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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 FRAMESREDUCED BEAM SECTIONS

LINK BEAMS IN ECCENTRICALLY BRACED FRAMESBUCKLING RESISTANT BRACES AS FUSES

RUBBER-LEAD BASE ISOLATORSHINGED BRIDGE COLUMNS

HINGES AT THE BASE LEVEL OF SHEAR WALLS ROCKING FOUNDATIONS

OVERDESIGNED COUPLING BEAMSOTHER SACRIFICIAL ELEMENTS


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STRUCTURAL COMPONENTS


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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 LOADSDESIGN CHECK PARAMETERS STRESS/BEAM MOMENTGET ALLOWABLE STRESSES/ULTIMATE– PHI FACTORSCALCULATE STRESSES – LOAD FACTORS (ST RS TH)

CALCULATE STRESS RATIOS

INELASTIC DEFORMATION BASED DESIGN ‐‐ KEY STEPS

CHOSE PERFORMANCE LEVEL AND DESIGN LOADS – ASCE 41DEMAND CAPACITY MEASURES – DRIFT/HINGE ROTATION/SHEAR

GET DEFORMATION AND FORCE CAPACITIESCALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH)

CALCULATE D/C RATIOS – LIMIT 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 lossResidual 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

Monotonic F-D relationship

Hysteresis loopsfrom 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 chord

rotation model.


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θi θj

Current θ

Elastic θPlastic (hinge) θ

Totalrotation

Zero length hinges

Elastic beam

Plasticrotation

Elasticrotation

Similar atthis end

Mi Mj

M or Mi j

M

θ θi j or

θ

BEAM END ROTATION 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.


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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).


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ASCE 41 CHORD ROTATION CAPACITIES

IO LS CPSteel Beam θp/θy = 1 θp/θy = 6 θp/θy = 8

RC BeamLow shearHigh shear

θp = 0.01θp = 0.005

θp = 0.02θp = 0.01

θp = 0.025θp = 0.02


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COLUMN AXIAL‐BENDING MODEL

Mi

P

PP

P

or

V

V

Mj

MM


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STEEL COLUMN AXIAL‐BENDING


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CONCRETE COLUMN AXIAL‐BENDING


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NodeZero-lengthshear "hinge"Elastic beam

SHEAR HINGE MODEL


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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).


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BUCKLING-RESTRAINED BRACE

The BRB element includes “isotropic” hardening.The D‐C measure is axial deformation.


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BEAM/COLUMN FIBER MODEL

The cross section is represented by a number of uni‐axial fibers.The M‐ψ relationship follows from the fiber properties, areas and locations. Stress‐strain relationships reflect the effects of confinement


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Steelfibers

σ

σ

ε

εConcrete

fibers

WALL FIBER MODEL


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ALTERNATIVE MEASURE - STRAIN


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∆ƒ

ƒ

∆ u u

1 2iteration

NEWTON – RAPHSONITERATION

∆ƒ

ƒ

∆ u u

1 2

iteration

CONSTANT STIFFNESSITERATION

3 4 56

NONLINEAR SOLUTION SCHEMES


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Y

+Y

-Y

0 Radius, R

θ

CIRCULAR FREQUENCY


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u.

t

ut1.

..

Slope = ut1..

t1

u

t1 t

Slope = ut1.

u

ut1..

t1 t

THE D, V & A RELATIONSHIP


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UNDAMPED FREE VIBRATION

)cos(0 tuut ω=

0=+ ukum &&

mkwhere =ω


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RESPONSE MAXIMA

ut )cos(0 tu ω=

)cos(02 tu ωω−=ut&&

)sin(0 tu ωω−=ut&

max2uω−=maxu&&


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BASIC DYNAMICS WITH DAMPING

guuuu &&&&& −=ω+ωξ+ 22guMKuuCuM

KuuCtuM

&&&&&

&&&

−=++

=++ 0

gu&&

M

K

C


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ug..

2ug2..

ug1..

1

t1 t2

Time t

&&A Bt ug= + = −&& &u u u+ +2 2ξω ω

RESPONSE FROM GROUND MOTION


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{ [& ] cos

[ (& )] sin }

e u B t

A u u B t B

tt d

dt t d

&ut = −

+ − − + +

−ξω

ωω

ωω ξω

ωω

ω

1 2

21 1 2 2

1

e A B t

u u A B t

A B Bt

tt d

dt t d

ut = − +

+ + − +−

+ − +

−ξω

ω

ξ

ωω

ωξω

ξω

ξ

ωω

ω

ξ

ω ω

{ [u ] cos

[& ( )] sin }

[ ]

1 2 3

1 1

2

2

2 3 2

2

1 2 1

2

DAMPED RESPONSE


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SDOF DAMPED RESPONSE


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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 6.00TIME, SECONDS

GR

OU

ND

AC

C, g

Earthquake Record

Displacement Response Spectrum

5% damping

-4.00-2.000.002.004.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

DIS

PL,

in.

