Nonlinear Analysis & Performance Based Design
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 FRAMESREDUCED BEAM SECTIONS
LINK BEAMS IN ECCENTRICALLY BRACED FRAMESBUCKLING RESISTANT BRACES AS FUSES
RUBBER-LEAD BASE ISOLATORSHINGED BRIDGE COLUMNS
HINGES AT THE BASE LEVEL OF SHEAR WALLS ROCKING FOUNDATIONS
OVERDESIGNED COUPLING BEAMSOTHER 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 17
ASCE 41 BEAM MODEL
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
Nonlinear Analysis & Performance Based Design
ELASTIC STRENGTH DESIGN - KEY STEPS
CHOSE DESIGN CODE AND EARTHQUAKE LOADSDESIGN CHECK PARAMETERS STRESS/BEAM MOMENTGET 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 41DEMAND CAPACITY MEASURES – DRIFT/HINGE ROTATION/SHEAR
GET DEFORMATION AND FORCE CAPACITIESCALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH)
CALCULATE D/C RATIOS – LIMIT STATES
STRENGTH vs DEFORMATION
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
Force (F)
In itia llylinear
Stra inH ardening
U ltim atestrength
Strength loss
R esidual strength
C om plete fa ilure
F irst y ie ld
D uctile lim it
H ysteresis loop
Stiffness, strength and ductile lim it m ayall degrade under cyclic deform ation
D eform ation (D )
F-D RELATIONSHIP
Nonlinear Analysis & Performance Based Design
LATERALLOAD
Partially DuctileBrittle Ductile
DRIFT
DUCTILITY
Nonlinear Analysis & Performance Based Design
ASCE 41 - DUCTILE AND BRITTLE
Nonlinear Analysis & Performance Based Design
FORCE AND DEFORMATION CONTROL
Nonlinear Analysis & Performance Based Design
F
R elationship a llow ingfor cyclic deform ation
M onotonic F-D re lationship
H ysteresis loopsfrom experim ent.
D
BACKBONE CURVE
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
ij
C urrent
Elastic Plastic (h inge)
Tota lrotation
Zero length h inges
E lastic beam
Plasticrotation
E lasticrotation
S im ilar a tth is end
M i M j
M or Mi j
M
i j or
BEAM END ROTATION MODEL
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.
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 qp/qy = 1 qp/qy = 6 qp/qy = 8
RC Beam Low shear High shear
qp = 0.01
qp = 0.005
qp = 0.02
qp = 0.01
qp = 0.025
qp = 0.02
Nonlinear Analysis & Performance Based Design
COLUMN AXIAL-BENDING MODEL
M i
P
PP
P
or
V
V
M j
MM
Nonlinear Analysis & Performance Based Design
STEEL COLUMN AXIAL-BENDING
Nonlinear Analysis & Performance Based Design
CONCRETE COLUMN AXIAL-BENDING
Nonlinear Analysis & Performance Based Design
N odeZero-lengthshear "h inge"E lastic beam
SHEAR HINGE MODEL
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
Steelfibers
C oncrete
fibers
WALL FIBER MODEL
Nonlinear Analysis & Performance Based Design
ALTERNATIVE MEASURE - STRAIN
Nonlinear Analysis & Performance Based Design
∆ƒ
ƒ
∆ u u
1 2
iteration
NEWTON – RAPHSONITERATION
∆ƒ
ƒ
∆ u u
1 2
iteration
CONSTANT STIFFNESSITERATION
3 4 56
NONLINEAR SOLUTION SCHEMES
Nonlinear Analysis & Performance Based Design
Y
+Y
-Y
0 Radius, R
q
CIRCULAR FREQUENCY
Nonlinear Analysis & Performance Based Design
u.
t
ut1
.
..
Slope = ut1
..
t1
u
t1 t
Slope = ut1
.
u
ut1
..
t1t
THE D, V & A RELATIONSHIP
Nonlinear Analysis & Performance Based Design
UNDAMPED FREE VIBRATION
)cos(0 tuut w=
0=+ ukum&&
mk
where =w
Nonlinear Analysis & Performance Based Design
RESPONSE MAXIMA
ut )cos(0 tu w=
)cos(02 tu ww-=ut&&
)sin(0 tu ww-=ut&
max2uw-=maxu&&
Nonlinear Analysis & Performance Based Design
BASIC DYNAMICS WITH DAMPING
guuuu &&&&& -=w+wx+ 22
guMKuuCuM
KuuCt
uM
&&&&&
&&&
-=++
=++ 0
gu&&
M
K
C
Nonlinear Analysis & Performance Based Design
ug..
2ug2
..
ug1
..
