Bridge Structure Response Spectrum Analysis and Seismic Design Midas Technical Seminar Ling Zhao March 14, 2013
Bridge Structure Response Spectrum
Analysis and Seismic Design
Midas Technical Seminar
Ling Zhao
March 14, 2013
OUTLINE
• Definition of Response Spectrum
• Multi-Modes Response Spectrum Analysis
• Mass, Stiffness, and Damping Modeling
• Bridge Modeling Issues
• Displacement-Based Seismic Design
• Example
DEFINITION OF RESPONSE SPECTRUM
Single-Degree-Of-Freedom (SDOF) Equation of Motion:
gxmkxxcxm
Or rewritten as:
gxxxx 22
m
k Circular Frequency
crc
c Damping Ratio
mccr 2 Critical Damping
DEFINITION OF RESPONSE SPECTRUM
Duhammel’s Integral:
dtextx t
g
sin1
)(
Taking derivative of x(t), with small damping ratio , it can be proven:
txtxtx
txtx
g maxmax
max)(max
2
Taking maximum of x(t) over the time history:
txmax
DEFINITION OF RESPONSE SPECTRUM
For a given earthquake txg
Response Spectra = Plots of peak response quantity (displacement, velocity, acceleration) of a SDOF system subjected to the given ground motion, versus the Fundamental Period T, and Damping Ratio of the SDOF system.
For a whole class of possible earthquake txg at a site
Design Response Spectra = Theoretically, Response Spectra constructed for a range of possible earthquake events, with its ordinate having a uniform probability of exceedance (e.g., 7% probability of exceedance in 75 years, 1000 year return period) over all periods. Practically, smoothed curve constructed by three-point method.
DEFINITION OF RESPONSE SPECTRUM
DEFINITION OF RESPONSE SPECTRUM R
esponse
Spec
tral
Acc
eler
atio
n,
Sa
As = Fpga PGA
0 0.2 1.0
Ts = SD1/SDS T0 = 0.2TS
SDS = FaSS
Sa = SD1/T
SD1 = FvSD1
Generic Design Response Spectrum
Constructed Using Three-Point Method
5% Critical Damping
You can define the Generic Design Response Spectrum for your project ONLINE! First get Latitude, Longitude, site classification of your project location, then go to:
http://geohazards.usgs.gov/designmaps/us/application.php
For example, the Design Response Spectrum for where I am now: (assuming Site Class D)
DEFINITION OF RESPONSE SPECTRUM
Seismic Design Category (2009 AASHTO Guide Specification for LRFD Seismic Bridge Design):
DEFINITION OF RESPONSE SPECTRUM
Seismic Design Category (SDC) Core Flow Chart (2009 AASHTO Guide Specification for LRFD Seismic Bridge Design):
DEFINITION OF RESPONSE SPECTRUM
Analysis Procedure (2009 AASHTO Guide Specification for LRFD Seismic Bridge Design):
Seismic Design Category Regular Bridges with 2 through 6 Spans
Not Regular Bridge with 2 or more Spans
A Not Required Not Required
B, C, or D Equivalent Static Analysis (ESA) or Elastic Dynamic Analysis (EDA)
Elastic Dynamic Analysis (EDA)
Nonlinear time history is generally not required unless: • P-D Effect too large to be neglected; • Damping provided by a base-isolation system is large; • Requested by Owner.
DEFINITION OF RESPONSE SPECTRUM
• Definition of Response Spectrum
• Multi-Modes Response Spectrum Analysis
• Mass, Stiffness, and Damping Modeling
• Bridge Modeling Issues
• Displacement-Based Seismic Design
• Example
OUTLINE
• Elastic Dynamic Analysis (EDA) = Multiple-Degree-Of-Freedom (MDOF) Response Spectrum Analysis: (Modal Superposition Method)
gx R[M]x[K]}x[C]{}x[M]{
Mass Matrix [M]
Stiffness Matrix [K]
Ground Motion Influence Coefficient Vector {R}
Damping Matrix [C]
Nodal Displacement Vector {x}
• Solve Eigenvalues i2 and Eigenvector Matrix [F] of the system first, then let:
qΦx
• Substitute {x} back into equation and left multiply [F]T
gx R[M]ΦqΦ[K]Φ}q{Φ[C]Φ}q{Φ[M]ΦTTTT
MULTI-MODES RESPONSE SPECTRUM ANALYSIS
• Because of the orthogonalities of eigenvectors, above Equation can be decoupled into a series of SDOF equation of motion (assuming proportional damping matrix), written as:
giiiiiii xqqq 2
2 NDOFi 1
• Where: i is the modal damping ratio of the i’th mode
i is the circular frequency of the i’th mode
i
T
i
T
ii
M
RM
i is the Modal Participation Coefficient of the i’th mode.
