calculated vs measured

Nonlinear Analysis & Performance Based Design

Nonlinear Analysis & Performance Based Design

Nonlinear Analysis & Performance Based Design

CALCULATED vs MEASURED

Nonlinear Analysis & Performance Based Design

ENERGY DISSIPATION

Nonlinear Analysis & Performance Based Design

Nonlinear Analysis & Performance Based Design

Nonlinear Analysis & Performance Based Design

Nonlinear Analysis & Performance Based Design

IMPLICIT NONLINEAR BEHAVIOR

Nonlinear Analysis & Performance Based Design

STEEL STRESS STRAIN RELATIONSHIPS

Nonlinear Analysis & Performance Based Design

INELASTIC WORK DONE!

Nonlinear Analysis & Performance Based Design

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

Nonlinear Analysis & Performance Based Design

STRUCTURAL COMPONENTS

Nonlinear Analysis & Performance Based Design

MOMENT ROTATION RELATIONSHIP

Nonlinear Analysis & Performance Based Design

IDEALIZED MOMENT ROTATION

Nonlinear Analysis & Performance Based Design

PERFORMANCE LEVELS

Nonlinear Analysis & Performance Based Design

IDEALIZED FORCE DEFORMATION CURVE

Nonlinear Analysis & Performance Based Design 17

ASCE 41 BEAM MODEL

Nonlinear Analysis & Performance Based Design

• 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

Nonlinear Analysis & Performance Based Design

ELASTIC STRENGTH DESIGN - KEY STEPS

CHOSE DESIGN CODE AND EARTHQUAKE LOADSDESIGN CHECK PARAMETERS STRESS/BEAM MOMENTGET ALLOWABLE STRESSES/ULTIMATE– PHI FACTORS

CALCULATE 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

Nonlinear Analysis & Performance Based Design

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.

Nonlinear Analysis & Performance Based Design

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.

Nonlinear Analysis & Performance Based Design

Force (F)

In itia llylinear

Stra inH ardening

U ltim atestrength

Strength loss

R esidual strength

C om plete fa ilure

F irst y ie ld

D uctile lim it

H ysteresis loop

Stiffness, strength and ductile lim it m ayall degrade under cyclic deform ation

D eform ation (D )

F-D RELATIONSHIP

Nonlinear Analysis & Performance Based Design

LATERALLOAD

Partially DuctileBrittle Ductile

DRIFT

DUCTILITY

Nonlinear Analysis & Performance Based Design

ASCE 41 - DUCTILE AND BRITTLE

Nonlinear Analysis & Performance Based Design

FORCE AND DEFORMATION CONTROL

Nonlinear Analysis & Performance Based Design

F

R elationship a llow ingfor cyclic deform ation

M onotonic F-D re lationship

H ysteresis loopsfrom experim ent.

D

BACKBONE CURVE

Nonlinear Analysis & Performance Based Design

HYSTERESIS LOOP MODELS

Nonlinear Analysis & Performance Based Design

STRENGTH DEGRADATION

Nonlinear Analysis & Performance Based Design

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.

Nonlinear Analysis & Performance Based Design

ij

C urrent

Elastic Plastic (h inge)

Tota lrotation

Zero length h inges

E lastic beam

Plasticrotation

E lasticrotation

S im ilar a tth is end

M i M j

M or Mi j

M

i j or

BEAM END ROTATION MODEL

Nonlinear Analysis & Performance Based Design

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.

Nonlinear Analysis & Performance Based Design

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).

Nonlinear Analysis & Performance Based Design

ASCE 41 CHORD ROTATION CAPACITIES

IO LS CP

Steel Beam qp/qy = 1 qp/qy = 6 qp/qy = 8

RC Beam Low shear High shear

qp = 0.01

qp = 0.005

qp = 0.02

qp = 0.01

qp = 0.025

qp = 0.02

Nonlinear Analysis & Performance Based Design

COLUMN AXIAL-BENDING MODEL

M i

P

PP

P

or

V

V

M j

MM

Nonlinear Analysis & Performance Based Design

STEEL COLUMN AXIAL-BENDING

Nonlinear Analysis & Performance Based Design

CONCRETE COLUMN AXIAL-BENDING

Nonlinear Analysis & Performance Based Design

N odeZero-lengthshear "h inge"E lastic beam

SHEAR HINGE MODEL

Nonlinear Analysis & Performance Based Design

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).

