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CALCULATED vs MEASURED
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ENERGY DISSIPATION
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IMPLICIT NONLINEAR BEHAVIOR
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STEEL STRESS STRAIN RELATIONSHIPS
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INELASTIC WORK DONE!
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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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STRUCTURAL COMPONENTS
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MOMENT ROTATION RELATIONSHIP
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IDEALIZED MOMENT ROTATION
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PERFORMANCE LEVELS
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IDEALIZED FORCE DEFORMATION CURVE
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ASCE 41 BEAM MODEL
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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)
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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
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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
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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
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F-D RELATIONSHIP Force (F) Strain Hardening
Ultimate strength Ductile limit Strength loss Residual strength
First yield Initially linear
Complete failure
Deformation (D) Hysteresis loop
Stiffness, strength and ductile limit may all degrade under cyclic deformation
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DUCTILITY
LATERAL LOAD
Brittle
Partially Ductile
Ductile
DRIFT Nonlinear Analysis & Performance Based Design
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ASCE 41 - DUCTILE AND BRITTLE
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FORCE AND DEFORMATION CONTROL
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BACKBONE CURVE
F
Monotonic F-D relationship
Hysteresis loops from experiment.
Relationship allowing for cyclic deformation
D
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HYSTERESIS LOOP MODELS
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STRENGTH DEGRADATION
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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
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BEAM END ROTATION MODEL
Mi
Mi or Mj
θj
θi
Mj θi or θj
Zero length hinges M
Elastic θ Plastic (hinge) θ
Elastic beam Total rotation Plastic rotation
Elastic rotation
Similar at this end
Current θ
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θ
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. Nonlinear Analysis & Performance Based Design
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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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ASCE 41 CHORD ROTATION CAPACITIES
IO
LS
CP
Steel Beam
θp/θy = 1
θp/θy = 6
θp/θy = 8
RC Beam Low shear High shear
θp = 0.01 θp = 0.005
θp = 0.02 θp = 0.01
θp = 0.025 θp = 0.02
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COLUMN AXIAL‐BENDING MODEL
Mj
P V P
P or M V
Mi P
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M
STEEL COLUMN AXIAL‐BENDING
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CONCRETE COLUMN AXIAL‐BENDING
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SHEAR HINGE MODEL
Node Elastic beam
Zero-length shear "hinge"
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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
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BUCKLING-RESTRAINED BRACE
The BRB element includes “isotropic” hardening. The D‐C measure is axial deformation.
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BEAM/COLUMN FIBER MODEL
The cross section is represented by a number of uni‐axial fibers. The M‐ψ relationship follows from the fiber properties, areas and locations. Stress‐strain relationships reflect the effects of confinement
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WALL FIBER MODEL
σ Steel fibers
ε σ
Concrete fibers
ε
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ALTERNATIVE MEASURE - STRAIN
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NONLINEAR SOLUTION SCHEMES
ƒ
iteration 2
1
iteration
ƒ
∆ƒ
1 2 3 4 56
∆ƒ
∆u
∆u
u
NEWTON – RAPHSON ITERATION
CONSTANT STIFFNESS ITERATION
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u
CIRCULAR FREQUENCY +Y
Y
θ 0
Radius, R
-Y
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THE D, V & A RELATIONSHIP u .
Slope = ut1
t1 . u . ut1
t
.. Slope = ut1
t1
t
t1
t
.. u .. ut1
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UNDAMPED FREE VIBRATION
&& + ku = 0 mu ut = u0 cos(ωt) k where ω = m Nonlinear Analysis & Performance Based Design
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RESPONSE MAXIMA ut = u0 cos(ωt) u&t = −ωu0 sin(ωt) u&&t = −ω u0 cos(ωt) 2
&u& max = −ω 2u max Nonlinear Analysis & Performance Based Design
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BASIC DYNAMICS WITH DAMPING && + Cu& + Ku = 0 Mu t && + Cu& + Ku = − Mu && Mu g &u& + 2ξωu& + ω2u = −u && g
M C K
&& u g Nonlinear Analysis & Performance Based Design
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RESPONSE FROM GROUND MOTION && ug u + 2ξωu& + ω u = A + Bt = −&& 2
.. ug 2
.. ug2
t1 .. ug1
t2 Time t
1
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DAMPED RESPONSE u& t =
e +
ut =
− ξωt
{ [u& t − 1
B ] cos ωd t 2 ω
1 B B [A − ω2ut − ξω(u& t + 2 )] sin ωd t } + 2 1 1 ωd ω ω
e
− ξωt
{ [ut
1
A 2ξB − + ] cos ωd t 2 3 ω ω
ξ A B(2ξ2 − 1) 1 ωd t } + + [u& t + ξωut − ] sin 1 1 ωd ω ω2 A 2ξB Bt +[ − + ] 2 3 2 ω ω ω
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SDOF DAMPED RESPONSE
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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
DISPLACEMENT, inches
GROUND ACC, g
0.40
2.00 0.00
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
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SPECTRAL PARAMETERS DISPLACEMENT, in.
