Ashraf Pbd Seminar

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 FRAMES REDUCED BEAM SECTIONS LINK BEAMS IN ECCENTRICALLY BRACED FRAMES BUCKLING RESISTANT BRACES AS FUSES RUBBER-LEAD BASE ISOLATORS HINGED BRIDGE COLUMNS HINGES AT THE BASE LEVEL OF SHEAR WALLS ROCKING FOUNDATIONS OVERDESIGNED COUPLING BEAMS OTHER SACRIFICIAL ELEMENTS


STRUCTURAL COMPONENTS


MOMENT ROTATION RELATIONSHIP


IDEALIZED MOMENT ROTATION


PERFORMANCE LEVELS


IDEALIZED FORCE DEFORMATION CURVE


ASCE 41 BEAM MODEL


17

ASCE 41 ASSESSMENT OPTIONS •

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)


STRENGTH vs DEFORMATION ELASTIC STRENGTH DESIGN - KEY STEPS CHOSE DESIGN CODE AND EARTHQUAKE LOADS DESIGN CHECK PARAMETERS STRESS/BEAM MOMENT GET 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 41 DEMAND CAPACITY MEASURES– DRIFT/HINGE ROTATION/SHEAR GET DEFORMATION AND FORCE CAPACITIES CALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH) CALCULATE D/C RATIOS– LIMIT STATES


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.


F-D RELATIONSHIP

Force (F) Strain Hardening

Ultimate strength Ductile limit Strength loss

First yield

Residual strength Complete failure

Initially linear

Deformation (D) Hysteresis loop

Stiffness, strength and ductile limit may all degrade under cyclic deformation


DUCTILITY

LATERAL LOAD

Brittle

Partially Ductile

Ductile

DRIFT Nonlinear Analysis & Performance Based Design

ASCE 41 - DUCTILE AND BRITTLE


FORCE AND DEFORMATION CONTROL


BACKBONE CURVE

F Hysteresis loops from experiment.

ono onc - rea ons p

Relationship allowing for cyclic deformation

D


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.


BEAM END ROTATION MODEL Mi or M j Mi

i

j

Mj or 



Zero length hinges

i

M

j

Elastic  Plastic (hinge) 

Elastic beam Total rotation Plastic rotation

Elastic rotation

Similar at this end

Current


 

PLASTIC HINGE MODEL

•

•

It is assumed that all inelastic deformation is concentrated in zerolength 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 zone length).


plastic

ASCE 41 CHORD ROTATION CAPACITIES

Steel Beam

IO

LS

CP

p / y = 1

p / y = 6

p /  y = 8

p = 0.01 p = 0.005

p = 0.02

p = 0.025

p = 0.01

p = 0.02

RC Beam Low shear High shear


COLUMN AXIAL-BENDING MODEL

V

Mj

P

P or M V

Mi P


M

STEEL COLUMN AXIAL-BENDING


CONCRETE COLUMN AXIAL-BENDING


SHEAR HINGE MODEL

Node Elastic beam

Zero-length shear "hinge"


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


WALL FIBER MODEL



Steel fibers



Concrete fibers






ALTERNATIVE MEASURE - STRAIN


NONLINEAR SOLUTION SCHEMES

ƒ

iteration

2

1

iteration

ƒ 1

∆

2 3 4 56

∆

ƒ

ƒ

∆u

NEWTON – RAPHSON ITERATION

u

∆u

CONSTANT STIFFNESS ITERATION


u

CIRCULAR FREQUENCY +Y

Y 0

Radius, R

-Y


THE D, V & A RELATIONSHIP u .

Slope = ut1

t1

t

. u .. Slope = u t1

u. t1 t1

t

t1

t

.. u .. ut1


UNDAMPED FREE VIBRATION

&& ku mu ut

0

= u0 cos(wt)

where

w=

k m


RESPONSE MAXIMA ut

&t u &&t u && max u

=

u0

cos(wt)

u

= -w

t 0

sin(w )

= -w 2u0 cos(wt) = -w 2u max


BASIC DYNAMICS WITH DAMPING

&& Mu t && Mu

&& u

2

Cu& Ku Cu& Ku

u&

2

u

0

&& Mu g && u g

M C K

&& u g


RESPONSE FROM GROUND MOTION

&u& + 2 x wu& + w2u = A + Bt = -&u&

g

.. ug .. u g2

2

t1 .. u g1

t2 Time t

1


DAMPED RESPONSE &t = u

e

+

ut =

- xwt

&t { [u 1

B

w2

] cos wd t

1 & t + B )] sin wd t } + B [A - w2ut - xw(u 1 1 wd w2 w2

e-

xw t { [ut

1

- A2 + 2x3B] cos wd t w w

x A B(2x2 - 1) 1 & + xwut + + [u ] sin wd t } 1 wd t 1 w w2 A 2xB Bt +[ + ] 2 3 w w w2