Nonlinear Analysis & Performance Based Design

BUCKLING-RESTRAINED BRACE

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

Nonlinear Analysis & Performance Based Design

BEAM/COLUMN FIBER MODEL

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

Nonlinear Analysis & Performance Based Design

Steelfibers

C oncrete

fibers

WALL FIBER MODEL

Nonlinear Analysis & Performance Based Design

ALTERNATIVE MEASURE - STRAIN

Nonlinear Analysis & Performance Based Design

∆ƒ

ƒ

∆ u u

1 2

iteration

NEWTON – RAPHSONITERATION

∆ƒ

ƒ

∆ u u

1 2

iteration

CONSTANT STIFFNESSITERATION

3 4 56

NONLINEAR SOLUTION SCHEMES

Nonlinear Analysis & Performance Based Design

Y

+Y

-Y

0 Radius, R

q

CIRCULAR FREQUENCY

Nonlinear Analysis & Performance Based Design

u.

t

ut1

.

..

Slope = ut1

..

t1

u

t1 t

Slope = ut1

.

u

ut1

..

t1t

THE D, V & A RELATIONSHIP

Nonlinear Analysis & Performance Based Design

UNDAMPED FREE VIBRATION

)cos(0 tuut w=

0=+ ukum&&

mk

where =w

Nonlinear Analysis & Performance Based Design

RESPONSE MAXIMA

ut )cos(0 tu w=

)cos(02 tu ww-=ut&&

)sin(0 tu ww-=ut&

max2uw-=maxu&&

Nonlinear Analysis & Performance Based Design

BASIC DYNAMICS WITH DAMPING

guuuu &&&&& -=w+wx+ 22

guMKuuCuM

KuuCt

uM

&&&&&

&&&

-=++

=++ 0

gu&&

M

K

C

Nonlinear Analysis & Performance Based Design

ug..

2ug2

..

ug1

..

1

t1 t2

Time t

&&A Bt ug= + = -&& &u u u+ +22xw w

RESPONSE FROM GROUND MOTION

Nonlinear Analysis & Performance Based Design

{ [& ] cos

[ (& )] sin }

e uB

t

A u uB

tB

tt d

dt t d

&ut = -

+ - - + +

-xw

ww

ww xw

ww

w

1 2

21 1 2 2

1

eA B

t

u uA B

t

A B Bt

tt d

dt t d

ut = - +

+ + - +-

+ - +

-xw

w

x

ww

wxw

xw

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

Nonlinear Analysis & Performance Based Design

SDOF DAMPED RESPONSE

Nonlinear Analysis & Performance Based Design

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

-2.00

0.00

2.00

4.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

DIS

PL,

in.

-8.00

-4.00

0.00

4.00

8.00

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 10

PERIOD, Seconds

DIS

PL

AC

EM

EN

T, in

ches

T= 0.6 sec

T= 2.0 sec

Nonlinear Analysis & Performance Based Design

0

4

8

12

16

0 2 4 6 8 10

PERIOD, sec

DIS

PL

AC

EM

EN

T, in

.

0

10

20

30

40

0 2 4 6 8 10

PERIOD, sec

VE

LO

CIT

Y, in

/se

c

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10

PERIOD, sec

AC

CE

LE

RA

TIO

N,

g

dV SPS w=va PSPS w=

SPECTRAL PARAMETERS

Nonlinear Analysis & Performance Based Design

THE ADRS SPECTRUM

ADRS Curve

Spectral Displacement, Sd

Spec

tral

Acc

eler

ation

, Sa

2.0 Seconds

1.0 Seconds0.5

Seco

nds

Period, T

Spec

tral

Acc

eler

ation

, Sa RS Curve

0.5

Seco

nds

1.0

Seco

nds

2.0

Seco

nds

Nonlinear Analysis & Performance Based Design

THE ADRS SPECTRUM

Nonlinear Analysis & Performance Based Design

ASCE 7 RESPONSE SPECTRUM

Nonlinear Analysis & Performance Based Design

PUSHOVER

Nonlinear Analysis & Performance Based Design

THE LINEAR PUSHOVER

Nonlinear Analysis & Performance Based Design

EQUIVALENT LINEARIZATION

How far to push? The Target Point!

