Nonlinear Static Pushover Analysis. Structural Analysis for Performance- Based Earthquake Engineering. Why Pushover Analysis? Why Pushover Analysis?

Structural Analysis for PerformanceBased Earthquake Engineering • Basic modeling concepts • Nonlinear static pushover analysis • Nonlinear dynamic re...
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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

Why Pushover Analysis?

• Performance-based methods require

• 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. Advanced Analysis 15-5b - 3

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

FEMA 451B Topic 15-5b Handouts

• Why pushover analysis? • Basic overview of method • Details of various steps • Discussion of assumptions • Improved methods

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 2

Why Pushover Analysis?

• Nonlinear static pushover analysis may

reasonable estimates of inelastic deformation or damage in structures.

Instructional Material Complementing FEMA 451, Design Examples

Nonlinear Static Pushover Analysis

Advanced Analysis 15-5b - 5

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

Basic Overview of Method

• Development of Capacity Curve • Prediction of “Target Displacement” ƒ Capacity-Spectrum Approach (ATC 40) ƒ Simplified Approach (FEMA 273, NEHRP) ƒ Uncoupled Modal Response History ƒ Modal Pushover

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 6

Advanced Analysis 1

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 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

Development of the Capacity Curve Pushover Curve

Capacity Curve Modal Acceleration

Base Shear

Roof Displacement

Modal Displacement

Instructional Material Complementing FEMA 451, Design Examples

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 Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 9

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 ⎪ ⎪ ⎪ y = ⎨ 2⎬ Φ = [φ1 φ 2 φ3 ... φ n ] ⎪.⎪ ⎪⎩ yn ⎪⎭ MΦ&y& + CΦy& + KΦy = − MRu&&g Instructional Material Complementing FEMA 451, Design Examples

FEMA 451B Topic 15-5b Handouts

Advanced Analysis 15-5b - 8

Spectral Displacement

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 10

Review of MDOF Dynamics (2) Use of Orthogonality Relationships:

Φ T MΦ&y& + Φ T CΦy& + Φ T KΦy = −Φ T MRu&&g Φ T MΦ = M *

φi T Mφi = mi *

Φ T CΦ = C *

φi T Cφi = ci *

Φ T KΦ = K *

φ i T Kφ i = k i *

SDOF equation in Mode i :

mi* &y&i + ci* y& i + ki* yi = −φi MRu&&g T

Advanced Analysis 15-5b - 11

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 12

Advanced Analysis 2

Review of MDOF Dynamics (3)

Review of MDOF Dynamics (4)

*

Simplify by dividing through by mi

&y&i + 2ξ iω i y& i + ω 2 yi = −

and noting

ci* = 2ξ iω i mi*

ki* = ω i2 mi*

&y&i + 2ξ iω i y& i + ω i2 yi = −

φi T MR u&&g = −Γi u&&g φiT Mφi

Modal Participation Factor:

Γi =

φi MR u&&g = −Γi u&&g φiT Mφi T

φi T MR φiT Mφi

Important Note: Γi depends on mode shape scaling Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 13

Variation of First Mode Participation Factor with First Mode Shape Γ1=1.0

Γ1=1.4 1.0

Γ1=1.6 1.0

1.0

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 14

Review of MDOF Dynamics (5) Any Mode of MDOF system

&y&i + 2ξ iω i y& i + ω i 2 yi = −Γi u&&g SDOF system

&& + 2ξ ω D& + ω 2 D = −u&& D 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 - 15

Review of MDOF Dynamics (6)

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 16

Review of MDOF Dynamics (7) In general

If we run a SDOF Response history analysis:

yi (t ) = Γi Di (t )

yi (t ) = Γi Di (t ) Recalling

ui (t ) = φi yi (t )

If we use a response spectrum:

yi ,max = Γi Di , max

Substituting

ui (t ) = Γiφi Di (t ) Instructional Material Complementing FEMA 451, Design Examples

FEMA 451B Topic 15-5b Handouts

Advanced Analysis 15-5b - 17

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 18

Advanced Analysis 3

Review of MDOF Dynamics (8)

Review of MDOF Dynamics (9) Total shear in mode:

