calculated vs measured

Nonlinear Analysis & Performance Based Design


CALCULATED vs MEASURED


ENERGY DISSIPATION


IMPLICIT NONLINEAR BEHAVIOR


STEEL STRESS STRAIN RELATIONSHIPS


INELASTIC WORK DONE!


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


STRUCTURAL COMPONENTS


MOMENT ROTATION RELATIONSHIP


IDEALIZED MOMENT ROTATION


PERFORMANCE LEVELS


IDEALIZED FORCE DEFORMATION CURVE

Nonlinear Analysis & Performance Based Design 17

ASCE 41 BEAM MODEL


• 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


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


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.


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.


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


LATERALLOAD

Partially DuctileBrittle Ductile

DRIFT

DUCTILITY


ASCE 41 - DUCTILE AND BRITTLE


FORCE AND DEFORMATION CONTROL


F

R elationship a llow ingfor cyclic deform ation

M onotonic F-D re lationship

H ysteresis loopsfrom experim ent.

D

BACKBONE CURVE


HYSTERESIS LOOP MODELS


STRENGTH DEGRADATION


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.


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


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

• 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

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


COLUMN AXIAL-BENDING MODEL

M i

P

PP

P

or

V

V

M j

MM


STEEL COLUMN AXIAL-BENDING


CONCRETE COLUMN AXIAL-BENDING


N odeZero-lengthshear "h inge"E lastic beam

SHEAR HINGE MODEL


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

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


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


Steelfibers

C oncrete

fibers

WALL FIBER MODEL


ALTERNATIVE MEASURE - STRAIN


∆ƒ

ƒ

∆ u u

1 2

iteration

NEWTON – RAPHSONITERATION

∆ƒ

ƒ

∆ u u

1 2

iteration

CONSTANT STIFFNESSITERATION

3 4 56

NONLINEAR SOLUTION SCHEMES


Y

+Y

-Y

0 Radius, R

q

CIRCULAR FREQUENCY


u.

t

ut1

.

..

Slope = ut1

..

t1

u

t1 t

Slope = ut1

.

u

ut1

..

t1t

THE D, V & A RELATIONSHIP


UNDAMPED FREE VIBRATION

)cos(0 tuut w=

0=+ ukum&&

mk

where =w


RESPONSE MAXIMA

ut )cos(0 tu w=

)cos(02 tu ww-=ut&&

)sin(0 tu ww-=ut&

max2uw-=maxu&&


BASIC DYNAMICS WITH DAMPING

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

guMKuuCuM

KuuCt

uM

&&&&&

&&&

-=++

=++ 0

gu&&

M

K

C


ug..

2ug2

..

ug1

..

1

t1 t2

Time t

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

RESPONSE FROM GROUND MOTION


{ [& ] 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


SDOF DAMPED RESPONSE


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


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


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


THE ADRS SPECTRUM


ASCE 7 RESPONSE SPECTRUM


PUSHOVER


THE LINEAR PUSHOVER


EQUIVALENT LINEARIZATION

How far to push? The Target Point!


DISPLACEMENT MODIFICATION


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)


LOAD CONTROL AND DISPLACEMENT CONTROL


P-DELTA ANALYSIS


P-DELTA DEGRADES STRENGTH


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


THE FAST NONLINEAR ANALYSIS METHOD (FNA)

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

STRUCTURAL DAMPERSSTRUCTURAL UPLIFT

STRUCTURAL POUNDINGBUCKLING RESTRAINED BRACES


RITZ VECTORS


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.


CREATING HISTORIES TO MATCH A SPECTRUM

FREQUENCY CONTENTS OF EARTHQUAKES

FOURIER TRANSFORMS

ARTIFICIAL EARTHQUAKES


ARTIFICIAL EARTHQUAKES


MESH REFINEMENT


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

APPROXIMATING BENDING BEHAVIOR


ACCURACY OF MESH REFINEMENT


PIER / SPANDREL MODELS


STRAIN & ROTATION MEASURES


Compare calculated strains and rotations for the 3 cases.

STRAIN CONCENTRATION STUDY


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


DISCONTINUOUS SHEAR WALLS


PIER AND SPANDREL FIBER MODELS

Vertical and horizontal fiber models


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.


• 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


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


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


DAMPING COEFFICIENT FROM HYSTERESIS


A BIG THANK YOU!!!

calculated vs measured

Documents

structural analysis

design loads asce

component fd relationships

corresponding deformation

structure fd relationship

braced beams

component type

symmetrical beam