SDOF DAMPED RESPONSE


RESPONSE SPECTRUM GENERATION Earthquake Record 0.40

g , C 0.20 C A D 0.00 N U O-0.20 R G

16

-0.40 0.00

. 4.00 n i , 2.00 L P 0.00 S I D-2.00

2.00

3.00 4.00 TIME, SECONDS

5.00

6.00

L 4 A P IS 2 D

T= 0.6 sec

0 0

2

4

6

8

10

PERIOD, Seconds

-4.00 0.00 . 8.00 n i , L 4.00 P S I 0.00 D-4.00

1.00

s14 e h c12 n i , T10 N E8 M E C6

1.00

2.00

3.00

4.00

5.00

6.00

T= 2.0 sec

-8.00 0.00

Displacement Response Spectrum 5% damping 1.00

2.00

3.00

4.00

5.00

6.00


SPECTRAL PARAMETERS 16 . in , T 12 N E M E 8 C A L P 4 S I D

PSV PSa

Sd PSv

0 0

2

4

6

8

10

PERIOD, sec 40

1.00

c e s / 30 in , Y T I 20 C O L E 10 V

g , 0.80 N O I T 0.60 A R E L 0.40 E C C A 0.20

0

0.00 0

2

4

6

PERIOD, sec

8

10

0

2

4

6

PERIOD, sec


8

10

THE ADRS SPECTRUM

ADRS Curve a S , n o i t a r

a S , n o i t a r e l e c c A l a r t c e p S

RS Curve s d n o c e S .5 0

s d n o c e S .0 1

s d n o c e S .0 2

Period, T

l e e c c A l a r t c e p S

Spectral Displacement, Sd


THE ADRS SPECTRUM


ASCE 7 RESPONSE SPECTRUM


PUSHOVER


THE LINEAR PUSHOVER


EQUIVALENT LINEARIZATION How far to push? The Target Point!


DISPLACEMENT MODIFICATION


DISPLACEMENT MODIFICATION

Calculating the Target Displacement

d=C

2

C C S T 0

1

2

a

e

2

/ (4p )

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


LOAD CONTROL AND DISPLACEMENT CONTROL


P-DELTA ANALYSIS


P-DELTA DEGRADES STRENGTH


P-DELTA EFFECTS ON F-D CURVES

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

STRUCTURE STRENGTH

P- effects will reduce the drift at which the structure loses strength. The "over-ductility" is reduced, and is more uncertain.

DRIFT


THE FAST NONLINEAR ANALYSIS METHOD (FNA)

NON LINEAR FRAME AND SHEAR WALL HINGES BASE ISOLATORS (RUBBER & FRICTION) STRUCTURAL DAMPERS STRUCTURAL UPLIFT STRUCTURAL POUNDING BUCKLING 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.


ARTIFICIAL EARTHQUAKES

CREATING HISTORIES TO MATCH A SPECTRUM FREQUENCY CONTENTS OF EARTHQUAKES FOURIER TRANSFORMS


ARTIFICIAL EARTHQUAKES


MESH REFINEMENT

71


APPROXIMATING BENDING BEHAVIOR

This is for a coupling beam. A slender pier is similar. 72


ACCURACY OF MESH REFINEMENT

73


PIER / SPANDREL MODELS

74


STRAIN & ROTATION MEASURES

75


STRAIN CONCENTRATION STUDY

Compare calculated strains and rotations for the 3 cases. 76


STRAIN CONCENTRATION STUDY

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%

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.

77


DISCONTINUOUS SHEAR WALLS


PIER AND SPANDREL FIBER MODELS

Vertical and horizontal fiber models

79


RAYLEIGH DAMPING

•

•

•

The aM dampers connect the masses to the ground. They exert external damping forces. Units of a are 1/T. The bK dampers act in parallel with the elements. They exert internal damping forces. Units of b are T. The damping matrix is C = aM + bK.

80


RAYLEIGH DAMPING

•

•

• •

For linear analysis, the coefficientsa 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. 81


RAYLEIGH DAMPING

M &u& t

& Cu

M &u& t

Ku

& K u

M

&u&

Cu & M

&u&

2

K u M

u &

Ku

0

0 2

C 2M

2

0

u

C 2M √K M

0

; 2 M C

2 √K M

C M 2 √K M

2 Nonlinear Analysis & Performance Based Design

K 2 √K M

RAYLEIGH DAMPING

Higher Modes (high w) = b Lower Modes (low w) = a To get z from a & b for any

√MK

;

T=

To get a & b from two values of z & z 1

z = 1

z = 2

a 2w

1

a 2w

2

bw 1 2 bw 2 2

Solve for a & b


2

2p w

DAMPING COEFFICIENT FROM HYSTERESIS


DAMPING COEFFICIENT FROM HYSTERESIS


A BIG THANK YOU!!!


Ashraf Pbd Seminar

Recommend Documents