Transcript Document

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
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
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
Hysteresis loops
from experiment.
Monotonic F-D relationship
Relationship allowing
for cyclic deformation
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
Mi
i
Mi or Mj
j
Mj
i or j
Zero length hinges
M
Elastic 
Plastic (hinge) 
Elastic beam
Total
rotation
Plastic
rotation
Elastic
rotation
Similar at
this end
Current 
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
Mj
P
V
P
P
or
M
V
Mi
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
Node
Elastic beam
Zero-length
shear "hinge"
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

Steel
fibers

Concrete
fibers

Nonlinear Analysis & Performance Based Design

ALTERNATIVE MEASURE - STRAIN
Nonlinear Analysis & Performance Based Design
NONLINEAR SOLUTION SCHEMES
ƒ
iteration
ƒ
2
1
iteration
1 2 3 4 56
∆ƒ
∆ƒ
∆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
t
..
Slope = ut1
t1
t
t1
t
..
u
..
ut1
Nonlinear Analysis & Performance Based Design
UNDAMPED FREE VIBRATION
&& + ku = 0
mu
ut = u0 cos(wt)
k
where w =
m
Nonlinear Analysis & Performance Based Design
RESPONSE MAXIMA
ut = u0 cos(wt)
u&t = -wu0 sin(wt)
u&&t = -w u0 cos(wt)
2
&u& max = -w 2u max
Nonlinear Analysis & Performance Based Design
BASIC DYNAMICS WITH DAMPING
&& + Cu& + Ku = 0
Mu
t
&& + Cu& + Ku = - Mu
&&
Mu
g
&u& + 2xwu& + w2u = - u
&&
g
M
C
K
&&
u
g
Nonlinear Analysis & Performance Based Design
RESPONSE FROM GROUND MOTION
&&
u + 2xwu& + w u = A + Bt = -&&
ug
2
..
ug
2
..
ug2
t1
..
ug1
t2
Time t
1
Nonlinear Analysis & Performance Based Design
DAMPED RESPONSE
u& t =
e
+
ut =
- xwt
{ [u& t 1
B
] cos wd t
2
w
1
B
B
[A - w2ut - xw(u& t + 2 )] sin wd t } + 2
1
1
wd
w
w
e
- xwt
{ [ut
1
A
2xB
+
] cos wd t
2
3
w
w
x A B(2x2 - 1)
1
+
+
wd t }
[u& t + xwut ]
sin
1
1
wd
w
w2
A
2xB Bt
+[
+
]
2
3
2
w
w
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
DISPLACEMENT, inches
GROUND ACC, g
0.40
14
12
10
8
6
4
2
0
0
0.00
2
4
6
8
10
PERIOD, Seconds
-2.00
-4.00
0.00
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
PSV = w Sd
PSa = w 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
PERIOD, sec
8
10
0
2
4
6
PERIOD, sec
Nonlinear Analysis & Performance Based Design
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
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
d = C0 C1 C2 Sa Te / (4p )
2
2
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
P- effects may reduce the drift at which the
"worst" component reaches its ductile limit.
STRUCTURE
STRENGTH
P- effects will reduce the drift at
which the structure loses strength.
The "over-ductility" is reduced,
and is more uncertain.
DRIFT
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
71
Nonlinear Analysis & Performance Based Design
APPROXIMATING BENDING BEHAVIOR
This is for a coupling beam. A slender pier is similar.
72
Nonlinear Analysis & Performance Based Design
ACCURACY OF MESH REFINEMENT
73
Nonlinear Analysis & Performance Based Design
PIER / SPANDREL MODELS
74
Nonlinear Analysis & Performance Based Design
STRAIN & ROTATION MEASURES
75
Nonlinear Analysis & Performance Based Design
STRAIN CONCENTRATION STUDY
Compare calculated strains and rotations for the 3 cases.
76
Nonlinear Analysis & Performance Based Design
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
Nonlinear Analysis & Performance Based Design
DISCONTINUOUS SHEAR WALLS
Nonlinear Analysis & Performance Based Design
PIER AND SPANDREL FIBER MODELS
Vertical and horizontal fiber models
79
Nonlinear Analysis & Performance Based Design
RAYLEIGH DAMPING
• The aM dampers connect the masses to the ground. They exert
external damping forces. Units of a are 1/T.
• The bK dampers act in parallel with the elements. They exert internal
damping forces. Units of b are 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 &u& t + Cu& + Ku = 0
M &u& t + (a M + b K ) u
& + Ku = 0
&u& +
Cu
&
M
+
K
u
M
= 0
&u& + 2x wu& + w2u = 0 ; 2x M w = C
x=
C
2Mw
=
C
2M √K
M
=
C
2 √K M
=
aM + bK
2 √K M
x = aw + b w
2
2
Nonlinear Analysis & Performance Based Design
2 √K M
RAYLEIGH DAMPING
Higher Modes (high w) = b
Lower Modes (low w) = a
K
To get z from a & b for any w = √M
; T = 2p
w
To get a & b from two values of z1& z2
z1 = a + b w1
2
2w
1
z2 = a + b w2
2
2w
Solve for a & 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