Direct Displacement-Based Design

7/30/2019 Direct Displacement-Based Design

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2001 5 15

Direct Displacement-Based Design:

Use of Inelastic vs. Elastic

Design Spectra

Anil K. Chopra & Rakesh K. Goel

Earthquake Spectra Vol.17 No.1, February 2001

Sungkyunkwan University, Korea

Choi Won Ho


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Introduction

Strength-Based Design

Displacement-Based Design

Design of Inelastic System

- Secant stiffness method (Jennings 1968)

- Substitute structure method (Shibata and Sozen 1976)

- Capacity Spectrum Method (Freeman 1975 ; Reinhorn 1995)

- Displacement Based Design (Sozen 1976 ; Moehle 1992 ; Wallace 1995 ; Priestley 2000)

Purpose

- Design produced using Elastic Design Spectra and Equivalent Linear SystemsDesign Criteria is not satisfied

- To demonstrate application of Inelastic Design Spectra to Direct Displacement-Based

Design(DBD) of structure


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DBD using Elastic Design Spectra

Equivalent Linear System

- Replacing Nonlinear System by Equivalent Linear System.

(Hudson 1965 , Jennings 1968 , Iwan and Gates 1979)

(1) Ductility Factor:

where,

um : peak of deformation of inelastic system

uy : peak of yield deformation

(2) For bilinear system, of equivalent linear

system with stiffness equal to

where,

Tn : natural vibration period

(3) Equivalent Viscous Damping Ratio

yu mu

y

f

)1(fy

1

1

k

k

1

seck

nDeformatio

Force


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Elastic Design Spectra

- Satisfactory Requirement Needed

1. Spectra intended for designs of new structures.

2. Seismic safety evaluation of existing structures to resist future earthquake.

Table1. Am lification factors : Elastic Desi n S ectra values of dam in ratio

Damping,

( % )

Median (50 percentile) One Sigma (84.1 percentile)

1 3.21 2.31 1.82 4.38 3.38 2.73

2 2.74 2.03 1.63 3.66 2.92 2.425 2.12 1.65 1.59 2.71 2.30 2.01

10 1.64 1.37 1.20 1.99 1.84 1.69

20 1.17 1.08 1.01 1.26 1.37 1.38

Table2. Amplification factors : Elastic Design Spectra (function of damping ratio)

Median (50 percentile) One Sigma (84.1 percentile)

3.21 - 0.68 ln 4.38 - 1.04 ln

2.31 - 0.41 ln 3.38 - 0.67 ln

1.82 - 0.27 ln 2.73 - 0.45 ln


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Elastic Design Spectrum

- Smooth spectrum curve

used to idealize the spectrum

for an individual ground

motion.

- Recommended period values

:

- Construction of Elastic

Design Spectrum for ground

motions with,

gug 10&& ,

sec/480 inug&

%5,360

inug

- Resulting Design Spectrum

as a Pseudo-Acceleration

Design Spectrum

- Acceleration Sensitive region

- Velocity Sensitive region

- Displacement Sensitive region


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Step-by-Step Procedure

1. Estimate yield deformation for the system.

2. Determine acceptable plastic rotation of hinge at base.

3. Determine design displacement

and design ductility factor

4. Estimate total equivalent viscous damping

, where

5. Enter deformation design spectrum for elastic systems with known

and to read .

6. Determine the required yield strength

7. Estimate member sizes and detailing to provide , and calculate initial elastic stiffness k and

.

8. Repeat steps 3 to 7 until a satisfactory solution is obtained.

h

yu

p


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Example 1

- inusinugu gogogo 18,/24,5.0&&&

- total weight : 190 kN/m

total heignt : 9m

1. Yield deformation

2. Determine acceptable plastic rotation .

3. Design displacement


4. For and , the Total equivalent viscous damping

is

idealized SDF

s stem

3

3

h

EIkh

kNw 7517

Calculation of eq for 1st

iteration

0.1 1 10

Tn, sec

0.1

1

10

100

um,cm

22.5 cm

%45

2.81 sec

example

sin le-column bent

9m


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5. For %45eq)

and this spectrum gives

6. Determine the required yield strength

7. Circular column is designed using ACI318-95 for Axial load.

- Superstructure Weight of 7517 + Column Self Weight 375kN

- Bending Moment due to Lateral force:

For the resulting column design,

- Flexural Strength: 7395

- Lateral strength = 821.7

- By using ,

- k obtained by

Yield deformation

.


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8. Since the yield deformation computed in step 7 differs from the initial estimate of iterations

are necessary.

No.(cm) (cm)

eq)

(%) (s) (kN/cm) (kN) (%)

Design

(kN) (kN/cm) (cm)

1 4.50 22.5 5.00 45 2.81 38.35 719.1 1.19 821.7 91.34 9.00

2 9.00 27.0 3.00 42 3.16 30.41 746.4 1.30 839.7 95.17 8.823 8.82 26.8 3.04 42 3.14 30.62 745.2 1.30 839.7 95.17 8.82

Example 2

The system of this example is identical to Ex1 (except bents = 4m )

- Initial elastic vibration period of system = 0.56s

- Initial stiffness k = 967.2 kN/cm

No.(cm) (cm)

eq

)

(%) (s) (kN/cm) (kN) (%)

Design

(kN) (kN/cm) (cm)

1 2.00 10.0 5.00 45 1.40 155.5 1296 1.00 1715 967.2 1.77

2 1.77 9.77 5.51 45 1.38 158.5 1264 1.00 1715 967.2 1.77


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DBD using Inelastic Design Spectra

- Using constant-ductility design spectra instead of elastic design spectra for equivalent linear system.Elastoplastic Idealization

Constant-ductility spectrum for elastoplastic hysteretic system

: Maximum displacement

: Yield deformation

: Yield strength

Elastoplastic force-deformation

relation

1

k

1

k

1

yf

mu

u

yf

k

yu

Yield force is same in 2 directions of deformation

Unloading from maximum deformation point is

parallel to the initial elastic branch.

