CAPACITY-DEMAND-DIAGRAM METHODS FOR ESTIMATING DEFORMATION ... · PDF fileCAPACITY-DEMAND-DIAGRAM METHODS FOR ESTIMATING DEFORMATION OF INELASTIC SYSTEMS ... Convert pushover curve

March 27, 2002 1 R. K. Goel

CAPACITY-DEMAND-DIAGRAM METHODS FOR ESTIMATING DEFORMATION OF INELASTIC SYSTEMS

Anil K. Chopraand

Rakesh K. Goel

In collaboration with Degenkolb Engineers


ATC-40 Nonlinear Static Procedure

1. Develop the pushover curve

uN

Vb

Pushover Curve

uN

Vb



2. Convert pushover curve to capacity diagram

Pushover Curve

uN

Vb

Capacity Diagram

D = Γ1φN1

uN

M1*A =

Vb



3. Plot elastic design spectrum in A-D format

D A4 π2

Tn2

= Demand DiagramA

Tn,1

D

Tn,2

A

Tn


ATC-40 Nonlinear Static Procedure4. Plot the demand diagram and capacity diagram

togetherIntersection point gives displacement demandAvoids nonlinear RHA; instead analyze equivalent linear systems

A

D

5% Demand Diagram

Capacity Diagram

Demand Diagram

Demand Point



5.Convert displacement demand to roof displacement and component deformation.

6.Compare to limiting values for specified performance goals.


Objectives

! Examine the procedure to determine the deformation of SDF systems"Does the iterative procedure converge?"How accurate is the estimated deformation? "Damping modification in ATC-40?

! Develop improved procedure"Based on inelastic design spectrum"Gives deformation consistent with design spectrum


Evaluation Method

! Define equivalent linear system"Period based on secant stiffness "Equivalent damping ratio

! Estimate deformation by ATC-40 procedures"Ground motion"Design spectrum

! Compare estimated deformation with “exact” value from nonlinear RHA or reference value from design spectrum


Equivalent Period

! For bilinear systems

! For elasto-plastic systems

µTT neq =

ααµµ

−+=

1TT neq

muyu

yf

Deformation

Forc

e

1

1

1

kα

ksec

k

( )ααµ −+1f y

March 27, 2002 10 R. K. Goel

Equivalent Damping

! For bilinear systems

! For elasto-plastic systems

( )( )( )ααµµ

αµπ

ζ−+−−=

1112

eq

( )µ

µπ

ζ 12 −=eq

DE

SE

Deformation

Forc

e

f y

( )ααµ −+1f y

umuy

March 27, 2002 11 R. K. Goel

Variation of Period and Damping of Equivalent Linear System with Ductility

March 27, 2002 12 R. K. Goel

ATC-40 Procedure

! Damping modification

! κ depends on"Structural behavior

! Type A! Type B! Type C

"Judgement

ζκζζ eqeq +=ˆ

March 27, 2002 13 R. K. Goel

ATC-40 Procedure A

1. Plot the capacity diagram and 5%-damped elastic demand diagram.2. Estimate deformation demand Di and determine Ai from the capacity

spectrum. Initially, assume Di=D(Tn,ζ=5%).3. Compute µ = Di÷Dy.

4. Compute ζeq = ζ + κ ζ eq.

5. Plot the elastic demand diagram for ζeq, intersection with capacity diagram gives displacement Dj.

6. Check for convergence.If (Dj-Di)÷Dj ≤ Tolerance then D = Dj.Otherwise, set Di = Dj and repeat steps 3 to 6.

March 27, 2002 14 R. K. Goel

Examples: Specified Ground Motion

March 27, 2002 15 R. K. Goel

ATC-40 Procedure-A Analysis of System 5

! Given: Tn = 1 s, ζ = 0.05, fy÷w = Ay÷g = 0.10.

! Input: 1940 El Centro Ground Motion

! Find: Dapprox

1. Capacity and 5%-damped elastic demand diagram

2. Start at D =11.3 cm

March 27, 2002 16 R. K. Goel


3. µ = 11.3 ÷2.56 = 4.40

4. ζeq = (2/π) ×(µ-1/µ)=0.49 = 0.45

k = 0.77

ζeq=0.05 + 0.77 ×0.45 = 0.397

5. Capacity diagram intersects 39.7%-damped elastic demand diagram at 3.73 cm

March 27, 2002 17 R. K. Goel


6. Error = 100×(3.73-11.3)÷3.72 = -202.6% > 5% tolerance

Repeat steps 3 to 6 till convergence is achieved

Dapprox = 4.46 cm

Dexact = 10.2 cm

Error = -56.1%

March 27, 2002 18 R. K. Goel


March 27, 2002 19 R. K. Goel


! The ATC-40 Procedure A converges to a deformation much smaller than the “exact” value.