Applied “static” forces required to produce ui(t):

Fi (t ) = Kui (t ) = Γi Kφi Di (t )

Vi (t ) = Γi (Mφi ) Rai (t ) = ΓiφiT MRai (t ) T

Kφi = ω i2 Mφi

Recall

ˆ a (t ) Vi ( t ) = M i i

Fi (t ) = Γi Mφiω Di (t ) = Γi Mφi ai (t ) 2 i

Fi ( t ) = S i ai ( t )

where

Effective Modal Mass:

[φ MR ] Mˆ i = i T φi Mφi T

Si = Γi Mφi

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 19

Variation of First Mode Effective Mass with First Mode Shape Mˆ 1 = 1 .0 M Total

1.0

Vi = FiT R

Mˆ 1 = 0 .8 M Total

Mˆ 1 = 0 .9 M Total

1.0

1.0

2

ˆ

Important Note: M i does NOT depend on mode shape scaling

Instructional Material Complementing FEMA 451, Design Examples

Review of MDOF Dynamics (10) S1 + S 2 + ... + S n = MR n

∑S k =1

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 21

Simple Numerical Example 0 ⎤ ⎡ 50 − 50 K = ⎢⎢− 50 110 − 60⎥⎥ − 60 130 ⎦⎥ ⎣⎢ 0 ⎧1.267 ⎫ ⎪ ⎪ S1 = ⎨1.060 ⎬ ⎪0.600⎪ ⎩ ⎭ 3

∑S k =1

1, k

= 2.927

∑S k =1

2,k

= 0.313

∑S k =1

3, k

FEMA 451B Topic 15-5b Handouts

Advanced Analysis 15-5b - 22

Displacement Response in single mode:

ui (t ) = Γiφi Di (t ) From Response-History or Response Spectrum Analysis

= 0.060

Total shear in single mode:

⎧1.0⎫ ⎪ ⎪ S1 + S 2 + S 3 = ⎨1.1⎬ ⎪1.2⎪ ⎩ ⎭ Instructional Material Complementing FEMA 451, Design Examples

= Mˆ i

Instructional Material Complementing FEMA 451, Design Examples

⎧ 0.071 ⎫ ⎪ ⎪ S 3 = ⎨− 0.183⎬ ⎪ 0.172 ⎪ ⎩ ⎭ 3

3

i ,k

Review of MDOF Dynamics (11)

0⎤ ⎡1.0 0 M = ⎢⎢ 0 1.1 0 ⎥⎥ ⎢⎣ 0 0 1.2⎦⎥

⎧− 0.338⎫ ⎪ ⎪ S 2 = ⎨ 0.223 ⎬ ⎪ 0.428 ⎪ ⎩ ⎭

Advanced Analysis 15-5b - 20

ˆ a (t ) Vi ( t ) = M i i Advanced Analysis 15-5b - 23

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 24

Advanced Analysis 4

First Mode Response as Function of System Response D1 (t ) =

Modal Displacement:

u1,roof (t ) = Γ1φ1,roof D1 (t ) D1(t)

u1,roof (t ) Γ1φ1,roof

a1 (t ) =

Modal Acceleration:

Converting Pushover Curve to Capacity Curve

V

V1 (t ) Mˆ 1

First Mode SDOF System (modal coords)

V First Mode System (natural coords)

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 25

Converting Pushover Curve to Capacity Curve

Instructional Material Complementing FEMA 451, Design Examples

Development of Pushover Curve

V (t ) Modal a (t ) = 1 ˆ Acceleration 1 M 1

Base Shear

Pushover Curve Roof Displacement

Advanced Analysis 15-5b - 26

Potential Plastic Hinge Location (Must be predicted and possibly corrected)

Capacity Curve

Modal Displacement u (t ) D1 (t ) = 1 Γ1φ1,roof

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 27

Event-to-Event Pushover Analysis

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 28

Initial Gravity Load Analysis

Create Mathematical Model Moment

Apply gravity load to determine initial nodal displacements and member forces

MG

A

Apply lateral load sufficient to produce single yield event

A,B Rotation

Update nodal displacements and member forces Modify structural stiffness to represent yielding

B

Continue Until Sufficient Load or Displacement is obtained.

