ashraf pbd seminar

8/11/2019 Ashraf Pbd Seminar

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Nonlinear Analysis & Performance Based Design


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CALCULATED vs MEASURED


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ENERGY DISSIPATION


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IMPLICIT NONLINEAR BEHAVIOR


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STEEL STRESS STRAIN RELATIONSHIPS


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INELASTIC WORK DONE!


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CAPACITY DESIGN

STRONG COLUMNS & WEAK BEAMS IN FRAMES

REDUCED BEAM SECTIONS

LINK BEAMS IN ECCENTRICALLY BRACED FRAMES

BUCKLING RESISTANT BRACES AS FUSES

RUBBER-LEAD BASE ISOLATORS

HINGED BRIDGE COLUMNSHINGES AT THE BASE LEVEL OF SHEAR WALLS

ROCKING FOUNDATIONS

OVERDESIGNED COUPLING BEAMS

OTHER SACRIFICIAL ELEMENTS


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MOMENT ROTATION RELATIONSHIP


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IDEALIZED MOMENT ROTATION


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PERFORMANCE LEVELS


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IDEALIZED FORCE DEFORMATION CURVE


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Nonlinear Analysis & Performance Based Design 17

ASCE 41 BEAM MODEL


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Linear Static Analysis

Linear Dynamic Analysis(Response Spectrum or Time History Analysis)

Nonlinear Static Analysis(Pushover Analysis)

Nonlinear Dynamic Time History Analysis

(NDI or FNA)

ASCE 41 ASSESSMENT OPTIONS


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ELASTIC STRENGTH DESIGN - KEY STEPS

CHOSE DESIGN CODE AND EARTHQUAKE LOADS

DESIGN CHECK PARAMETERS STRESS/BEAM MOMENT

GET ALLOWABLE STRESSES/ULTIMATEPHI FACTORS

CALCULATE STRESSESLOAD FACTORS (ST RS TH)

CALCULATE STRESS RATIOS

INELASTIC DEFORMATION BASED DESIGN -- KEY STEPS

CHOSE PERFORMANCE LEVEL AND DESIGN LOADSASCE 41

DEMAND CAPACITY MEASURESDRIFT/HINGE ROTATION/SHEARGET DEFORMATION AND FORCE CAPACITIES

CALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH)

CALCULATE D/C RATIOSLIMIT STATES

STRENGTH vs DEFORMATION


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STRUCTURE and MEMBERS

For a structure, F = load, D = deflection.

For a component, F depends on the component type, D is the

corresponding deformation.

The component F-D relationships must be known.

The structure F-D relationship is obtained by structural analysis.


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FRAME COMPONENTS

For each component type we need :

Reasonably accurate nonlinear F-D relationships.

Reasonable deformation and/or strength capacities. We must choose realistic demand-capacity measures, and it must be

feasible to calculate the demand values.

The best model is the simplest model that will do the job.


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Force (F)

Initiallylinear

StrainHardening

Ultimatestrength

Strength loss

Residual strength

Complete failure

First yield

Ductile limit

Hysteresis loop

Stiffness, strength and ductile limit mayall degrade under cyclic deformation

Deformation (D)

F-D RELATIONSHIP


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LATERALLOAD

Partially DuctileBrittle Ductile

DRIFT

DUCTILITY


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ASCE 41 - DUCTILE AND BRITTLE


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FORCE AND DEFORMATION CONTROL


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F

Relationship allowingfor cyclic deformation

ono on c - re a ons p

Hysteresis loops

from experiment.

D

BACKBONE CURVE


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HYSTERESIS LOOP MODELS


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STRENGTH DEGRADATION


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ASCE 41 DEFORMATION CAPACITIES

This can be used for components of all types.

It can be used if experimental results are available.

ASCE 41 gives capacities for many different components.

For beams and columns, ASCE 41 gives capacities only for the chordrotation model.


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i j

Current

Elastic

Plastic (hinge)

Total

rotation

Zero length hinges

Elastic beam

Plasticrotation

Elasticrotation

Similar atthis end

Mi Mj

M or Mi j

M

i jor

BEAM END ROTATION MODEL

PLASTIC HINGE MODEL


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PLASTIC HINGE MODEL

It is assumed that all inelastic deformation is concentrated in zero-

length plastic hinges.

The deformation measure for D/C is hinge rotation.

PLASTIC ZONE MODEL


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PLASTIC ZONE MODEL

The inelastic behavior occurs in finite length plastic zones.

Actual plastic zones usually change length, but models that have

variable lengths are too complex.

The deformation measure for D/C can be :

- Average curvature in plastic zone.- Rotation over plastic zone ( = average curvature x plastic

zone length).