! Convergence is deceptive because it may leave the erroneous impression that calculated deformation is accurate.

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! Input: 1940 El Centro Ground Motion

! Find: Dapprox

! Procedure-A fails to converge

March 27, 2002 21 R. K. Goel


! ATC-40 Procedure A fails to converge.

! Intersection point oscillates between 11.7 and 3.52 cm.

! Procedure diverges even when restarted with deformation close to “exact” value.

March 27, 2002 22 R. K. Goel

Examples: Design Spectrum

March 27, 2002 23 R. K. Goel

Newmark-Hall Elastic Design Spectrum

0.02 0.05 0.1 0.2 0.5 1 2 5 10 20 500.002

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

2

5

10

Tn, s

A, g

a

Ta = 1/33 sec

b

Tb = 0.125 sec

c

Tc

d

Td

e

Te = 10 sec

f

Tf = 33 sec

AccelerationSensitive

VelocitySensitive

DisplacementSensitive

March 27, 2002 24 R. K. Goel



! Input: Newmark-Hall (1982) design spectrum

! Find: Dapprox

1. Capacity and 5%-damped elastic demand diagram

2. Start at D =44.6 cm

March 27, 2002 25 R. K. Goel


3. µ = 44.6 ÷11.6 = 4.0

4. ζeq = (2/π) ×(µ-1/µ)=0.48 = 0.45

k = 0.77

ζeq=0.05 + 0.77 ×0.45 = 0.397

5. Capacity diagram intersects 39.7%-damped elastic demand diagram at 28.2 cm

March 27, 2002 26 R. K. Goel


6. Error = 100×(28.2-44.6)÷44.6 = -58.4% > 5% tolerance

Repeat steps 3 to 6 till convergence is achieved

Dapprox = 30.4 cm

Dexact = 44.6 cm

Error = -31.8%

March 27, 2002 27 R. K. Goel


March 27, 2002 28 R. K. Goel


! ATC-40 Procedure A gives deformation smaller than inelastic design spectrum.

! Convergence leaves an erroneous impression that calculated deformation is accurate.

March 27, 2002 29 R. K. Goel



! Input: Newmark-Hall (1982) design spectrum

! Find: Dapprox

! Procedure-A fails to converge

March 27, 2002 30 R. K. Goel


! ATC-40 Procedure A fails to converge.

! Intersection point oscillates between 13.72 and 89.28 cm.

! Procedure diverges even when restarted with deformation close to value determined from inelastic design spectrum

March 27, 2002 31 R. K. Goel

ATC-40 Procedure B1. Plot the capacity diagram.

2. Estimate deformation demand Di. Initially, assume Di=D(Tn,ζ=5%).

3. Compute µ = Di ÷Dy.

4. Compute Teq and ζeq.

5. Compute D(Teq,ζeq) and A(Teq,ζeq) of an equivalent SDF system with Teq and ζeq.

6. Plot D(Teq,ζeq) and A(Teq,ζeq).

7. Check if the curve generated by connecting points plotted in step 6 intersects the capacity diagram.

If yes, intersection point gives D.

Otherwise, repeat steps 3 to 7.

