bridge structure response spectrum

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


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


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 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)






Seismic Design Category (2009 AASHTO Guide Specification for LRFD Seismic Bridge Design):


Seismic Design Category (SDC) Core Flow Chart (2009 AASHTO Guide Specification for LRFD Seismic Bridge Design):


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

• 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


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:


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


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)



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.


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.


STIFFNESS Modeling (Continued):

Bi-Linear Moment Curvature Approximation:

N.A.

M

F

EIe

EIe depends on axial load level and reinforcement ratio.


STIFFNESS Modeling (Continued):

EIe / EIg ratio recommended by 2009 AASHTO Guide Specification for LRFD Seismic Bridge Design


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


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


OUTLINE






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


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.


• 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


N.A.

M

F

e F y-c

c

y

x



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



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



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



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%




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




Calculated Pushover Curve


Displacement D (in)

Force E (kip)

2.02 2.30 11.95

332

355


Implicit Displacement Capacity for SDC B & C (2009 AASHTO Guide Spec):


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


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.


Abutment Modeling:


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







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


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


Capacity Protection: Capacity-Protected Member shall equal to or

exceed the Over-Strength Capacity of the Ductile Member

• Column Shear • Pier Cap • Foundation • Joint







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.

bridge structure response spectrum

Documents