Moments plotted on tension side.

Instructional Material Complementing FEMA 451, Design Examples

FEMA 451B Topic 15-5b Handouts

Advanced Analysis 15-5b - 29

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 30

Advanced Analysis 5

Lateral Load Analysis (Acting Alone)

Analysis 1: Gravity Analysis Base Shear

ML

Moment

A

Total Load = V

B B

A Rotation

Roof Displacement Instructional Material Complementing FEMA 451, Design Examples

Moments plotted on tension side.

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 31

Advanced Analysis 15-5b - 32

Analysis 2a First Lateral Analysis

Combined Load Analysis Including Total Load V

Base Shear MG+ML Moment

B Total Load = V

A B

A

Rotation

V

Roof Displacement Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 33

Instructional Material Complementing FEMA 451, Design Examples

Combined Load Analysis:

Analysis 2b Adjust Load to First Yield

Determine amount of Lateral Load Required to Produce First Yield MG+ψ ML

Advanced Analysis 15-5b - 34

Base Shear

Moment

Old Tangent Stiffness

Total Load = ψV

New Tangent Stiffness

Rotation

1 1

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

FEMA 451B Topic 15-5b Handouts

Roof Displacement Advanced Analysis 15-5b - 35

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 36

Advanced Analysis 6

Analysis 3a

Determine amount of Lateral Load Required to Produce Second Yield

Modify System Stiffness Apply Remainder of Load

Base Shear

Total Load = (1-ψ1)V

VR=V(1-ψ1) 1 1

Roof Displacement Instructional Material Complementing FEMA 451, Design Examples

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 37

Analysis 3b Adjust Load to Second Yield

Advanced Analysis 15-5b - 38

Analysis 4a Modify System Stiffness Apply Remainder of Load

Base Shear

Base Shear Old Tangent Stiffness New Tangent Stiffness

VR 2

2

1

1

2

2

1

1

Roof Displacement Instructional Material Complementing FEMA 451, Design Examples

Roof Displacement Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 39

Analysis 4b Adjust Load to Third Yield

Advanced Analysis 15-5b - 40

Analysis 5a….. Base Shear

Base Shear

Old Tangent Stiffness New Tangent Stiffness

3

VR

3 2

2 1

2 3

FEMA 451B Topic 15-5b Handouts

2

1

Roof Displacement Instructional Material Complementing FEMA 451, Design Examples

1

Advanced Analysis 15-5b - 41

3

1

Roof Displacement Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 42

Advanced Analysis 7

Equivalent Viscous Damping

Convert Pushover Curve to Capacity Curve

Base Shear/Weight or Pseudoacceleration (g)

V (t ) Modal a (t ) = 1 ˆ Acceleration 1 M 1

Base Shear

Pushover Curve

Elastic Spectrum based demand curve for X% equivalent viscous damping

Capacity Curve

Point on capacity curve representing X% Equivalent Viscous Damping.

ACTUAL SIMPLIFIED

Roof Displacement

Modal Displacement u (t ) D1 ( t ) = 1

Γ1φ1,roof

Instructional Material Complementing FEMA 451, Design Examples

Target Displacement

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 43

FD

F Area=ED

Area=ES

FS

F F

K

u

FS

Area=ED

FD

u

Advanced Analysis 15-5b - 44

Computing Damping Ratio from Damping Force and Elastic Force

Computing Damping Ratio from Damping Energy and Strain Energy

F

Spectral Displacement

Area=ES

u

ED = π FD u

= πCu 2ω = 2πξmω 2u 2

ES = 0.5Fs u = 0.5 Ku 2 = 0.5mω u 2

2

ED = π FD u

E ξ= D 4πES Instructional Material Complementing FEMA 451, Design Examples

ξ=

ES =

1 Fs u 2

ED F = D 4πES 2 FS

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 45

Computing “True” Viscous Damping Ratio from Damping Energy and Strain Energy

u

Advanced Analysis 15-5b - 46

Harmonic Resonant Response from NONLIN 15000

10000

5000

ED F = D 4πES 2 FS

Force, Kips

ξ=

0

-5000

Note: System must be in steady state harmonic RESONANT response for this equation to work.