Nonlinear Analysis & Performance Based Design

DISPLACEMENT MODIFICATION

Nonlinear Analysis & Performance Based Design

Calculating the Target Displacement

DISPLACEMENT MODIFICATION

C0 Relates spectral to roof displacement C1 Modifier for inelastic displacement C2 Modifier for hysteresis loop shape

d=C0 C

1 C

2 S

a T

e

2 / (4p2)

Nonlinear Analysis & Performance Based Design

LOAD CONTROL AND DISPLACEMENT CONTROL

Nonlinear Analysis & Performance Based Design

P-DELTA ANALYSIS

Nonlinear Analysis & Performance Based Design

P-DELTA DEGRADES STRENGTH

Nonlinear Analysis & Performance Based Design

STR U C TU R ESTR EN G TH

The "over-ductility" is reduced,and is m ore uncerta in.

P - effects m ay reduce the drift a t which the"worst" com ponent reaches its ductile lim it.

P- effects w ill reduce the drift a twhich the structure loses strength.

D R IFT

P-DELTA EFFECTS ON F-D CURVES

Nonlinear Analysis & Performance Based Design

THE FAST NONLINEAR ANALYSIS METHOD (FNA)

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

STRUCTURAL DAMPERSSTRUCTURAL UPLIFT

STRUCTURAL POUNDINGBUCKLING RESTRAINED BRACES

Nonlinear Analysis & Performance Based Design

RITZ VECTORS

Nonlinear Analysis & Performance Based Design

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.

Nonlinear Analysis & Performance Based Design

CREATING HISTORIES TO MATCH A SPECTRUM

FREQUENCY CONTENTS OF EARTHQUAKES

FOURIER TRANSFORMS

ARTIFICIAL EARTHQUAKES

Nonlinear Analysis & Performance Based Design

ARTIFICIAL EARTHQUAKES

Nonlinear Analysis & Performance Based Design 71

MESH REFINEMENT

Nonlinear Analysis & Performance Based Design 72

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

APPROXIMATING BENDING BEHAVIOR

Nonlinear Analysis & Performance Based Design 73

ACCURACY OF MESH REFINEMENT

Nonlinear Analysis & Performance Based Design 74

PIER / SPANDREL MODELS

Nonlinear Analysis & Performance Based Design 75

STRAIN & ROTATION MEASURES

Nonlinear Analysis & Performance Based Design 76

Compare calculated strains and rotations for the 3 cases.

STRAIN CONCENTRATION STUDY

Nonlinear Analysis & Performance Based Design 77

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

Nonlinear Analysis & Performance Based Design

DISCONTINUOUS SHEAR WALLS

Nonlinear Analysis & Performance Based Design 79

PIER AND SPANDREL FIBER MODELS

Vertical and horizontal fiber models

Nonlinear Analysis & Performance Based Design 80

RAYLEIGH DAMPING

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

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

• The damping matrix is C = aM + bK.

Nonlinear Analysis & Performance Based Design 81

• For linear analysis, the coefficients a and b 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 a and b 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

Nonlinear Analysis & Performance Based Design

KuuC&tuM&& =++ 0

Kuu&++ = 0 (

a

M + b

K )

u&& + CM

u&M

u+ K = 02uu& =w+wx+ 2 0 wx2 M = C

w2x= a

+ b

w2

w2x= C

M 2= C

M√MK

=2

C

√ MK 2√ MK

a

M + b

K=

tuM&&

u&& ;

2√ MK

RAYLEIGH DAMPING

Nonlinear Analysis & Performance Based Design

Higher Modes (high )=w bLower Modes (low )=w a

To get z from a & b for any

w √MK=

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

RAYLEIGH DAMPING

Nonlinear Analysis & Performance Based Design

DAMPING COEFFICIENT FROM HYSTERESIS

Nonlinear Analysis & Performance Based Design

DAMPING COEFFICIENT FROM HYSTERESIS

Nonlinear Analysis & Performance Based Design

A BIG THANK YOU!!!