Reloading from minimum deformation is parallel to

initial elastic branch.

Force-deformation relation is no longer

single-valued if the system is underloading or

reloading


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Inelastic Design Spectrum

1. Normalized yield strength :

2. Yield reduction factor

3. Ductility factor :

cf. in case of elastic range :

0f

yf

yu

mu

0u

u

sf systemingcorrespond

systemticelastoplas

Reduction Factor & Inelastic Design Spectrum :

elastic design spectrum x normalized strength

>


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Construction of Inelastic Design Spectrum by Newmark & Hall

In Equal Maximum Deflection Assumption

-

-

L

ateralInertia

Load

Lateral Deflection

O

B

A

y u

In Equal Energy Concept

LateralInertia

Loa

d

Lateral Deflection

O

B

E

A

y uD G

F

C


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Deformation Design Spectrum

Computed by the peak deformation um of the inelastic system

Inelastic Design Spectra

a Newmark & Hall

0 1 2 3 4 5

Tn

0

1

2

3

A

y

ductility = 1ductility = 2

ductility = 4

ductility = 8

b Krawinkler & Nassar

0 1 2 3 4 5

Tn

0

1

2

3

A

y


ductility = 4

ductility = 8

c Fa far & Fischin er

0 1 2 3 4 5

Tn

0

1

2

3

A

y


ductility = 4

ductility = 8


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Step-by-Step Procedure

1. Yield deformation

2. Determine acceptable plastic rotation .

3. Design displacement


4. Corresponding to and ,

and

5. Yield strength

6. Circular column is then designed using ACI318-95 for

axial load due to superstructure weight of 7517 plus column self weight 375kN and the

bending moment due to lateral force is

For the resulting column design, the flexural strength = 12976 and

lateral strength = 1441 . By using , .

0.1 1 10

Tn, sec

0.1

1

10

100

um,cm

22.5 cm

1.01 sec

5

50

200


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And k can be obtained by

Yield deformation : .

7. Since the yield deformation computed in step 7 differs from the initial estimate of iterations

are necessary. The results of such iterations are summarized in Table3.

No.(cm) (cm) (s) (kN/cm) (kN) (%)

Design

(kN) (kN/cm) (cm)

1 4.50 22.5 5.00 1.01 298.7 1344 3.62 1441 174.4 8.27

2 8.27 26.3 3.18 1.18 219.1 1812 5.55 1912 240.3 7.96

3 7.96 26.0 3.26 1.16 224.4 1786 5.43 1899 236.2 8.04

4 8.04 26.0 3.24 1.17 223.0 1793 5.50 1907 238.6 7.99

5 7.99 26.0 3.25 1.16 223.8 1789 5.50 1907 238.6 7.99


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Example 2

- For this system, the procedure converged after 4 iterations giving a column design .

- Lateral yield strength

- Initial stiffness k = 1784 kN/cm

No. (cm) (cm) (s) (kN/cm) (kN) (%)

Design

(kN)

Design

(kN/cm) (cm)

1 2.00 10.0 5.00 0.45 1512 3024 3.60 3226 1979 1.63

2 1.63 9.63 5.91 0.43 1630 2658 3.00 2907 1745 1.67

3 1.67 9.67 5.80 0.43 1618 2696 3.10 2965 1784 1.66

4 1.66 9.66 5.81 0.43 1620 2692 3.10 2965 1784 1.66


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Evaluation of Example Designs

Whether a design is satisfactory, it will be judged by the next demands.

1) Deformation Demand

2) Plastic Rotation Demand

3) Ductility Demand

These Demands can be computed with known properties by the following procedure:

1. Calculate initial elastic period

2. Determine Pseudo-Acceleration A from Elastic Design Spectrum;

3. Calculate Yield-Strength Reduction Factor

4. Determine Ductility Demand using the relation.

5. Calculate from and from .


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Structural Design Using Elastic Design Spectra

Example 1.Designing the structure by using the elastic

design spectra for equivalent linear system

Analyzing the structure using equations in

this paper

26.8 39.7

0.02 0.0343Underestimated the deformation demand by

Overestimated the plastic rotation by

Unsatisfactory design !!!

Example 2.

Designing the structure by using the elastic

design spectra for equivalent linear system

Analyzing the structure using equations in

this paper

9.77 12.6

0.02 0.0271

Underestimated the deformation demand by

Overestimated the plastic rotation by

Also an unsatisfactory design without any warning to the designer !!


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Structural Design Using Inelastic Design Spectra

Example 1.Designing the structure by using the

inelastic design spectra for equivalent linear

system

Analyzing the structure using

equations in this paper

26.0 25.9

0.02 0.0199

Clearly Satisfying the Design !!

Example 2.Designing the structure by using the

inelastic design spectra for equivalent linear

system

Analyzing the structure using

equations in this paper

9.70 9.66

0.02 0.0201

Also an Satisfactory Design !!!


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Conclusions

Followings are demonstrated for the procedure

1. Provides displacement estimates consistent with those predicted by the well-established

concepts of inelastic design spectra

2. Produces structural design for acceptable plastic rotation

DBD procedure uses elastic design spectra for

1. Equivalent linear systems based on the secant stiffness method

2. Variations like the substitute structure method.

It is demonstrated that,

1. Displacement and ductility factor that are estimated by this procedure are much smaller

than that by nonlinear analysis using inelastic design spectra

2. Plastic rotation demand on structures designed by this procedure may exceed acceptable

value of plastic rotation.

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Direct Displacement-Based Design