ENERGY DIAGRAM
ENERGY DIAGRAM w/ HYSTERETIC
IMPLIED NONLINEAR BEHAVIOR
STEEL STRESS STRAIN RELATIONSHIPS
INELASTIC WORK DONE
HYSTERETIC BEHAVIOR
MOMENT ROTATION RELATIONSHIP
IDEALIZED MOMENT ROTATION
DUCTILITY LATERAL LOAD Brittle Partially Ductile Ductile DRIFT
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
PERFORMANCE LEVELS Restaurant Restaurant Restaurant Operational Immediate Occupancy Life Safety Collapse Prevention Less Damage More Damage Ref: FEMA 451 B
PERFORMANCE LEVELS
IDEALIZED FORCE DEFORMATION CURVE
ASCE 41 BEAM MODEL
ASCE 41 MOMENT HINGE
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
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)
PERFORMANCE PARAMETERS BRIDGE CATEGORIES LIFELINE BRIDGES MAJOR-ROUTE BRIDGES OTHER BRIDGES PERFORMANCE LEVELS SERVICE IMMEDIATE DAMAGE MINIMAL DAMAGE GROUND MOTION LEVELS PROBABILITY IN 50 YEARS 10% RETURN PERIOD 475 YEARS LIMITED REPAIRABLE DAMAGE 5% 975 YEARS SERVICE DISRUPTION EXTENSIVE DAMAGE 2% 2475 YEARS LIFE SAFETY PROBABLE REPLACEMENT
STRUCTURAL COMPONENTS
F-D RELATIONSHIP
DUCTILITY LATERAL LOAD Brittle Partially Ductile Ductile DRIFT
ASCE 41 DUCTILE AND BRITTLE
FORCE AND DEFORMATION CONTROL
BACKBONE CURVE
HYSTERESIS LOOP MODELS
STRENGTH AND 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.
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.
ASCE 41 ROTATION 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..
STEEL COLUMN AXIAL-BENDING
COLUMN AXIAL-BENDING MODEL
CONCRETE COLUMN AXIAL-BENDING
FEMA PMM HINGE
CONCRETE COLUMN FIBER HINGE MODEL Reinforced Concrete Column Steel Rebar Fibers Confined Concrete Fibers Unconfined Concrete Fibers
SHEAR WALL FIBER HINGE MODEL Reinforcement Layout Steel Fibers Confined Concrete Fibers Unconfined Concrete Fibers
MATERIAL STRESS-STRAIN CURVES Unconfined and Confined Concrete ( Compared ) Confined Concrete Steel
STRAIN AS PERFORMANCE MEASURE Strain Limit Fully confined concrete compressive strain 0.015 Unconfined concrete compressive strain 0.005 Rebar tensile strain 0.05 Rebar compressive strain 0.02
PIER AND SPANDREL FIBER MODELS
BRIDGE SECTIONS AND FIBER MODELS B R I D G E C R O S S S E C T I O N S F I B E R M O D E L S OF C R O S S S E C T I O N S
SHEAR HINGE MODEL
PANEL ZONE ELEMENT
NONLINEAR SOLUTION SCHEMES ƒ iteration 1 2 ƒ iteration 1 2 3 4 5 6 ƒ ƒ u u u u NEWTON RAPHSON ITERATION CONSTANT STIFFNESS ITERATION
THE POWER OF RITZ VECTORS APPROXIMATELY THREE TIMES FASTER THAN THE CALCULATION OF EXACT EIGENVECTORS IMPROVED ACCURACY WITH A SMALLER NUMBER OF VECTORS CAN BE USED FOR NONLINEAR ANALYSIS TO CAPTURE LOCAL RESPONSE
FAST NONLINEAR ANALYSIS (FNA) DISCRETE NONLINEARITY FRAME AND SHEAR WALL HINGES BASE ISOLATORS (RUBBER & FRICTION) STRUCTURAL DAMPERS STRUCTURAL UPLIFT STRUCTURAL POUNDING BUCKLING RESTRAINED BRACES
RITZ VECTORS
FNA ADVANTAGES MODAL SOLUTION - NO STIFFNESS REDUCTION CLOSED FORM SOLUTION VERY FAST TIME STEP INDEPENDENT CAPTURES HIGH FREQUENCY RESPONSE RITZ VECTORS CALCULATED ONCE MULTIPLE TIME HISTORIES ARE FAST
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.
