Fy
0.8 0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
u/uy Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 62
Pseudoacceleration Spectrum in Traditional Format 1.2
SDS Pseudoacceleration, g
1.0
0.8
0.6
SD1/T
0.4
0.2
0.0 0.0
1.0
2.0
3.0
4.0
Period, Seconds
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 63
Pseudoacceleration (Demand) Spectrum in ADRS Format (5% Damping) 1.2 T=.10
T=.50
T=1.0
Pseudoacceleration, g
1.0
T=1.5
0.8 0.6
T=2.0
0.4 T=3.0
0.2 T=4.0
0.0 0.0
5.0
10.0
15.0
20.0
Spectral Displacement, Inches Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 64
Spectral Reduction Factors for Increased Equivalent Damping 1.2 Robust Degrading Severely Degrading
Spectral Reduction Factor
1.0
0.8
0.6
0.4
0.2
0.0 0
10
20
30
40
50
Equivalent Viscous Damping, Percent Critical Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 65
Demand Spectra for Various Damping Values 1.2
T=1.0
T=.50 5% Damping
1.0 Pseudoacceleration, g
T=1.5 10%
0.8 20% 30%
0.6
T=2.0
40%
0.4 T=3.0
0.2 T=4.0
0.0 0
5
10
15
20
Spectral Displacement, Inches Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 66
Combined Capacity-Demand Spectra 1.2 5% Damping
Pseudoacceleration, g
1.0 10%
0.8 20% 5%
30%
0.6
10%
40%
20% 30%
0.4
40%
0.2
0.0 0
5
10
15
20
Spectral Displacement, Inches Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 67
Finding the Target Displacement 0.4
40%
30% 20%
10%
5%
5% Damping
Pseudoacceleration, g
10% 20%
0.3
30% 0.2
40% 0.1
Target Disp = 6.2 in.
0.0 0
2
4
6
8
10
Spectral Displacement, Inches Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 68
You are Not Done Yet! • Note: The target displacement from the Capacity-Demand diagram corresponds to a first mode SDOF system. It must be multiplied by the first mode modal participation factor and the modal amplitude of the first mode mode shape at the roof to determine displacements or deformations in the original system. Hinge rotations may then be obtained for comparison with performance criterion.
• Knowing the target displacement, the base shear can be found from the original pushover curve.
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 69
“There is sometimes cause to fear that scientific technique, that proud servant of engineering arts, is trying to swallow its master� Professor Hardy Cross
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 70
Simplified Pushover Approaches: 2003 NEHRP Recommended Provisions
• • • • •
•
Procedure is presented in Appendix to Chapter 5 Gravity Loads include 25% of live load (but Provisions are not specific on P-Delta Modeling Requirements) Lateral Loads Applied in a “First Mode Pattern” Structure is pushed to 150% of target displacement Target displacement is assumed equal to the displacement computed from a first mode response spectrum analysis, multiplied by the factor Ci Ci adjusts for “error” in equal displacement theory when structural period is low
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 71
Simplified Pushover Approaches: 2003 NEHRP Provisions (2)
(1 − Ts / T1 ) Ci = + (Ts / T1 ) Rd
TS = S D1/ S DS
1.5 R Rd = Ω0
Ci=1 if Ts/T1 <1
Sa
Ts
Instructional Material Complementing FEMA 451, Design Examples
T Advanced Analysis 15-5b - 72
Simplified Pushover Approaches: 2003 NEHRP Provisions (3)
• Member strengths need not be evaluated • Component deformation acceptance based on • •
laboratory tests Maximum story drift may be as high as 1.25 times standard limit Nonlinear Analysis must be Peer Reviewed
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 73
Simplified Pushover Approaches: FEMA 356*. (Also used in FEMA 350) â&#x20AC;˘ â&#x20AC;˘
Procedure presented in Chapter 3 More detailed (thoughtful) treatment than in NEHRP Recommended Provisions Principal Differences: > Apply 25% of unreduced Gravity Load > Use of two different lateral load patterns > P-Delta effects included > Consideration of Hysteretic Behavior * FEMA 273 in Prestandard Format
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 74
Simplified Pushover Approaches: FEMA 356 (2) 2
Te δ t = C0C1C2C3 S a 2 g 4π
Spectral Displacement
δt = Target Displacement C0 = Modification factor to relate roof displacement to first mode spectral displacement. C1 = Modification factor to relate expected maximum inelastic displacement to displacement calculated from elastic response (similar to NEHRP Provisions Ci)
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 75
Simplified Pushover Approaches: FEMA 356 (3) 2
Te δ t = C0C1C2C3 S a 2 g 4Ď&#x20AC; C2 = Modification factor to represent effect of pinched hysteretic loop, stiffness degradation, and strength loss. C3 = Modification factor to represent increased displacements due to dynamic P-Delta effect
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 76
Discussion of Assumptions 1. Dynamic effects are ignored 2. Duration effects are ignored 3. Choice of lateral load pattern 4. Only first mode response included 5. Use of elastic response spectrum 6. Use of equivalent viscous damping 7. Modification of response spectrum for higher damping
Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 77
True Acceleration vs Pseudoacceleration 30% Critical Damping 200
Acceleration, in/sec/sec
True Total Acceleration Pseudo Acceleration 150
100
50
0 0.0
1.0
2.0
3.0
Undamped Period, Seconds Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 78
Relative Error Between True Acceleration and Pseudoacceleration 50% 5% Damping 10% Damping 15% Damping 20% Damping 25% Damping 30% Damping
Error, Percent
40%
30%
20%
10%
0% 0.00
1.00
2.00
3.00
Period of Vibration, Seconds Instructional Material Complementing FEMA 451, Design Examples
Advanced Analysis 15-5b - 79
“Equivalent” Elastic System m
Posin(ωt)
Initial Stiffness k
Resonant Frequency:
ω sec
ksec = m
Maximum Steady State Resonant Response: ksec umax = 2ξ sec Po
F Fy ksec umax
u
Equivalent Damping: uy ξ sec = 0.637(1 − ) umax
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These systems have the same hysteretic Energy Dissipation, the same AVERAGE (+/-) displacement, but considerably DIFFERENT maximum displacement.