Spring Damper

-10000

-15000 -6

-4

-2

0

2

4

6

Displacement, Inches

Instructional Material Complementing FEMA 451, Design Examples

FEMA 451B Topic 15-5b Handouts

Advanced Analysis 15-5b - 47

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 48

Advanced Analysis 8

Harmonic Non-Resonant Response from NONLIN

Results from NONLIN Using: ξ =

2000

1500

System Period = 0.75 seconds Harmonic Loading Target Damping Ratio 5% Critical

1000

500 Force, Kips

ED F = D 4πES 2 FS

0

Loading Period (sec) 0.50 0.75 1.00

-500

-1000

Spring Damper

-1500

-2000 -1.00

-0.50

0.00

0.50

Damping Force (k) 118 984 197

Spring Force (k) 787 9828 2251

Damping Ratio % 7.50 5.00 3.75

Resonant

1.00

Displacement, Inches Instructional Material Complementing FEMA 451, Design Examples

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 49

Results from NONLIN Using: ξ =

ED F = D 4πES 2 FS

Computing Equivalent Viscous Damping Ratio from Yield-Based Hysteretic Energy and Strain Energy

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

Viscous System

ξ≡

EH 4πES

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 51

Initial Frequency:

Posin(ωInelt) ω =

System Stiffness

Initial Stiffness k

k m

Resonant Frequency:

ω Inelastic =

Rigid

u

Instructional Material Complementing FEMA 451, Design Examples

FEMA 451B Topic 15-5b Handouts

Fy

ES ED

k u

Advanced Analysis 15-5b - 53

ω2 (θ − 0.5 sin 2θ ) π

θ = cos −1 (1 − 2

F

ES

Advanced Analysis 15-5b - 52

Original Yielding System

“Equivalent” Elastic System

System Energy Dissipation

u

Yielding System

Resonant

Rigid

EH

EH

u

m

System Stiffness and Energy Dissipation

ES

ES ED

Instructional Material Complementing FEMA 451, Design Examples

Actual Yielding System

Advanced Analysis 15-5b - 50

uy

)

Maximum Steady State Response (loaded at ωInelastic):

ksec umax

uy umax

u

umax =

4u y Pπ 4− o ku y

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 54

Advanced Analysis 9

“Equivalent” Elastic System when Strain Hardening is Included

“Equivalent” Elastic System m

F

Resonant Frequency:

Posin(ωt)

ω sec =

Initial Stiffness k

αK

ksec m

Fmax Fy

K u

Maximum Steady State Resonant Response: u max =

F

uy

umax

ξ sec

u

umax

Po 2ξ sec ksec

Fy ksec

ksec uy

Equivalent Damping: uy 1 = 0.637(1 − ) = 0.637(1 − ) umax μΔ

Instructional Material Complementing FEMA 451, Design Examples

ξ sec ≡ 0.637

( Fy umax − Fmax u y ) Fmax umax

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 55

Advanced Analysis 15-5b - 56

Effect of Secondary Stiffness On Equivalent Viscous Damping

“Equivalent” Elastic System when Strain Hardening is Included F

0.6

Fmax Fy

K u

ksec uy

umax

uy ⎤ ⎡ ⎡ Fy 1 1 ⎤ − − ⎥ ⎥ = 0.637 ⎢ ⎣ Fmax u max ⎦ ⎣α ( μ Δ − 1) + 1 μ Δ ⎦

ξ Equiv ≡ 0.637 ⎢

Equivalent Viscous Damping Ratio

α=0.0

αK

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

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 57

Reduction in Ductility Demand with Strain Hardening Ratio

Total System Damping (% Critical)

ξTotal = 5 + κξ Equiv

(W = 11250 k, K = 918 k/in., T=1.0 sec, El Centro Ground Motion) 8 Yield Strength