16
PSV = ω Sd PSa = ω PSv
12
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
8
10
0
PERIOD, sec
2
4
6
PERIOD, sec
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8
10
Spectral Acceleration, Sa
2.0 Seconds
1.0 Seconds
RS Curve 0.5 Seconds
ADRS Curve
Spectral Acceleration, Sa
THE ADRS SPECTRUM
Period, T
Spectral Displacement, Sd
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THE ADRS SPECTRUM
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ASCE 7 RESPONSE SPECTRUM
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PUSHOVER
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THE LINEAR PUSHOVER
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EQUIVALENT LINEARIZATION How far to push? The Target Point!
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DISPLACEMENT MODIFICATION
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DISPLACEMENT MODIFICATION
Calculating the Target Displacement
δ=
2 C0 C1 C2 Sa Te
/ (4π ) 2
C0 Relates spectral to roof displacement C1 Modifier for inelastic displacement C2 Modifier for hysteresis loop shape
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LOAD CONTROL AND DISPLACEMENT CONTROL
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P-DELTA ANALYSIS
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P-DELTA DEGRADES STRENGTH
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P-DELTA EFFECTS ON F-D CURVES
P-Δ effects may reduce the drift at which the "worst" component reaches its ductile limit.
STRUCTURE STRENGTH
P-Δ effects will reduce the drift at which the structure loses strength. The "over-ductility" is reduced, and is more uncertain.
DRIFT
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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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RITZ VECTORS
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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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ARTIFICIAL EARTHQUAKES
CREATING HISTORIES TO MATCH A SPECTRUM FREQUENCY CONTENTS OF EARTHQUAKES FOURIER TRANSFORMS
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ARTIFICIAL EARTHQUAKES
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MESH REFINEMENT
71
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APPROXIMATING BENDING BEHAVIOR
This is for a coupling beam. A slender pier is similar. 72
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ACCURACY OF MESH REFINEMENT
73
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PIER / SPANDREL MODELS
74
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STRAIN & ROTATION MEASURES
75
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STRAIN CONCENTRATION STUDY
Compare calculated strains and rotations for the 3 cases. 76
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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
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DISCONTINUOUS SHEAR WALLS
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PIER AND SPANDREL FIBER MODELS
Vertical and horizontal fiber models
79
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RAYLEIGH DAMPING
• • •
The αM dampers connect the masses to the ground. They exert external damping forces. Units of α are 1/T. The βK dampers act in parallel with the elements. They exert internal damping forces. Units of β are T. The damping matrix is C = αM + βK.
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80
RAYLEIGH DAMPING
• •
• •
For linear analysis, the coefficients α and β 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 α and β 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.
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81
RAYLEIGH DAMPING
M &u& t + Cu& + Ku = 0 M &u& t + (α M + β K ) u & + Ku = 0
&u& +
Cu & M
+
K u M
= 0
&u& + 2ξ ωu& + ω2u = 0 ; 2ξ M ω = C ξ=
C 2Mω
=
C 2M K M
=
C 2 KM
=
αM + βK
2 KM
ξ = αω + β ω 2 2
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2 KM
RAYLEIGH DAMPING
Higher Modes (high ω) = β Lower Modes (low ω) = α To get ζ from α & β for any ω =
K M
; Τ = 2π ω
To get α & β from two values of ζ1& ζ2
ζ1 = α + β ω1 2 2ω 1
ζ2 = α + β ω2 2 2ω
Solve for α & β
2
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DAMPING COEFFICIENT FROM HYSTERESIS
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DAMPING COEFFICIENT FROM HYSTERESIS
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A BIG THANK YOU!!!
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