• Equation of Motion for each mode is essentially a SDOF equation of motion with the ground motion being scaled by a factor i.
Or RMT
ii
when F is mass-normalized
MULTI-MODES RESPONSE SPECTRUM ANALYSIS
Therefore, procedure for solving peak response of the j’th DOF at i’th mode is:
1. Using the i’th mode period Ti and modal damping ratio i, read the response acceleration Sai from the design response spectra curve.
2. Scale Sai by a factor i , written as:
aiii Sq
3. Multiply mode shape ordinate ji with qi, resulting:
jiaiiji SRa
Response acceleration:
ji
i
aiiji
SRv
Response velocity:
ji
i
aiiji
SRd
2
Response displacement:
MULTI-MODES RESPONSE SPECTRUM ANALYSIS
Look at Base Shear from the i’th mode, Vi:
i
T
aii
jiaiij
jiji
S
Sm
RamV
MR
Recall:
Therefore: ai
i
T
i
ii SV
M
2
represents the Mass Participation for each mode.
mass totalof %90
1
2
i
NMODE
i i
T
i M
i
T
i
T
ii
M
RM
i
T
i
i
M
2
or Simply: aiii SV2
when F is mass-normalized
MULTI-MODES RESPONSE SPECTRUM ANALYSIS
Combination of Modal Maxima Ri of each mode:
• Absolute Maximum:
i
iRR
• SRSS (Square root of Sum of Square):
i
iRR2
(Too conservative)
(May lead to erroneous results when modes are not well separated)
• CQC (Complete Quadratic Combination):
i j
jiji RRR
Where:
222222
2/3
)(4)1(4)1(
8
tttt
tt
jiji
jiji
ij
represents the cross-modal coefficients which depends on the modal damping ratio i, j, and the modal period ratio t = Ti/Tj .
(Required by AASHTO)
MULTI-MODES RESPONSE SPECTRUM ANALYSIS
MULTI-MODES RESPONSE SPECTRUM ANALYSIS
Directional Combination of Rx, Ry, Rz:
• Very little cross-correlation between two horizontal perpendicular response Rx, Ry, therefore SRSS rule can apply:
22 )()( yx RRR
• Which produce results within 5% of those obtained with the commonly used 30% rule:
yx RRR 3.0
• Vertical response should not be combined with horizontal motion due to time separation in the maximum intensity of ground shaking in the vertical and horizontal motion. However, some DOTs may have different requirement from AASHTO on this regard.
MULTI-MODES RESPONSE SPECTRUM ANALYSIS
Definition of Response Spectrum
Multi-Modes Response Spectrum Analysis
Mass, Stiffness, and Damping Modeling
Bridge Modeling Issues
Displacement-Based Seismic Design
Example
OUTLINE
MASS, STIFFNESS AND DAMPING MODELING
Modeling means to build mathematical representations, mass [M], stiffness [K], and damping [C], of your structure.
MASS Modeling:
1. All components that create inertia forces • Foundation Mass?
2. Translational Mass and Mass Moment of Inertia
STIFFNESS Modeling:
1. Linear Elastic Stiffness for all components that are not expected to yield during seismic event: superstructure, prestressed concrete, conventional bearing, foundation.
2. For cracked reinforced concrete, use effective stiffness:
y
y
e
MEI
F
Where My and Fy represent the yield moment and curvature for a bi-linear moment-curvature approximation.
MASS, STIFFNESS AND DAMPING MODELING
STIFFNESS Modeling (Continued):
Bi-Linear Moment Curvature Approximation:
N.A.
M
F
EIe
EIe depends on axial load level and reinforcement ratio.