ASCE 41 CHORD ROTATION CAPACITIES


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ASCE 41 CHORD ROTATION CAPACITIES

IO LS CP

Steel Beam p/y= 1 p/y= 6 p/y= 8

RC Beam

Low shear

High shear

p= 0.01

p= 0.005

p= 0.02

p= 0.01

p= 0.025

p= 0.02


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COLUMN AXIAL-BENDING MODEL

Mi

PP

P

or

V

V

Mj

MM


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STEEL COLUMN AXIAL-BENDING

CONCRETE COLUMN AXIAL BENDING


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CONCRETE COLUMN AXIAL-BENDING

SHEAR HINGE MODEL


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NodeZero-lengthshear "hinge"Elastic beam

SHEAR HINGE MODEL

PANEL ZONE ELEMENT


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PANEL ZONE ELEMENT

Deformation, D = spring rotation = shear strain in panel zone.

Force, F = moment in spring = moment transferred from beam to

column elements.

Also, F = (panel zone horizontal shear force) x (beam depth).

BUCKLING RESTRAINED BRACE


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BUCKLING-RESTRAINED BRACE

The BRB element includes isotropic hardening.

The D-C measure is axial deformation.

BEAM/COLUMN FIBER MODEL


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BEAM/COLUMN FIBER MODEL

The cross section is represented by a number of uni-axial fibers.

The M-yrelationship follows from the fiber properties, areas and locations.Stress-strain relationships reflect the effects of confinement

WALL FIBER MODEL


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Steelfibers

Concretefibers

WALL FIBER MODEL

ALTERNATIVE MEASURE STRAIN


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ALTERNATIVE MEASURE - STRAIN

NONLINEAR SOLUTION SCHEMES


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u u

1 2iteration

NEWTONRAPHSON

ITERATION

u u

1 2iteration

CONSTANT STIFFNESS

ITERATION

3 4 56

NONLINEAR SOLUTION SCHEMES

CIRCULAR FREQUENCY


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Y

+Y

-Y

0 Radius, R

CIRCULAR FREQUENCY

THE D V & A RELATIONSHIP


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u.

t

ut1.

..

Slope = ut1..

t1

u

t1 t

Slope = ut1.

u

ut1

..

t1 t

THE D, V & A RELATIONSHIP

UNDAMPED FREE VIBRATION


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UNDAMPED FREE VIBRATION

)cos(0 tuut w=

0ukum&&

m

k

where =w

RESPONSE MAXIMA


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RESPONSE MAXIMA

ut )cos(0 tu w=

)cos(02

tu ww-=ut&&

)sin(0t

u ww-=

ut

&

max2uw-=maxu&&

BASIC DYNAMICS WITH DAMPING


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BASIC DYNAMICS WITH DAMPING

guuuu &&&&& -22

g

uMKuuCuM

KuuCt

uM

&&&&&

&&&

-

= 0

gu&&

M

K

C

RESPONSE FROM GROUND MOTION


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ug..

2ug2..

ug1..

1

t1 t2

Time t

&&A Bt ug= + =-&& &u u u+ +2

2xw w

RESPONSE FROM GROUND MOTION

DAMPED RESPONSE


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{[& ]cos

[ (& )]sin }

e uB

t

A u uB

tB

t

t d

dt t d

&ut

= -

+ - - + +

-xw

ww

ww xw

ww

w

1 2

2

1 1 2 2

1

e A B t

u uA B

t

A B Bt

t t d

dt t d

ut = - +

+ + - +-

+ - +

-xw

wx

ww

wxw

x

w

x

ww

w

x

w w

{[u ]cos

[&( )

]sin }

[ ]

1 2 3

1 1

2

2

2 3 2

2

1 2 1

2

DAMPED RESPONSE

SDOF DAMPED RESPONSE


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SDOF DAMPED RESPONSE

RESPONSE SPECTRUM GENERATION


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RESPONSE SPECTRUM GENERATION

-0.40

-0.20

0.00

0.20

0.40

0.00 1.00 2.00 3.00 4.00 5.00 6.00

TIME, SECONDS

GROUND

ACC,g

Earthquake Record

DisplacementResponse Spectrum

5% damping

-4.00

-2.00

0.00

2.00

4.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

DISPL,in.

-8.00

-4.00

0.00

4.008.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

DISPL,in.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10

PERIOD,Seconds

DISPLA

CEMENT,inches

T= 0.6 sec

T= 2.0 sec

SPECTRAL PARAMETERS


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0

4

8

12

16

0 2 4 6 8 10

PERIOD, sec

DISPLACEMENT

,in.