umin
umax
umin
umax
umin
umax
The equivalent viscous damping (see previous slide) is good at predicting the AVERAGE displacement, but CAN NOT predict the true maximum displacement. Instructional Material Complementing FEMA 451, Design Examples
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“Improved” Pushover Methods
• Use of Inelastic Response Spectrum • Adaptive Load Patterns • Use of SDOF Response History Analysis • Inclusion of Higher Mode Effects
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Elastic Spectrum Based Target Displacement Base Shear/Weight or Pseudoacceleration (g) Elastic Spectrum based demand curve for X% equivalent viscous damping
Point on capacity curve representing X% equivalent viscous damping.
Target Displacement
Spectral Displacement
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Inelastic Response Spectrum Based Target Displacement
Base Shear/Weight or Pseudoacceleration (g) Inelastic Spectrum based Demand Curve for ductility demand of X.
Point on capacity curve representing ductility demand of X.
Target Displacement
Spectral Displacement
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Inelastic Spectrum Based Target Displacement
â&#x20AC;˘ Gives the same results as the equal displacement theory for (longer period) EPP systems
â&#x20AC;˘ When compared to inelastic response history analysis, the use of inelastic spectra gives better results than ATC 40 procedure.
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Computing Target Displacements from Response History Analysis of SDOF Systems
• Method called “Uncoupled Modal Response History Analysis” (UMRHA) is described by Chopra and Goel. See, for example, Appendix A of PEER Report 2001/03, entitled Modal Pushover Analysis Procedure to Estimate Seismic Demands for Buildings.
• In the UMHRA method, the undamped mode shapes are used to determine a static load pattern for each mode.
• Using these static lateral loads, a series of pushover curves and corresponding bilinear capacity curves are obtained for the first few modes. This is done using the procedures described earlier for the ATC 40 approach.
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Computing Target Displacements from Response History Analysis of SDOF Systems (2)
• Using an appropriate ground motion, a nonlinear dynamic response history analysis is computed for each modal bilinear system. This may be accomplished using NONLIN or NONLIN-Pro.
• The modal response histories are transformed to system coordinates and displacement (and deformation) response histories are obtained for each mode.
• The modal response histories are added algebraically to determine the final displacement (deformations). In the Modal Pushover approach, the individual response histories are combined using SRSS. Instructional Material Complementing FEMA 451, Design Examples
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Computing Target Displacements from Response History Analysis of SDOF Systems (3)
• Results from such an analysis are detailed in PEER Report 2001/16, entitled Statistics of SDF-System Estimate of Roof Displacement for Pushover Analysis of Buildings. Conclusions from above report (paraphrased by F. Charney): For larger ductility demands the SDOF method, using only the first mode, overestimates roof displacements and the bias increases for longer period buildings. For small ductility demand systems, the SDOF system, using only the first mode, underestimates displacement, and the bias increases for longer period systems. Instructional Material Complementing FEMA 451, Design Examples
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Conclusions (continued) First mode SDOF estimates of roof displacements due to individual ground motions can be alarmingly small (as low as 0.31 to 0.82 times “exact”) to surprisingly large (1.45 to 2.15 times exact). Errors increase when P-Delta effects are included. (Note: the method includes P-Delta effects only in the first mode). The large errors arise because for individual ground motions the first mode SDOF system may underestimate or overestimate the residual deformation due to yield-induced permanent drift. The error is not improved significantly by including higher mode contributions. However, the dispersion is reduced when elastic or nearly elastic systems are considered. Instructional Material Complementing FEMA 451, Design Examples
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Computing Target Displacements from Response History Analysis of SDOF Systems Problems with the method:
• No rational basis • Does not include P-Delta effects in higher modes • Can not consider differences in hysteretic behavior of individual components
• No reduction in effort compared to full time-history analysis • Problem of ground motion selection and scaling still exists
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