MASS, STIFFNESS AND DAMPING MODELING
STIFFNESS Modeling (Continued):
EIe / EIg ratio recommended by 2009 AASHTO Guide Specification for LRFD Seismic Bridge Design
MASS, STIFFNESS AND DAMPING MODELING
DAMPING Modeling:
Convert hysteresis damping to equivalent viscous damping:
Figure Referenced from “Seismic Design and Retrofit of Bridges” by M.J.N. Priestley et. al.
e
heq
A
A
4
MASS, STIFFNESS AND DAMPING MODELING
Back Ground: Logarithmic Attenuation Rate of Amplitude:
2
1
eAA NN
Logarithmic Attenuation Rate of Energy:
4
1
eEE NN
Then:
4
1
1/
eAA
EA
EEA
eh
Ne
NNh
Expanding Taylor Series and neglect second-order term.
DAMPING Modeling (Continued): Steel Structure: 2~5% Concrete Structure: 2~7% Commonly Assumed: 5%
Consideration of Damping > 5% (maximum 10%) only valid when: 1. Substantial energy dissipation through soil at the abutments; 2. Special energy absorption devices are employed; 3. Predominate response as SDOF system.
Damping Reduction Factor (2009 AASHTO Guide Spec):
4.0
05.0
DR
MASS, STIFFNESS AND DAMPING MODELING
OUTLINE
Definition of Response Spectrum
Multi-Modes Response Spectrum Analysis
Mass, Stiffness, and Damping Modeling
Bridge Modeling Issues
Displacement-Based Seismic Design
Example
OUTLINE
Inclusion Limit of Model:
• Global Model: From Abutment to Abutment (rarely used)
• Frame Model:
Between Movement Joints (most commonly used)
• Bent Model:
Individual Bents, or Multiple Bents with Rigid-Body Constraint that represents superstructure (mostly used to develop effective bent stiffness characteristics and displacement limit)
BRIDGE MODELING ISSUES
Superstructure Modeling (in ascending order of modeling complicity and descending order of preference):
• Spine Model: Whole superstructure modeled as a spine member following the superstructure center of gravity. Spine member properties represent the overall superstructure section properties. Superstructure mass moment of inertia modeled by splitting the section mass to half and placing them at a distance equal to Radius of Gyration rG from the center of gravity.
• Grillage Model:
A grillage of beam elements connected transversely by dummy members that represents the transverse stiffness of deck and diaphragms.
• Prototype Model:
Exact member to member modeling
BRIDGE MODELING ISSUES
Substructure Modeling (most important modeling elements):
• Moment Curvature Analysis and Collapse Mechanism Analysis
(Pushover Analysis) can produce valuable information on substructure stiffness characteristics, displacement capacity and ductility capacity.
BRIDGE MODELING ISSUES
• For Moment Curvature Analysis and Push Over Analysis, use Expected instead of Nominal Material Properties:
'' 3.1:Concrete cce ff
Bar 60 Gradefor ksi68:Rebar yef
Substructure Modeling (Continued) – Moment Curvature Analysis
BRIDGE MODELING ISSUES
N.A.
M
F
e F y-c
c
y
x
Substructure Modeling (Continued) – Moment Curvature Analysis
BRIDGE MODELING ISSUES
Actual M-F Curve
Idealized Perfect
Elastic-Plastic M-F
Curve
Identical Area
Elastic Response
Line through (0,0)
and the point
representing the 1st
Rebar Yield
Fy Fu
Substructure Modeling (Continued) – Moment Curvature Analysis
BRIDGE MODELING ISSUES
Plastic Hinge Rotational Capacity:
pyup L)( FF
Lp = Analytical Plastic Hinge Length
blyeblyep dfdfLL 3.015.008.0
L = Length of column from point of maximum moment to point of contraflexure (in.) fye = Expected yield strength of longitudinal column rebar (ksi) dbl = Nominal diameter of longitudinal column rebar (in.)