-8.00-4.000.004.008.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

DIS

PL,

in.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10PERIOD, Seconds

DIS

PLA

CEM

ENT,

inch

es

T= 0.6 sec

T= 2.0 sec


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0

4

8

12

16

0 2 4 6 8 10

PERIOD, sec

DIS

PLA

CE

ME

NT,

in.

0

10

20

30

40

0 2 4 6 8 10

PERIOD, sec

VE

LOC

ITY,

in/s

ec

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10

PERIOD, sec

AC

CE

LER

ATIO

N, g

dV SPS ω=va PSPS ω=

SPECTRAL PARAMETERS


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THE ADRS SPECTRUM

ADRS Curve

Spectral Displacement, Sd

Spectral Acceleration, Sa

Period, T

Spectral Acceleration, Sa RS Curve

0.5 Second

s

1.0 Second

s

2.0 Second

s


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THE ADRS SPECTRUM


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ASCE 7 RESPONSE SPECTRUM


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PUSHOVER


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THE LINEAR PUSHOVER


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EQUIVALENT LINEARIZATION

How far to push? The Target Point!


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DISPLACEMENT MODIFICATION


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Calculating the Target Displacement

DISPLACEMENT MODIFICATION

C0 Relates spectral to roof displacementC1 Modifier for inelastic displacementC2 Modifier for hysteresis loop shape

δ = C0C1C2SaTe2 / (4π2)


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LOAD CONTROL AND DISPLACEMENT CONTROL


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P-DELTA ANALYSIS


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P-DELTA DEGRADES STRENGTH


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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.

Δ

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

Δ

DRIFT

P-DELTA EFFECTS ON F-D CURVES


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THE FAST NONLINEAR ANALYSIS METHOD (FNA)

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

STRUCTURAL DAMPERSSTRUCTURAL UPLIFT

STRUCTURAL POUNDINGBUCKLING RESTRAINED BRACES


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RITZ VECTORS


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FNA KEY POINT

The Ritz modes generated by the nonlinear deformation loads are used to modify the basic structural modes whenever the nonlinear elements go nonlinear.


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CREATING HISTORIES TO MATCH A SPECTRUM

FREQUENCY CONTENTS OF EARTHQUAKES

FOURIER TRANSFORMS

ARTIFICIAL EARTHQUAKES


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ARTIFICIAL EARTHQUAKES


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Nonlinear Analysis & Performance Based Design71

MESH REFINEMENT


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

APPROXIMATING BENDING BEHAVIOR


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ACCURACY OF MESH REFINEMENT


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PIER / SPANDREL MODELS


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STRAIN & ROTATION MEASURES


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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%

STRAIN CONCENTRATION STUDY


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DISCONTINUOUS SHEAR WALLS


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PIER AND SPANDREL FIBER MODELS

Vertical and horizontal fiber models


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RAYLEIGH DAMPING

• The αM dampers connect the masses to the ground. They exert external damping forces. Units of α are 1/T.

• The βK dampers act in parallel with the elements. They exert internal damping forces. Units of β are T.

• The damping matrix is C = αM + βK.


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• For linear analysis, the coefficients α and β can be chosen to give essentially constant 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 α and β to 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 &tuM && =++ 0

Kuu& ++ = 0(α M + β K )

u&& + CM

u&M

u+ K = 02uu& =ω+ωξ+ 2 0 ωξ2 M = C

ω2ξ = α + βω2

ω2ξ = CM 2

= CM

MK =

2C

MK 2 MKα M + β K=

tuM &&

u&& ;

2 MK

RAYLEIGH DAMPING


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Higher Modes (high ω) = βLower Modes (low ω) = α

To get ζ from α & β for any ω MK=

To get α & β from two values of ζ1& ζ

2

; Τ 2πω=

ζ1

= α2ω

1

+ β ω12

ζ2

= α2ω

2

+ β ω22

Solve for α & β

RAYLEIGH DAMPING


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DAMPING COEFFICIENT FROM HYSTERESIS


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A BIG THANK YOU!!!


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ashraf pbd seminarcivil808.com/sites/default/files/publication/ashraf pbd seminar.pdf · calculate...

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