DYNAMIC EQUILIBRIUM EQUATIONS.. M u.. M u t +. C u + Ku = 0 +. C u + Ku = - u.. + x w. 2 2 u + w u = - M u g.. u g.. M K.. u g C
RESPONSE FROM GROUND MOTION. u.. + 2xwu + w 2 u = A + B t = - u.. g.. ug.. u g 2 2 t 1.. u g 1 1 t 2 t
CLOSED FORM DAMPED RESPONSE. t e { [ u. B u t = - ] cos - xw t1 2 w 1 [ (. B + A - 2 w ut - xw ut + )] sin wd t } + w w d - t A B u t = e xw 2x { [u t - + ] cos 1 2 3 w w w d t 1 1 2 2 1 xa B x - + [ u. ( 2 1) t + xwu t - + ] sin wd t } w 1 1 d w 2 w A 2xB Bt + [ - + ] 2 3 2 w w w w d t 2 B w
UNDAMPED RESPONSE ) ( ] [ Bt A t sin B u u t 1 t + + - + = 2 2 1 1 w w w w. ] [ t cos A u 1 t - 2 2 2 2 1 B t sin u A t cos B u u 1 1 t t t + - + - = ] [ ] [. w w w w w w w w
STEP BY STEP DYNAMIC ANALYSIS Ground Accn = u g K C Effective load = M R = -Mu g Displ = u Veloc = u Accn = u At any point in time, dynamic equilibrium is : Mu + Cu + Ku = R Over a time step, Dt, dynamic equilibrium is : M Du + C Du + K Du = DR This equation can be solved by step-by-step methods. There is one equation with three unknowns (Du, Du, Du), so assumptions must be made and the solution is approximate.
STEP-BY-STEP INTEGRATION (CAA) u 1 u 0 u 0 u 0 u 1 u 0 R 1 R 0 Dt Du Du Du DR Equilibrium : M Du + C Du + K Du = DR From Kinematics : Du = Dt (u + u ) = Dt 2 (2u + Du ) 0 1 0 2 Du = Dt (u + u ) = Dt 2 (2u + Du ) 0 1 0 2 Hence get effective stiffness and load : K 4 Dt M + 2 eff = 2 C + K Dt DR 4 eff = -Mu g + M(2u + u 0 ) + 2C Dt 0 u 0 Solve K eff Du = Then : DR eff Du = -2u + 2 0 Du Dt Du = -2u + 2 Du 0 Dt
BASIC DYNAMICS WITH DAMPING Mu&& + Cu& + Ku = 0 t Mu&& + Cu& + Ku = - Mu&& g u & + xwu& + w 2 2 u = -u& g M K C u& & g
RESPONSE MAXIMA u = t u cos( w t) 0 u& t = -w u sin( w t 0 ) u&& t = -w 2 u cos( w t) 0 u& & max = -w 2 u max
DISPL, in. DISPL, in. GROUND ACC, g DISPLACEMENT, inches RESPONSE SPECTRUM GENERATION 0.40 Earthquake Record 16 0.20 14 0.00-0.20-0.40 0.00 1.00 2.00 3.00 4.00 5.00 6.00 TIME, SECONDS 12 10 8 6 4.00 T= 0.6 sec 4 2 2.00 0.00-2.00-4.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 T= 2.0 sec 8.00 4.00 0.00-4.00-8.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0 0 2 4 6 8 10 PERIOD, Seconds Displacement Response Spectrum 5% damping
VELOCITY, in/sec ACCELERATION, g DISPLACEMENT, in. SPECTRAL PARAMETERS 16 12 8 4 PS PS = = w w V S d a PS v 0 0 2 4 6 8 10 PERIOD, sec 40 1.00 30 0.80 0.60 20 0.40 10 0.20 0 0.00 0 2 4 6 8 10 0 2 4 6 8 10 PERIOD, sec PERIOD, sec
Spectral Acceleration, Sa Spectral Acceleration, Sa 0.5 Seconds 1.0 Seconds 2.0 Seconds THE ADRS SPECTRUM RS Curve ADRS Curve Period, T Spectral Displacement, Sd
THE ADRS SPECTRUM
ASCE 7 RESPONSE SPECTRUM
PUSHOVER
THE LINEAR PUSHOVER
EQUIVALENT LINEARIZATION How far to push? The Target Point!
DAMPING COEFICIENT FROM HYSTERESIS
DAMPING COEFICIENT FROM HYSTERESIS
DISPLACEMENT MODFICATION Calculating the Target Displacement d = C C C S T 2 / (4p 2 ) 0 1 2 a e C 0 Relates spectral to roof displacement C 1 Modifier for inelastic displacement C 2 Modifier for hysteresis loop shape
ARTIFICIAL EARTHQUAKES CREATING HISTORIES TO MATCH A SPECTRUM FREQUENCY CONTENTS OF EARTHQUAKES FOURIER TRANSFORMS
MATCHING THE SPECTRUM
FOURIER TRANSFORMS
ENERGY DISSIPATION DEVICES Friction Isolator Rubber Isolator Oil Damper Friction Damper Buckling-Restrained Brace (BRB)
RATING FOR SEISMIC PERFORMANCE CoRE Rating Safety Reparability Functionality 5-Star Life Safe Loss <5% 4-Star Life Safe Loss <10% 3-star Life Safe Loss <20% Occupiable Immediately Functional < 72 hours Occupiable Immediately Functional < 1 month Occupiable < 1 month Functional < 6 months Certified Life Safe Not estimated Not estimated Not Certified Life Safety Hazard Not estimated Not estimated
DAMAGE ANALYSIS Servers/Network, 7% Computers, 6% Cabinets, 1% Bookcases, 1% Roof Equipment, 1% Cladding, 2% Partitions, 27% Elevators, 21% Moment Frame, 2% Ceiling, 32%
NONLINEAR ANALYSIS SURVEY