For Reinforced Concrete Columns
Substructure Modeling (Continued) – Pushover Analysis Example
Column Type P (kip) Fy (in-1) Fu (in-1) Myi
(kip-ft) EIeff
(kip-in2)
Compression 1026 112x10-6 1360x10-6 3779 4.05x108
Gravity Load 750 111x10-6 1460x10-6 3549 3.84x108
Tension 474 113x10-6 1650x10-6 3319 3.52x108
36’
20’
28’
Ws =1500 kip
E =2Myi/H = 2(3549)/20 = 354.9 kip
DP =±EH’/B = ±28/36 E = ±0.778E = ±276 kip
Moment Curvature Analysis Results:
Hinge
B
A
C
D EIeff = Average of Tension and Compression Column = 3.79x108 kip-in2
Plastic Hinge Length Lp = 0.08 L + 0.15 fye dbl = 0.08(20x12)+0.15(68)(1.27) = 32.2 in
BRIDGE MODELING ISSUES
Substructure Modeling (Continued) – Pushover Analysis Example
B C
D A
3319
- 3319
0
3779
- 3319
460
E1 =331.9 kip E =1 kip
10 10
E1 = (1) (331.9) = 331.9 kip
Event 1
D1 = 2.02 in
D1 = E1H3/(6EIeff) = 331.9 (20x12)3/(6(3.79x108)) = 2.02 in
S1 =331.9
Scale S1 = 3319 / 10 = 331.9
BRIDGE MODELING ISSUES
Substructure Modeling (Continued) – Pushover Analysis Example
B C
D A
0 460
E2 =23 kip E =1 kip
20
E2 = 1 (23) = 23 kip
Event 2
D2 = 0.28 in
D2 = E2H3/(3EIeff) = 23 (20x12)3/(3(3.79x108)) = 0.28 in
S2 =23
Scale S2 = 460 / 20 = 23
B2 = D2/H =0.117%
BRIDGE MODELING ISSUES
BRIDGE MODELING ISSUES
Substructure Modeling (Continued) – Pushover Analysis Example
B C
D A
0 0
E3 = 0 kip
Event 3
D3 = 9.65 in D3 = p
C x H = 4.02% x (20’x12) = 9.65 in
pB= Lp (Fu-Fy) = 32.2 in x (1650-113)x10-6 in-1 = 4.95% (Tension Column)
p
C= Lp (Fu-Fy) = 32.2 in x (1360-112)x10-6 in-1 = 4.02% (Compression Column)
B3 = pC = 4.02%
pB - B2 = 4.95% - 0.117% = 4.83% still larger than p
C . Therefore, plastic hinge of compression column reaches rotational capacity first.
E = SEi = 331.9 + 23 = 354.9 kip
Du = SDi = 2.02 + 0.28 + 9.65 = 11.95 in
Dy = 2.02 in
Ductility Capacity mc = Du/Dy= 11.95 / 2.02 = 5.9
BRIDGE MODELING ISSUES
BRIDGE MODELING ISSUES
Substructure Modeling (Continued) – Pushover Analysis Example
Calculated Pushover Curve
BRIDGE MODELING ISSUES
Displacement D (in)
Force E (kip)
2.02 2.30 11.95
332
355
BRIDGE MODELING ISSUES
Implicit Displacement Capacity for SDC B & C (2009 AASHTO Guide Spec):
BRIDGE MODELING ISSUES
o
o
oo
L
c
oo
L
c
H
Bx
HxH
HxH
D
D
12.0)22.1)ln(32.2(12.0 :C SDCFor
12.0)32.0)ln(27.1(12.0 :B SDCFor
Ho = Clear Height of Column (ft) Bo = Column Diameter of width measured parallel to the direction of displacement under consideration (ft) = factor for column end restraint condition = 1 for fixed – free (pinned on one end) = 2 for fixed top and bottom
Foundation Modeling - Common Foundation Type: • Spread Footing • Pile Supported Cap Footing • Drilled Shaft
Foundation Modeling Method (FMM) Requirement by AASHTO: • FMM I is permitted for SDCs B and C for foundation located at Site Class
A, B, C, or D; otherwise, FMM II is required; • FMM II is required for SDC D.
