ENERGY DIAGRAM w/ HYSTERETIC

Transcription

1

2

3 ENERGY DIAGRAM

4 ENERGY DIAGRAM w/ HYSTERETIC

5 IMPLIED NONLINEAR BEHAVIOR

6 STEEL STRESS STRAIN RELATIONSHIPS

7 INELASTIC WORK DONE

8 HYSTERETIC BEHAVIOR

9 MOMENT ROTATION RELATIONSHIP

10 IDEALIZED MOMENT ROTATION

11 DUCTILITY LATERAL LOAD Brittle Partially Ductile Ductile DRIFT

12 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

13 PERFORMANCE LEVELS Restaurant Restaurant Restaurant Operational Immediate Occupancy Life Safety Collapse Prevention Less Damage More Damage Ref: FEMA 451 B

14 PERFORMANCE LEVELS

15 IDEALIZED FORCE DEFORMATION CURVE

16 ASCE 41 BEAM MODEL

17 ASCE 41 MOMENT HINGE

18 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

19 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)

20 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

21 STRUCTURAL COMPONENTS

22 F-D RELATIONSHIP

23 DUCTILITY LATERAL LOAD Brittle Partially Ductile Ductile DRIFT

24 ASCE 41 DUCTILE AND BRITTLE

25 FORCE AND DEFORMATION CONTROL

26 BACKBONE CURVE

27 HYSTERESIS LOOP MODELS

28 STRENGTH AND DEGRADATION

29 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.

30 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.

31 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..

32 STEEL COLUMN AXIAL-BENDING

33 COLUMN AXIAL-BENDING MODEL

34 CONCRETE COLUMN AXIAL-BENDING

35 FEMA PMM HINGE

36 CONCRETE COLUMN FIBER HINGE MODEL Reinforced Concrete Column Steel Rebar Fibers Confined Concrete Fibers Unconfined Concrete Fibers

37 SHEAR WALL FIBER HINGE MODEL Reinforcement Layout Steel Fibers Confined Concrete Fibers Unconfined Concrete Fibers

38 MATERIAL STRESS-STRAIN CURVES Unconfined and Confined Concrete ( Compared ) Confined Concrete Steel

39 STRAIN AS PERFORMANCE MEASURE Strain Limit Fully confined concrete compressive strain Unconfined concrete compressive strain Rebar tensile strain 0.05 Rebar compressive strain 0.02

40 PIER AND SPANDREL FIBER MODELS

41 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

42 SHEAR HINGE MODEL

43 PANEL ZONE ELEMENT

44 NONLINEAR SOLUTION SCHEMES ƒ iteration 1 2 ƒ iteration ƒ ƒ u u u u NEWTON RAPHSON ITERATION CONSTANT STIFFNESS ITERATION

45 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

46 FAST NONLINEAR ANALYSIS (FNA) DISCRETE NONLINEARITY FRAME AND SHEAR WALL HINGES BASE ISOLATORS (RUBBER & FRICTION) STRUCTURAL DAMPERS STRUCTURAL UPLIFT STRUCTURAL POUNDING BUCKLING RESTRAINED BRACES

47 RITZ VECTORS

48 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

49 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.

50 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

51 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

52 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 w w w d t xa B x - + [ u. ( 2 1) t + xwu t - + ] sin wd t } w 1 1 d w 2 w A 2xB Bt + [ - + ] w w w w d t 2 B w

53 UNDAMPED RESPONSE ) ( ] [ Bt A t sin B u u t 1 t = w w w w. ] [ t cos A u 1 t B t sin u A t cos B u u 1 1 t t t = ] [ ] [. w w w w w w w w

54 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.

55 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 ) Du = Dt (u + u ) = Dt 2 (2u + Du ) 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 Du Dt Du = -2u + 2 Du 0 Dt

56 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

57 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

58 DISPL, in. DISPL, in. GROUND ACC, g DISPLACEMENT, inches RESPONSE SPECTRUM GENERATION 0.40 Earthquake Record TIME, SECONDS T= 0.6 sec T= 2.0 sec PERIOD, Seconds Displacement Response Spectrum 5% damping

59 VELOCITY, in/sec ACCELERATION, g DISPLACEMENT, in. SPECTRAL PARAMETERS PS PS = = w w V S d a PS v PERIOD, sec PERIOD, sec PERIOD, sec

60 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

61 THE ADRS SPECTRUM

62 ASCE 7 RESPONSE SPECTRUM

63 PUSHOVER

64 THE LINEAR PUSHOVER

65 EQUIVALENT LINEARIZATION How far to push? The Target Point!

66 DAMPING COEFICIENT FROM HYSTERESIS

67 DAMPING COEFICIENT FROM HYSTERESIS

68 DISPLACEMENT MODFICATION Calculating the Target Displacement d = C C C S T 2 / (4p 2 ) a e C 0 Relates spectral to roof displacement C 1 Modifier for inelastic displacement C 2 Modifier for hysteresis loop shape

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

70 MATCHING THE SPECTRUM

71 FOURIER TRANSFORMS

72 ENERGY DISSIPATION DEVICES Friction Isolator Rubber Isolator Oil Damper Friction Damper Buckling-Restrained Brace (BRB)

73 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

74 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%

75 NONLINEAR ANALYSIS SURVEY

76