March 27, 2002 32 R. K. Goel

Examples: Ground Motion

March 27, 2002 33 R. K. Goel

ATC-40 Procedure-B Analysis of Systems 4 to 6

March 27, 2002 34 R. K. Goel

Application of ATC-40 Procedure B to Systems 5 and 6! System 5

Dexact = 10.16 cm

Darrox = 4.45 cm

Error = -56.2%

! System 6Dexact = 8.53 cm

Darrox = 5.32 cm

Error = -37.7%

March 27, 2002 35 R. K. Goel

Examples: Design Spectrum

March 27, 2002 36 R. K. Goel

ATC Procedure-B Analysis of Systems 4 to 6

March 27, 2002 37 R. K. Goel

ATC-40 Procedure-B Analysis of Systems 5 and 6! System 5

Dexact = 44.6 cm

Darrox = 30.4 cm

Error = -31.7%

! System 6Dexact = 44.6 cm

Darrox = 29.8 cm

Error = -33.1%

March 27, 2002 38 R. K. Goel

Evaluation of ATC-40 Procedure: Ground Motion

March 27, 2002 39 R. K. Goel

Comparison of deformations: µµµµ = 2

! ATC-40 method is inaccurate"Underestimates deformation

significantly over a wide period range

"D ≅ Half of “Exact”! Damping modification factor,

κ, is not attractive"Results improve marginally" κ based on judgement

March 27, 2002 40 R. K. Goel


March 27, 2002 41 R. K. Goel

Errors in deformations: µµµµ = 2

March 27, 2002 42 R. K. Goel


! ATC-40 method is inaccurate"Underestimates deformation

significantly over a wide period range

! Damping modification factor, κ, is not attractive"Results improve marginally" κ based on judgement

March 27, 2002 43 R. K. Goel

Errors for Six Ground Motions

March 27, 2002 44 R. K. Goel

Evaluation of ATC-40 Procedure: Design Spectrum

March 27, 2002 45 R. K. Goel

Deformation Spectra for Inelastic Systems

! Newmark-Hall (NH)! Krawinkler-Nassar (KN)! Vidic, Fajfar, and Fischinger

(VFF)! NH and KN give similar

results except for Tn < 0.3 sec

March 27, 2002 46 R. K. Goel

Comparison of Deformations (NH)! Large errors in ATC-40! Discrepancy depends on

ductility and period region"Underestimates deformation

in acc. and disp. regions; discrepancy increases with increasing µ

"Underestimates deformation in velo. region for µ = 2 and 4, but overestimates for µ = 8.

! ATC-40 method is deficient compared to elasticspectrum in velo. and disp. regions

March 27, 2002 47 R. K. Goel

Comparison of Deformations (NH)

March 27, 2002 48 R. K. Goel

Errors in Deformations (NH)

March 27, 2002 49 R. K. Goel

Results for NH, KN and VFF Design Spectra

March 27, 2002 50 R. K. Goel

Improved Procedures

! Existing procedures use equivalent linear systems"Convergence of iterative procedure?"Excessive damping?"Large errors

! Develop procedures based on inelastic design spectrum"Eliminate errors (or discrepancies)"Retain graphical appeal

March 27, 2002 51 R. K. Goel

Definitions

ff

Ry

oy =

! fo = Strength demand for elastic behavior

! fy = Yield strength

! Normalized yield strength

! Yield strength reduction factor

ff

fo

yy =

u

f

uy

f y

f o

March 27, 2002 52 R. K. Goel

Inelastic Design Spectrum

! Yield strength required for structure to remain elastic

! Yield strength of inelastic structure

! Yield strength reduction factor: Ry(µ,Tn)

! Plot of Ay v’s Tn

! Ry−µ−Tn relations

"Newmark-Hall (1982)

"Krawinkler-Nassar (1992)

"Vidic-Fajfar-Fischinger (1994)

"Others

wgAf o =

wgA

f yy =

yy R

AA =

March 27, 2002 53 R. K. Goel

Inelastic Design Spectra

March 27, 2002 54 R. K. Goel

Inelastic Design Spectra Using Three Different Ry−−−−µµµµ−−−−Tn Equations

March 27, 2002 55 R. K. Goel

Inelastic Demand Diagram

! Inelastic design spectrum plotted in A-D format

! Deformation from spectrum: D = m Dy = µ(Tn÷2π)2Ay

! Plot Ay v’s D for constant µ

March 27, 2002 56 R. K. Goel

Improved Procedure-A Analysis of System 5

! Plot capacity and demand diagrams in A-D format

! Yielding branch of capacity diagram intersects the demand diagram for several µ

! At relevant intersection point, µ µ µ µ from the two diagrams should match

! Interpolate between two µvalues or plot demand diagrams at finer µ values if necessary

March 27, 2002 57 R. K. Goel


March 27, 2002 58 R. K. Goel

Improved Procedure-A Analysis of Systems 4 to 6

March 27, 2002 59 R. K. Goel


! Plot capacity and demand diagrams in A-D format

! Yielding branch of capacity diagram intersects the demand diagram for several µ