7

600 800 1000

6

Ductility Demand

Advanced Analysis 15-5b - 58

5 4

Robust

Moderately Robust

Short

κ=1

κ = .7

κ = .7

Long

κ = .7

κ = .33

κ = .33

3

Shaking Duration

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

FEMA 451B Topic 15-5b Handouts

Pinched Or Brittle

See ATC 40 for Exact Values Advanced Analysis 15-5b - 59

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 60

Advanced Analysis 10

Equivalent Viscous Damping Values for EPP System

Equivalent Viscous Damping Values for System

(Values Shown are Percent Critical)

With 5% Strain Hardening Ratio (Values Shown are Percent Critical)

1.2

5 10 20

1.0

30

1.2

40

0.8 F/Fy

F/Fy

0.8

40

30

5 10 20

1.0

0.6

0.6 0.4

0.4

0.2

0.2

0.0 0.0

0.0 0.0

0.5

1.0

1.5

2.0

0.5

1.0

2.5

U/Uy Instructional Material Complementing FEMA 451, Design Examples

2.5

3.0

Advanced Analysis 15-5b - 62

Pseudoacceleration (Demand) Spectrum in ADRS Format (5% Damping) 1.2

1.2

T=.10

SDS

T=.50

T=1.0

1.0 Pseudoacceleration, g

1.0

Pseudoacceleration, g

2.0

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 61

Pseudoacceleration Spectrum in Traditional Format

1.5 u/uy

0.8

0.6

SD1/T

0.4

T=1.5

0.8 0.6

T=2.0

0.4 T=3.0

0.2

0.2 T=4.0

0.0 0.0

0.0 0.0

1.0

2.0

3.0

5.0

20.0

Spectral Displacement, Inches

Instructional Material Complementing FEMA 451, Design Examples

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 63

Spectral Reduction Factors for Increased Equivalent Damping

Advanced Analysis 15-5b - 64

Demand Spectra for Various Damping Values 1.2

1.2

T=1.0

T=.50 5% Damping

Robust Degrading

1.0

15.0

4.0

Period, Seconds

1.0 T=1.5

Severely Degrading

Pseudoacceleration, g

Spectral Reduction Factor

10.0

0.8

0.6

0.4

10%

0.8 20% 30%

0.6

T=2.0

40%

0.4 T=3.0

0.2

0.2 T=4.0

0.0

0.0 0

10

20

30

40

50

Equivalent Viscous Damping, Percent Critical Instructional Material Complementing FEMA 451, Design Examples

FEMA 451B Topic 15-5b Handouts

0

5

10

15

20

Spectral Displacement, Inches Advanced Analysis 15-5b - 65

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 66

Advanced Analysis 11

Finding the Target Displacement

Combined Capacity-Demand Spectra 1.2

0.4

40%

30% 20%

10%

5%

5% Damping

10% 5% Damping

10%

Pseudoacceleration, g

Pseudoacceleration, g

1.0

0.8 20% 5%

30%

0.6

10%

40%

20% 30%

0.4

40%

20%

0.3

30% 0.2

40% 0.1

0.2

Target Disp = 6.2 in.

0.0

0.0

0

0

5

10

15

2

20

4

6

8

10

Spectral Displacement, Inches

Spectral Displacement, Inches Instructional Material Complementing FEMA 451, Design Examples

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 67

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.

“There is sometimes cause to fear that scientific technique, that proud servant of engineering arts, is trying to swallow its master”

Hinge rotations may then be obtained for comparison with performance criterion.

Professor Hardy Cross

• Knowing the target displacement, the base shear can be found from the original pushover curve.

Instructional Material Complementing FEMA 451, Design Examples

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

FEMA 451B Topic 15-5b Handouts

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 69

Advanced Analysis 15-5b - 71

Advanced Analysis 15-5b - 70

Simplified Pushover Approaches: 2003 NEHRP Provisions (2)

Ci =

(1 − Ts / T1 ) + (Ts / T1 ) Rd

TS = S D1/ S DS Rd =

1.5R Ω0

Ci=1 if Ts/T1 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

Simplified Pushover Approaches: FEMA 356 (2)

