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Nonlinear Analysis & Performance Based Design

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CALCULATED vs MEASURED

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Nonlinear Analysis & Performance Based Design ENERGY DISSIPATION

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Nonlinear Analysis & Performance Based Design

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IMPLICIT NONLINEAR BEHAVIOR

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Nonlinear Analysis & Performance Based Design STEEL STRESS STRAIN RELATIONSHIPS

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Nonlinear Analysis & Performance Based Design INELASTIC WORK DONE!

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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

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Nonlinear Analysis & Performance Based Design STRUCTURAL COMPONENTS

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Nonlinear Analysis & Performance Based Design MOMENT ROTATION RELATIONSHIP

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Nonlinear Analysis & Performance Based Design IDEALIZED MOMENT ROTATION

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Nonlinear Analysis & Performance Based Design PERFORMANCE LEVELS

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Nonlinear Analysis & Performance Based Design IDEALIZED FORCE DEFORMATION CURVE

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Nonlinear Analysis & Performance Based Design 17 ASCE 41 BEAM MODEL

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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

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Nonlinear Analysis & Performance Based Design 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 STRENGTH vs DEFORMATION

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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.

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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.

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Nonlinear Analysis & Performance Based Design F-D RELATIONSHIP

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Nonlinear Analysis & Performance Based Design LATERAL LOAD Partially Ductile Brittle Ductile DRIFT DUCTILITY

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Nonlinear Analysis & Performance Based Design ASCE 41 - DUCTILE AND BRITTLE

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Nonlinear Analysis & Performance Based Design FORCE AND DEFORMATION CONTROL

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Nonlinear Analysis & Performance Based Design BACKBONE CURVE

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Nonlinear Analysis & Performance Based Design HYSTERESIS LOOP MODELS

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Nonlinear Analysis & Performance Based Design STRENGTH DEGRADATION

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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.

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Nonlinear Analysis & Performance Based Design BEAM END ROTATION MODEL

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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.

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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).

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Nonlinear Analysis & Performance Based Design ASCE 41 CHORD ROTATION CAPACITIES IOLSCP Steel Beam p / y = 1 p / y = 6 p / y = 8 RC Beam Low shear High shear p = 0.01 p = p = 0.02 p = 0.01 p = p = 0.02

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Nonlinear Analysis & Performance Based Design COLUMN AXIAL-BENDING MODEL

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Nonlinear Analysis & Performance Based Design STEEL COLUMN AXIAL-BENDING

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Nonlinear Analysis & Performance Based Design CONCRETE COLUMN AXIAL-BENDING

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Nonlinear Analysis & Performance Based Design SHEAR HINGE MODEL

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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).

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Nonlinear Analysis & Performance Based Design BUCKLING-RESTRAINED BRACE The BRB element includes isotropic hardening. The D-C measure is axial deformation.

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Nonlinear Analysis & Performance Based Design BEAM/COLUMN FIBER MODEL The cross section is represented by a number of uni-axial fibers. The M- relationship follows from the fiber properties, areas and locations. Stress-strain relationships reflect the effects of confinement

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Nonlinear Analysis & Performance Based Design WALL FIBER MODEL

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Nonlinear Analysis & Performance Based Design ALTERNATIVE MEASURE - STRAIN

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Nonlinear Analysis & Performance Based Design ƒ ƒ u u 1 2 iteration NEWTON – RAPHSON ITERATION ƒ ƒ u u 12 iteration CONSTANT STIFFNESS ITERATION NONLINEAR SOLUTION SCHEMES

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Nonlinear Analysis & Performance Based Design Y +Y -Y 0 Radius, R CIRCULAR FREQUENCY

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Nonlinear Analysis & Performance Based Design u. t u t1... Slope = u t1.. t1t1 u t1t1 t Slope = u t1. u u t1.. t1t1 t THE D, V & A RELATIONSHIP

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Nonlinear Analysis & Performance Based Design UNDAMPED FREE VIBRATION )cos( 0 tuu t 0 ukum m k where

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Nonlinear Analysis & Performance Based Design RESPONSE MAXIMA u t )cos( 0 tu )cos( 0 2 tu u t )sin( 0 tu u t max 2 u max u

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Nonlinear Analysis & Performance Based Design BASIC DYNAMICS WITH DAMPING g uuuu 2 2 g uMKuuCuM uC t uM 0 g u M K C

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Nonlinear Analysis & Performance Based Design ug.. 2 ug2ug2 ug1ug1 1 t1t1 t2t2 Time t ABtu g u u u 2 2 RESPONSE FROM GROUND MOTION