! At relevant intersection point, µ µ µ µ from the two diagrams should match

! Interpolate between two µvalues or plot demand diagrams at finer µ values if necessary

March 27, 2002 60 R. K. Goel


March 27, 2002 61 R. K. Goel

Improved Procedure-A Analysis of Systems 1 to 3

March 27, 2002 62 R. K. Goel

Improved Procedure-A Applied to Other Inelastic Design Spectra

March 27, 2002 63 R. K. Goel

Improved Procedure-A Applied to Other Inelastic Design Spectra

March 27, 2002 64 R. K. Goel

Improved Procedure-A

! Gives deformations consistent with the selected design spectrum

! Retains graphical feature of ATC-40 Procedure-A"Desired deformation is at intersection of capacity and

demand diagrams! Demand diagram used is different

"Constant-µ demand diagram in improved procedure"Elastic demand diagram in ATC-40

March 27, 2002 65 R. K. Goel

Improved Procedure-B

! Plot capacity diagram! Plot D − Ay pairs to generate

curve A-B"Assume expected µ"Determine Ay from

inelastic design spectrum"Calculate D

! Find intersection of A-B and the capacity diagram

March 27, 2002 66 R. K. Goel

Improved Procedure-B

! Gives deformations consistent with the selected design spectrum

! Retains graphical feature of ATC-40 Procedure-B"Desired deformation is at intersection of curve A-B and the

capacity diagram! Different systems are analyzed to obtain a point on

curve A-B"Inelastic system in improved procedure"Equivalent elastic system in ATC-40

March 27, 2002 67 R. K. Goel

Numerical Version of Improved Procedure! Compute Ry = f0 ÷ fy = A ÷ Ay

! Determine µ from available Ry−µ−Tn relations"Newmark-Hall

"Krawinkler-Nassar

"Vidic-Fajfar-Fischinger

"Others

! Estimate deformation demand: D = µDy

March 27, 2002 68 R. K. Goel

Three Different Ry−−−−µµµµ−−−−Tn Relations

! Ry versus Tn for selected µ

! µ versus Tn for selected Ry

March 27, 2002 69 R. K. Goel

Numerical Version of Improved Procedure Analysis of System 3

! Given: Tn = 0.5s, ζ = 5%, Ay

= 1.56 g

"A = 2.71 g for system to remain elastic

"Dy = 9.7 cm

! Newmark-Hall Inelastic Design Spectrum

! Find: D

1. Ry = 2.71 ÷ 1.56 = 1.73.

2. µ = (1+1.732) ÷ 2 = 2.0

Obtained from Newmark-Hall Ry−µ−Tn relations

3. D = 2.0×9.7 = 19.4 cm

March 27, 2002 70 R. K. Goel

Conclusions

! ATC-40 method is inaccurate"Underestimates deformation significantly over a

wide range of Tn and µ values"D ≅ Half of “Exact”

! ATC-40 method is deficient compared to elastic spectrum in velocity and displacement regions

March 27, 2002 71 R. K. Goel

Conclusions

! Improved capacity-demand-diagram methods, based on well-known constant ductility design spectrum, have been developed"Improved Procedure-A"Improved Procedure-B"Numerical Version of Improved Procedure

! Improved procedures give deformation consistent with the selected design spectrum

March 27, 2002 72 R. K. Goel

Conclusions

! Improved Procedure-A retains graphical feature of ATC-40 Procedure-A"Desired deformation is at intersection of capacity

and demand diagrams! Demand diagram used is different

"Constant-µ demand diagram in improved procedure

"Elastic demand diagram in ATC-40

March 27, 2002 73 R. K. Goel

Conclusions

! Improved Procedure-B retains graphical feature of ATC-40 Procedure-B"Desired deformation is at intersection of curve A-B

and the capacity diagram

! Different systems are analyzed to obtain a point on curve A-B"Inelastic system in improved procedure"Equivalent elastic system in ATC-40

March 27, 2002 74 R. K. Goel

Conclusions

! Numerical Version of Improved Procedure is convenient if graphical feature is not needed "Based on Ry−µ−Tn relations"Gives essentially the same values of deformation

as graphical implementation

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CAPACITY-DEMAND-DIAGRAM METHODS FOR ESTIMATING DEFORMATION ... · PDF fileCAPACITY-DEMAND-DIAGRAM METHODS FOR ESTIMATING DEFORMATION OF INELASTIC SYSTEMS ... Convert pushover curve