δ t = C0C1C2C3 S a

2

Te g 4π 2

Spectral Displacement

Advanced Analysis 15-5b - 74

Simplified Pushover Approaches: FEMA 356 (3) 2

T δ t = C0C1C2C3 S a e 2 g 4π

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

Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 73

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

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Discussion of Assumptions

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True Acceleration vs Pseudoacceleration 30% Critical Damping 200 True Total Acceleration Pseudo Acceleration Acceleration, in/sec/sec

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

150

100

50

0 0.0

1.0

2.0

3.0

Undamped Period, Seconds Instructional Material Complementing FEMA 451, Design Examples

FEMA 451B Topic 15-5b Handouts

Advanced Analysis 15-5b - 77

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Advanced Analysis 15-5b - 78

Advanced Analysis 13

Relative Error Between True Acceleration and Pseudoacceleration

“Equivalent” Elastic System m

50% 5% Damping 10% Damping 15% Damping 20% Damping 25% Damping 30% Damping

Error, Percent

40%

Initial Stiffness k

20%

Resonant Frequency:

ω sec =

ksec m

Maximum Steady State Resonant Response: k umax = sec 2ξ sec Po

30%

F Fy

10%

0% 0.00

ksec 1.00

2.00

3.00

umax

Period of Vibration, Seconds Instructional Material Complementing FEMA 451, Design Examples

Advanced Analysis 15-5b - 79

These systems have the same hysteretic Energy Dissipation, the same AVERAGE (+/-) displacement, but considerably DIFFERENT maximum displacement.

umin

Posin(ωt)

umax

umin

umax

umin

umax

u

Equivalent Damping: u ξ sec = 0.637(1 − y ) umax

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Advanced Analysis 15-5b - 80

“Improved” Pushover Methods

• Use of Inelastic Response Spectrum • Adaptive Load Patterns • Use of SDOF Response History Analysis • Inclusion of Higher Mode Effects

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

Advanced Analysis 15-5b - 81

Elastic Spectrum Based Target Displacement

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Inelastic Response Spectrum Based Target Displacement

Base Shear/Weight or Pseudoacceleration (g)

Base Shear/Weight or Pseudoacceleration (g) Elastic Spectrum based demand curve for X% equivalent viscous damping

Inelastic Spectrum based Demand Curve for ductility demand of X.

Point on capacity curve representing X% equivalent viscous damping.

Target Displacement

Spectral Displacement

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FEMA 451B Topic 15-5b Handouts

Advanced Analysis 15-5b - 82

Advanced Analysis 15-5b - 83

Point on capacity curve representing ductility demand of X.

Target Displacement

Spectral Displacement

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Advanced Analysis 15-5b - 84

Advanced Analysis 14

Inelastic Spectrum Based Target Displacement

• Gives the same results as the equal displacement theory for (longer period) EPP systems

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

• When compared to inelastic response history analysis, the use of inelastic spectra gives better results than ATC 40 procedure.

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Advanced Analysis 15-5b - 85

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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Advanced Analysis 15-5b - 86

Computing Target Displacements from Response History Analysis of SDOF Systems (2)

Computing Target Displacements from Response History Analysis of SDOF Systems (3)

• Using an appropriate ground motion, a nonlinear dynamic

• Results from such an analysis are detailed in PEER Report

response history analysis is computed for each modal bilinear system. This may be accomplished using NONLIN or NONLIN-Pro.

2001/16, entitled Statistics of SDF-System Estimate of Roof Displacement for Pushover Analysis of Buildings. Conclusions from above report (paraphrased by F. Charney):

• 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

Advanced Analysis 15-5b - 87

ƒ 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). ƒ Errors increase when P-Delta effects are included. (Note: the method includes P-Delta effects only in the first mode). ƒ 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. ƒ The error is not improved significantly by including higher mode contributions. However, the dispersion is reduced when elastic or nearly elastic systems are considered.

FEMA 451B Topic 15-5b Handouts

ƒ 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

Advanced Analysis 15-5b - 88

Computing Target Displacements from Response History Analysis of SDOF Systems

Conclusions (continued)

Instructional Material Complementing FEMA 451, Design Examples

ƒ For larger ductility demands the SDOF method, using only the first mode, overestimates roof displacements and the bias increases for longer period buildings.

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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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Advanced Analysis 15

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