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Nonlinear Analysis & Performance Based Design DAMPED RESPONSE

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Nonlinear Analysis & Performance Based Design SDOF DAMPED RESPONSE

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Nonlinear Analysis & Performance Based Design RESPONSE SPECTRUM GENERATION TIME, SECONDS GROUND ACC, g Earthquake Record Displacement Response Spectrum 5% damping DISPL, in DISPL, in PERIOD, Seconds DISPLACEMENT, inches T= 0.6 sec T= 2.0 sec

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Nonlinear Analysis & Performance Based Design SPECTRAL PARAMETERS

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Nonlinear Analysis & Performance Based Design THE ADRS SPECTRUM ADRS Curve Spectral Displacement, Sd Spectral Acceleration, Sa 2.0 Seconds 1.0 Seconds 0.5 Seconds Period, T Spectral Acceleration, Sa RS Curve 0.5 Seconds1.0 Seconds2.0 Seconds

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Nonlinear Analysis & Performance Based Design THE ADRS SPECTRUM

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Nonlinear Analysis & Performance Based Design ASCE 7 RESPONSE SPECTRUM

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Nonlinear Analysis & Performance Based Design PUSHOVER

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Nonlinear Analysis & Performance Based Design THE LINEAR PUSHOVER

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Nonlinear Analysis & Performance Based Design EQUIVALENT LINEARIZATION How far to push? The Target Point!

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Nonlinear Analysis & Performance Based Design DISPLACEMENT MODIFICATION

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Nonlinear Analysis & Performance Based Design Calculating the Target Displacement DISPLACEMENT MODIFICATION C 0 Relates spectral to roof displacement C 1 Modifier for inelastic displacement C 2 Modifier for hysteresis loop shape C 0 C 1 C 2 S a T e 2 / 2

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Nonlinear Analysis & Performance Based Design LOAD CONTROL AND DISPLACEMENT CONTROL

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Nonlinear Analysis & Performance Based Design P-DELTA ANALYSIS

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Nonlinear Analysis & Performance Based Design P-DELTA DEGRADES STRENGTH

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Nonlinear Analysis & Performance Based Design P-DELTA EFFECTS ON F-D CURVES

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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

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Nonlinear Analysis & Performance Based Design RITZ VECTORS

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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.

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Nonlinear Analysis & Performance Based Design CREATING HISTORIES TO MATCH A SPECTRUM FREQUENCY CONTENTS OF EARTHQUAKES FOURIER TRANSFORMS ARTIFICIAL EARTHQUAKES

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Nonlinear Analysis & Performance Based Design ARTIFICIAL EARTHQUAKES

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Nonlinear Analysis & Performance Based Design 71 MESH REFINEMENT

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

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Nonlinear Analysis & Performance Based Design 73 ACCURACY OF MESH REFINEMENT

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Nonlinear Analysis & Performance Based Design 74 PIER / SPANDREL MODELS

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Nonlinear Analysis & Performance Based Design 75 STRAIN & ROTATION MEASURES

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Nonlinear Analysis & Performance Based Design 76 Compare calculated strains and rotations for the 3 cases. STRAIN CONCENTRATION STUDY

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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 12.32%2.39% 1.99% 22.32%3.66%2.36%1.97% 32.32%4.17%2.35%1.96% STRAIN CONCENTRATION STUDY

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Nonlinear Analysis & Performance Based Design DISCONTINUOUS SHEAR WALLS

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Nonlinear Analysis & Performance Based Design 79 PIER AND SPANDREL FIBER MODELS Vertical and horizontal fiber models

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

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Nonlinear Analysis & Performance Based Design 81 For linear analysis, the coefficients and can be chosen to give essentially constant damping over a range of mode periods, as shown. A possible method is as follows : Choose T B = 0.9 times the first mode period. Choose T A = 0.2 times the first mode period. Calculate and 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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Nonlinear Analysis & Performance Based Design KuuC t uM 0 Kuu 0 M K u C M u M u K 0 2 u u M C C M 2 C M M K 2 C M K 2 M K M K t uM u ; 2 M K RAYLEIGH DAMPING

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Nonlinear Analysis & Performance Based Design Higher Modes high Lower Modes low To get from & for any M K To get & from two values of ; Solve for & RAYLEIGH DAMPING

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Nonlinear Analysis & Performance Based Design DAMPING COEFFICIENT FROM HYSTERESIS

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Nonlinear Analysis & Performance Based Design DAMPING COEFFICIENT FROM HYSTERESIS

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Nonlinear Analysis & Performance Based Design A BIG THANK YOU!!!

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