1
t1 t2
Time t
&&A Bt ug= + = -&& &u u u+ +22xw w
RESPONSE FROM GROUND MOTION
Nonlinear Analysis & Performance Based Design
{ [& ] cos
[ (& )] sin }
e uB
t
A u uB
tB
tt d
dt t d
&ut = -
+ - - + +
-xw
ww
ww xw
ww
w
1 2
21 1 2 2
1
eA B
t
u uA B
t
A B Bt
tt d
dt t d
ut = - +
+ + - +-
+ - +
-xw
w
x
ww
wxw
xw
x
ww
w
x
w w
{ [u ] cos
[&( )
] sin }
[ ]
1 2 3
1 1
2
2
2 3 2
2
1 2 1
2
DAMPED RESPONSE
Nonlinear Analysis & Performance Based Design
SDOF DAMPED RESPONSE
Nonlinear Analysis & Performance Based Design
RESPONSE SPECTRUM GENERATION
-0.40
-0.20
0.00
0.20
0.40
0.00 1.00 2.00 3.00 4.00 5.00 6.00TIME, SECONDS
GR
OU
ND
AC
C, g
Earthquake Record
Displacement Response Spectrum
5% damping
-4.00
-2.00
0.00
2.00
4.00
0.00 1.00 2.00 3.00 4.00 5.00 6.00
DIS
PL,
in.
-8.00
-4.00
0.00
4.00
8.00
0.00 1.00 2.00 3.00 4.00 5.00 6.00
DIS
PL,
in.
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10
PERIOD, Seconds
DIS
PL
AC
EM
EN
T, in
ches
T= 0.6 sec
T= 2.0 sec
Nonlinear Analysis & Performance Based Design
0
4
8
12
16
0 2 4 6 8 10
PERIOD, sec
DIS
PL
AC
EM
EN
T, in
.
0
10
20
30
40
0 2 4 6 8 10
PERIOD, sec
VE
LO
CIT
Y, in
/se
c
0.00
0.20
0.40
0.60
0.80
1.00
0 2 4 6 8 10
PERIOD, sec
AC
CE
LE
RA
TIO
N,
g
dV SPS w=va PSPS w=
SPECTRAL PARAMETERS
Nonlinear Analysis & Performance Based Design
THE ADRS SPECTRUM
ADRS Curve
Spectral Displacement, Sd
Spec
tral
Acc
eler
ation
, Sa
2.0 Seconds
1.0 Seconds0.5
Seco
nds
Period, T
Spec
tral
Acc
eler
ation
, Sa RS Curve
0.5
Seco
nds
1.0
Seco
nds
2.0
Seco
nds
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
Calculating the Target Displacement
DISPLACEMENT MODIFICATION
C0 Relates spectral to roof displacement C1 Modifier for inelastic displacement C2 Modifier for hysteresis loop shape
d=C0 C
1 C
2 S
a T
e
2 / (4p2)
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
STR U C TU R ESTR EN G TH
The "over-ductility" is reduced,and is m ore uncerta in.
P - effects m ay reduce the drift a t which the"worst" com ponent reaches its ductile lim it.
P- effects w ill reduce the drift a twhich the structure loses strength.
D R IFT
P-DELTA EFFECTS ON F-D CURVES
Nonlinear Analysis & Performance Based Design
THE FAST NONLINEAR ANALYSIS METHOD (FNA)
NON LINEAR FRAME AND SHEAR WALL HINGESBASE ISOLATORS (RUBBER & FRICTION)
STRUCTURAL DAMPERSSTRUCTURAL UPLIFT
STRUCTURAL POUNDINGBUCKLING 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
CREATING HISTORIES TO MATCH A SPECTRUM
FREQUENCY CONTENTS OF EARTHQUAKES
FOURIER TRANSFORMS
ARTIFICIAL EARTHQUAKES
Nonlinear Analysis & Performance Based Design
ARTIFICIAL EARTHQUAKES
Nonlinear Analysis & Performance Based Design 71
MESH REFINEMENT
Nonlinear Analysis & Performance Based Design 72
This is for a coupling beam. A slender pier is similar.
APPROXIMATING BENDING BEHAVIOR
Nonlinear Analysis & Performance Based Design 73
ACCURACY OF MESH REFINEMENT
Nonlinear Analysis & Performance Based Design 74
PIER / SPANDREL MODELS
Nonlinear Analysis & Performance Based Design 75
STRAIN & ROTATION MEASURES
Nonlinear Analysis & Performance Based Design 76
Compare calculated strains and rotations for the 3 cases.
STRAIN CONCENTRATION STUDY
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
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%
STRAIN CONCENTRATION STUDY
Nonlinear Analysis & Performance Based Design
DISCONTINUOUS SHEAR WALLS
Nonlinear Analysis & Performance Based Design 79
PIER AND SPANDREL FIBER MODELS
Vertical and horizontal fiber models
Nonlinear Analysis & Performance Based Design 80
RAYLEIGH DAMPING
• The aM dampers connect the masses to the ground. They exert external damping forces. Units of a are 1/T.
• ThebK 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 81
• 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.
RAYLEIGH DAMPING
Nonlinear Analysis & Performance Based Design
KuuC&tuM&& =++ 0
Kuu&++ = 0 (
a
M + b
K )
u&& + CM
u&M
u+ K = 02uu& =w+wx+ 2 0 wx2 M = C
w2x= a
+ b
w2
w2x= C
M 2= C
M√MK
=2
C
√ MK 2√ MK
a
M + b
K=
tuM&&
u&& ;
2√ MK
RAYLEIGH DAMPING
Nonlinear Analysis & Performance Based Design
Higher Modes (high )=w bLower Modes (low )=w a
To get z from a & b for any
w √MK=
To geta & b from two values of z1&z
2
; T 2pw=
z1= a
2w1
+ b w12
z2= a
2w2
+ b w22
Solve fora & b
RAYLEIGH DAMPING
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!!!
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