0

10

20

30

40

0 2 4 6 8 10

PERIOD, sec

VELOCITY,in/sec

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10

PERIOD, sec

ACCE

LERATION,g

dV SPS w

va PSPS w

SPECTRAL PARAMETERS

THE ADRS SPECTRUM


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THE ADRS SPECTRUM

ADRS Curve

Spectral Displacement,Sd

SpectralAccele

ration,

Sa

Period, T

SpectralAccelerat

ion,

Sa

RS Curve

0.5

Seco

nds

1.0

Seco

nds

2.0

Seco

nds

THE ADRS SPECTRUM


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THE ADRS SPECTRUM

ASCE 7 RESPONSE SPECTRUM


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ASCE 7 RESPONSE SPECTRUM

PUSHOVER


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PUSHOVER

THE LINEAR PUSHOVER


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THE LINEAR PUSHOVER

EQUIVALENT LINEARIZATION


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EQUIVALENT LINEARIZATION

How far to push? The Target Point!

DISPLACEMENT MODIFICATION


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Calculating the Target Displacement


C0 Relates spectral to roof displacement

C1Modifier for inelastic displacement

C2Modifier for hysteresis loop shape

d= C0C

1C

2S

aT

e

2/ (4p

2)

LOAD CONTROL AND DISPLACEMENT CONTROL


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LOAD CONTROL AND DISPLACEMENT CONTROL

P-DELTA ANALYSIS


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P DELTA ANALYSIS

P-DELTA DEGRADES STRENGTH


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P DELTA DEGRADES STRENGTH

P-DELTA EFFECTS ON F-D CURVES


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STRUCTURESTRENGTH

The "over-ductility" is reduced,and is more uncertain.

P- effects may reduce the drift at which the"worst" component reaches its ductile limit.

P- effects will reduce the drift atwhich the structure loses strength.

DRIFT

P DELTA EFFECTS ON F D CURVES

THE FAST NONLINEAR ANALYSIS METHOD (FNA)


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( )

NON LINEAR FRAME AND SHEAR WALL HINGESBASE ISOLATORS (RUBBER & FRICTION)

STRUCTURAL DAMPERS

STRUCTURAL UPLIFT

STRUCTURAL POUNDING

BUCKLING RESTRAINED BRACES

RITZ VECTORS


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FNA KEY POINT


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The Ritz modes generated by thenonlinear deformation loads are used

to modify the basic structural modes

whenever the nonlinear elements go

nonlinear.

ARTIFICIAL EARTHQUAKES


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CREATING HISTORIES TO MATCH A SPECTRUM

FREQUENCY CONTENTS OF EARTHQUAKES

FOURIERTRANSFORMS




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MESH REFINEMENT


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Nonlinear Analysis & Performance Based Design71

APPROXIMATING BENDING BEHAVIOR


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This is for a coupling beam. A slender pier is similar.

ACCURACY OF MESH REFINEMENT


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PIER / SPANDREL MODELS


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STRAIN & ROTATION MEASURES


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STRAIN CONCENTRATION STUDY


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Compare calculated strains and rotations for the 3 cases.

STRAIN CONCENTRATION STUDY


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The strain over the story height is insensitive to the number of elements.

Also, the rotation over the story height is insensitive to the number of elements.

Therefore these are good choices for D/C measures.

No. of

elems

Roof

drift

Strain in bottom

element

Strain over story

height

Rotation over

story height

1 2.32% 2.39% 2.39% 1.99%

2 2.32% 3.66% 2.36% 1.97%

3 2.32% 4.17% 2.35% 1.96%

DISCONTINUOUS SHEAR WALLS


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PIER AND SPANDREL FIBER MODELS


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Vertical and horizontal fiber models

RAYLEIGH DAMPING


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The aM dampers connect the masses to the ground. They exertexternal damping forces. Units of a are 1/T.

ThebK dampers act in parallel with the elements. They exert internaldamping forces. Units of b are T.

The damping matrix is C= aM+ bK.

RAYLEIGH DAMPING


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For linear analysis, the coefficients aand bcan be chosen to give essentiallyconstant damping over a range of mode periods, as shown.

A possible method is as follows :

Choose TB= 0.9 times the first mode period.

Choose TA= 0.2 times the first mode period.

Calculate aand bto give 5% damping at these two values.

The damping ratio is essentially constant in the first few modes.

The damping ratio is higher for the higher modes.

RAYLEIGH DAMPING


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KuuC&t

uM&& = 0

Kuu&+ = 0M+b

K)

u&&+

CM

u&M

uK = 0

2uu& =2 0 w2 M = C

w2= +b

2

w2=CM 2

= CM

MK

=2

C

MK 2

MK

M+

b

K=

tuM&&

u&& ;

2

MK

RAYLEIGH DAMPING


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Higher Modes (high w) =b

Lower Modes (low w) = a

To get z from a&bfor anyw M

K=

To geta&b from two values of z1& z

2

; T 2pw

=

z1= a

2w1

+

b w12

z2= a

2w2

+

b w22

Solve fora&b

DAMPING COEFFICIENT FROM HYSTERESIS


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DAMPING COEFFICIENT FROM HYSTERESIS


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A BIG THANK YOU!!!


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ashraf pbd seminar

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