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 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 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 ASCE 41 BEAM MODEL Nonlinear Analysis & Performance Based Design 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) Nonlinear Analysis & Performance Based Design 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 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 F-D RELATIONSHIP F o rc e (F ) S tr a in H a r d e n in g U ltim a te s tre n g th D u c tile lim it S tr e n g th lo s s R e s id u a l s tr e n g th

F ir s t y ie ld In itia lly lin e a r C o m p le te fa ilu r e D e fo r m a tio n ( D ) H y s te r e s is lo o p S tiffn e s s , s tr e n g th a n d d u c tile lim it m a y a ll d e g r a d e u n d e r c y c lic d e fo r m a tio n Nonlinear Analysis & Performance Based Design

DUCTILITY LATERAL LOAD Brittle Partially Ductile Ductile DRIFT Nonlinear Analysis & Performance Based Design ASCE 41 - DUCTILE AND BRITTLE

Nonlinear Analysis & Performance Based Design FORCE AND DEFORMATION CONTROL Nonlinear Analysis & Performance Based Design BACKBONE CURVE F H y s te r e s is lo o p s fr o m e x p e r im e n t. M o n o t o n ic F - D r e la tio n s h ip R e la t io n s h ip a llo w in g

fo r c y c lic d e fo r m a t io n D 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 BEAM END ROTATION MODEL M ior M M

i j i M j i o r j Z e r o le n g t h h in g e s M

E la s tic b e a m T o ta l r o ta t io n P la s tic r o ta t io n E la s tic r o ta tio n S im ila r a t th is e n d j E la s tic

P la s tic ( h in g e ) C u rre n t Nonlinear Analysis & Performance Based Design 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. 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 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

Nonlinear Analysis & Performance Based Design COLUMN AXIAL-BENDING MODEL P V M j P P M V

or M i P Nonlinear Analysis & Performance Based Design M STEEL COLUMN AXIAL-BENDING Nonlinear Analysis & Performance Based Design CONCRETE COLUMN AXIAL-BENDING

Nonlinear Analysis & Performance Based Design SHEAR HINGE MODEL N ode E la s tic b e a m Z e r o - le n g th s h e a r " h in g e " 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 WALL FIBER MODEL S te e l fib e r s C o n c re te

fib e r s Nonlinear Analysis & Performance Based Design ALTERNATIVE MEASURE - STRAIN Nonlinear Analysis & Performance Based Design NONLINEAR SOLUTION SCHEMES

iteration 2 1 iteration 1 2 3 4 56

u NEWTON RAPHSON ITERATION u u CONSTANT STIFFNESS ITERATION Nonlinear Analysis & Performance Based Design u CIRCULAR FREQUENCY

+Y Y 0 Radius, R -Y Nonlinear Analysis & Performance Based Design THE D, V & A RELATIONSHIP u

. Slope = ut1 t1 . u . ut1 .. u .. ut1 t

.. Slope = ut1 t1 t t1 t Nonlinear Analysis & Performance Based Design UNDAMPED FREE VIBRATION

&+ ku = 0 m& u ut = u0 cos(wt) k where w = m Nonlinear Analysis & Performance Based Design RESPONSE MAXIMA ut = u0 cos(wt) & ut = - wu0 sin(wt) 2

= w & & ut u0 cos(wt) 2 = w & u&max u max Nonlinear Analysis & Performance Based Design

BASIC DYNAMICS WITH DAMPING & + C& M& u u + Ku = 0 t & + C& & M& u u + Ku = - M& u g 2 &

& & + x w + w & u 2 u u = -& u g M C

K & & u g Nonlinear Analysis & Performance Based Design RESPONSE FROM GROUND MOTION & &+ 2xw& & u u u + w u = A + Bt = -&

g 2 .. ug 2 .. ug 2 t1 .. ug1 t2

Time t 1 Nonlinear Analysis & Performance Based Design DAMPED RESPONSE & ut = e + ut = - xwt { [&

ut 1 - B ] cos wd t 2 w 1 B B 2

& w xw + w + [A ut (ut )] sin d t } 2 1 1 wd w

w2 e - xwt { [ut 1 A 2xB + ] cos wd t 2

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

w +[ A w 2 - 2xB 3 w

+ Bt 2 w ] Nonlinear Analysis & Performance Based Design SDOF DAMPED RESPONSE Nonlinear Analysis & Performance Based Design

RESPONSE SPECTRUM GENERATION Earthquake Record 0.20 16 0.00 -0.20 DISPL, in. -0.40 0.00