Foundation Type Modeling Method I Modeling Method II
Spread Footing Rigid Rigid for Site Class A and B, foundation spring required if footing flexibility contributes >20% to pier displacement
Pile Footing with Pile Cap
Rigid Foundation Spring required if footing flexibility contributes >20% to pier displacement
Pile Bent/Drilled Shaft
Estimated depth to fixity
Estimated depth to fixity or soil spring based on P-y curve
BRIDGE MODELING ISSUES
Foundation Flexibility Calculation:
• Spread Footing: Reference FEMA 273, best estimated strain
compatible shear modulus G, and ± 25%.
• Pile Foundation:
1. Evaluate single pile axial stiffness kp
2. Evaluate single pile lateral stiffness kL (p-y curve analysis
such as LPILE)
3. Integrate: Kv = Skp; Kr = Skpxi2; KL = SkL
• Drilled Shaft Foundation: estimated point of fixity.
BRIDGE MODELING ISSUES
Abutment Modeling:
BRIDGE MODELING ISSUES
Force
Displacement
Exp. Jt.
Gap Passive Soil
Resistance
Beyond soil
failure
Push
Direction Pull
Direction
Kpile
Kpile+Ksoil
Kpile
Bearing Modeling:
• Seismic - isolated?
• Isolated ---- all other elements of bridge remain elastic except
the seismic-isolation bearings
• Non-isolated ---- bearing remains elastic; protect bearings
BRIDGE MODELING ISSUES
Definition of Response Spectrum
Multi-Modes Response Spectrum Analysis
Mass, Stiffness, and Damping Modeling
Bridge Modeling Issues
Displacement-Based Seismic Design
Example
OUTLINE
Traditional Force-Based Design:
E
D
D
E
Du
EE
Dy
Elastic Analysis
Inelastic Response
EP
Design Load EP = EE / R
R
1
R = Response Modification Factor R = Du / Dy, represent the ductility capacity of the ERS
DISPLACEMENT-BASED SEISMIC DESIGN
Displacement-Based Design: E D
D
E
Du
Dy
Elastic Analysis
Inelastic Response
EP Equal Displacement Assumption: Displacements resulted from inelastic response is approximately equal to displacement obtained from linear elastic response spectrum analysis.
Design Load is EP.
What to be checked:
DD ≤ Du
DD
DISPLACEMENT-BASED SEISMIC DESIGN
Comparison of two Design Approaches:
Force • AASHTO LRFD Bridge Design
Specification • Complete design for STR, SERV limit
state first • Elastic demand forces divided by
Response Modification Factor “R” • Ductile response is assumed to be
adequate without verification • Capacity protection assumed
Displacement • AASHTO Guide Specification for
LRFD Seismic Bridge Design • Complete design for STR, SERV limit
state first • Displacement demands checked
against displacement capacity • Ductile response is assured with
limitations prescribed for each SDC • Capacity protection assured
DISPLACEMENT-BASED SEISMIC DESIGN
Capacity Protection: Capacity-Protected Member shall equal to or
exceed the Over-Strength Capacity of the Ductile Member
• Column Shear • Pier Cap • Foundation • Joint
DISPLACEMENT-BASED SEISMIC DESIGN
Definition of Response Spectrum
Multi-Modes Response Spectrum Analysis
Mass, Stiffness, and Damping Modeling
Bridge Modeling Issues
Displacement-Based Seismic Design
Example
OUTLINE
Example 1: 3-Span Steel Girder Bridge – Transverse Mode
Example
Example 1: 3-Span Steel Girder Bridge – Torsion Mode
Example
Example 1: 3-Span Steel Girder Bridge – Longitudinal Mode
Example
Example 1: 3-Span Steel Girder Bridge – Pier 1 Displacement Demand 0.3EX + 1.0EY
Example
Example 1: 3-Span Steel Girder Bridge – Pier 1 Displacement Check
Example
Example 2: 4-Span Concrete Box Girder Bridge – Transverse Mode
Example
Example 2: 4-Span Concrete Box Girder Bridge – Torsion Mode
Example
Example 2: 4-Span Concrete Box Girder Bridge – Longitudinal Mode
Example
Example 2: 4-Span Concrete Box Girder Bridge – Pier 1 Displacement Demand, 0.3EX + 1.0EY
Example
Example 2: 4-Span Concrete Box Girder Bridge – Pier 1 Displacement Check
Example
Thank you.