4.00 1.00 2.00 3.00 4.00 TIME, SECONDS 5.00 6.00 T= 0.6 sec

2.00 0.00 DISPLACEMENT, inches GROUND ACC, g 0.40 14 12 10 8 6

4 2 0 0 -2.00 -4.00 0.00 2 4 6

8 10 PERIOD, Seconds 1.00 2.00 3.00 4.00 5.00

6.00 DISPL, in. T= 2.0 sec 8.00 4.00 0.00 -4.00 -8.00 0.00 Displacement Response Spectrum

5% damping 1.00 2.00 3.00 4.00 5.00 6.00 Nonlinear Analysis & Performance Based Design

SPECTRAL PARAMETERS DISPLACEMENT, in. 16 12 PSV = w Sd PSa = w PSv 8 4 0 0 2

4 6 8 10 PERIOD, sec 1.00 ACCELERATION, g VELOCITY, in/sec

40 30 20 10 0 0.80 0.60 0.40 0.20 0.00 0

2 4 6 PERIOD, sec 8 10 0 2

4 6 PERIOD, sec Nonlinear Analysis & Performance Based Design 8 10 0.5 Se conds

Spectral Acceleration, Sa 1. s nd o ec S 0 2.0 Seconds 1.0 Seconds

RS Curve 0.5 Seconds ADRS Curve Spectral Acceleration, Sa THE ADRS SPECTRUM Period, T nds 2.0 Seco

Spectral Displacement, Sd 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 DISPLACEMENT MODIFICATION

Calculating the Target Displacement 2 2 d=C0 C1 C2 Sa Te / (4p ) C0 Relates spectral to roof displacement C1 Modifier for inelastic displacement C2 Modifier for hysteresis loop shape 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 P-DELTA EFFECTS ON F-D CURVES STR U C TUR E STR EN G TH

P - e ffe c ts m a y r e d u c e th e d r ift a t w h ic h th e " w o r s t" c o m p o n e n t r e a c h e s its d u c tile lim it. P - e ffe c ts w ill r e d u c e th e d r ift a t w h ic h t h e s t r u c t u r e lo s e s s t r e n g t h . T h e " o v e r - d u c t ilit y " is r e d u c e d , a n d is m o r e u n c e r ta in . D R IF T Nonlinear Analysis & Performance Based Design 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 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 ARTIFICIAL EARTHQUAKES CREATING HISTORIES TO MATCH A SPECTRUM FREQUENCY CONTENTS OF EARTHQUAKES FOURIER TRANSFORMS Nonlinear Analysis & Performance Based Design ARTIFICIAL EARTHQUAKES

Nonlinear Analysis & Performance Based Design MESH REFINEMENT Nonlinear Analysis & Performance Based Design 71 APPROXIMATING BENDING BEHAVIOR This is for a coupling beam. A slender pier is similar. Nonlinear Analysis & Performance Based Design 72

ACCURACY OF MESH REFINEMENT Nonlinear Analysis & Performance Based Design 73 PIER / SPANDREL MODELS Nonlinear Analysis & Performance Based Design 74 STRAIN & ROTATION MEASURES

Nonlinear Analysis & Performance Based Design 75 STRAIN CONCENTRATION STUDY Compare calculated strains and rotations for the 3 cases. Nonlinear Analysis & Performance Based Design 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.

Nonlinear Analysis & Performance Based Design 77 DISCONTINUOUS SHEAR WALLS Nonlinear Analysis & Performance Based Design PIER AND SPANDREL FIBER MODELS Vertical and horizontal fiber models Nonlinear Analysis & Performance Based Design

79 RAYLEIGH DAMPING The aM dampers connect the masses to the ground. They exert external damping forces. Units of aare 1/T. ThebK dampers act in parallel with the elements. They exert internal damping forces. Units of bare T. The damping matrix is C = aM + bK. Nonlinear Analysis & Performance Based Design 80 RAYLEIGH DAMPING

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. Nonlinear Analysis & Performance Based Design 81 RAYLEIGH DAMPING & M& ut + C& u + Ku = 0 & M&

ut + (a M + bK )& u + Ku = 0 K C & & & u+ u+ u = 0 M M &

& u + 2x w& u + w2u = 0 ; 2x M w = C b K C a M x= C = C = = + M

w 2 2M K 2 K M KM KM 2 2 M x = aw + bw 2 2

Nonlinear Analysis & Performance Based Design RAYLEIGH DAMPING Higher Modes (high w)=b Lower Modes (low w)=a K To get zfrom a & b for w= M ; T = 2p w any To geta & b from two values of z1&z2 z1 = a + bw1 2

2w 1 z2 = a + bw2 2 2w Solve fora & b 2 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!!! Nonlinear Analysis & Performance Based Design