bentley rm bridge seismic design and analysis alexander mabrich, pe, msc

Bentley RM Bridge Seismic Design and Analysis

Alexander Mabrich, PE, Msc

AGENDA

Kobe, Japan (1995)

AGENDA

Loma Prieta, California (1989)

RM Bridge Seismic Design and Analysis

• Critical infrastructures require:

– Sophisticated design methods

– Withstand collapse in earthquake occurrences

RM Bridge Seismic Design and Analysis

• AASTHO, Simple Seismic Load

• Basic concepts for Dynamic Analysis:

- Eigenvalues

- Eigenshapes

• Two non-linear dynamic options:

- Response Spectrum

- Time-History

AASHTO Bridge Design Specifications

• 7% probability of exceedence in 75years

• Seismic Design Categories

– Soil

– Site / location

– Importance

• Earthquake Resistant System

• Demand/Capacity

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• Site Location


• Type of Seismic Analysis Required


Static Seismic Load

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Equivalent Static Analysis

• Uniform Load Analysis

• Orthogonal Displacements

• Simultaneously

• Fundamental mode


Equivalent Static Analysis• Direction, Factor


Fundamental Mode


Results



Basic Concepts used inDynamic Analysis

Basic Concepts

• Vibration of Systems with one or more DOF

• Eigen values and Eigen modes

• Forced Vibration– Harmonic and Stochastic Simulation

• Linear and Non-linear behavior of the structure

Dynamic Vibration

T 2T 1

2

3

-1

-2

-3

0 TIME [sec]

Oscillation deflection [m] Oscillation velocity [m/s] Oscillation acceleration [m/s²]

Maximum Amplitude

Period T

d,v,a

M

K

2

ff

T1

Damped Vibration

)0(

)(

v

v t

n0t

tevv tbt 0)0()( cos

tbev )0(

-1

1

0.5

-0.5

0.0

A1 A2

2ln

3

1

A

A

Single Mass Oscillator

spring constant k damping constant c

xF

mass m

external Force F(t)amplitude x(t)

EQUILIBRIUM

EQUATION OF MOTION

Damping Ratio

c0: 20

2

22

m

c

m

cs

t

u

=100%

t

u

t

u

=200%

t

u

=0%

=10%

Critic damping

Not damped

Below critic damping Above critic damping

0 kuucum

tutu

tu 00 sinsin)(

Solution:

Free Vibration

M

K

0c …no damping…and dividing by m…

0 um

ku But..

02 uu

Multi Degree of Freedom System

)()()()( tttt PUKUCUM

nm

m

m

M

...00

......

0...0

0...0

2

1

nnnn

n

n

ccc

ccc

ccc

C

...

......

...

...

21

22221

11211

nnnn

n

n

kkk

kkk

kkk

K

...

......

...

...

21

22221

11211

nu

u

u

U...

2

1

nu

u

u

U

...

2

1

nu

u

u

U

...

2

1

)(

...

)(

)(

2

1

)(

tp

tp

tp

P

n

t

Numerical Methods for Dynamic Analysis

• Calculation of Eigen frequency

• Modal Analysis

• Direct Time integration, linear and non-linear

• System of dynamic equations :

)(tFXKXCXM

• Free vibration motion:

02 nMK

• Non trivial solution:

02 MK n

Modal Analysis

Eigen Calculation

• Eigen values

• Eigen shapes

• Unique nature

• Differential equations


Eigen Shapes


MASS PARTICIPATION FACTORS [%] MODE phi*M*phi X Y Z SUM-X SUM-Y SUM-Z HERTZ

-------------------------------------------------------------------------

1 0.3768E+04 88.33 0.00 3.14 88.33 0.00 3.14 0.905

2 0.1653E+04 2.35 0.00 71.45 90.68 0.00 74.59 1.704

3 0.8292E+03 0.00 5.03 0.04 90.68 5.03 74.63 3.111

4 0.1770E+04 1.14 0.01 0.05 91.82 5.04 74.68 3.809

5 0.1055E+04 0.28 0.01 0.01 92.10 5.05 74.69 5.425

6 0.1101E+04 0.00 57.35 0.01 92.10 62.40 74.69 6.300

7 0.1675E+04 0.43 0.01 7.31 92.54 62.41 82.00 7.145

8 0.9072E+03 0.17 0.00 0.05 92.70 62.41 82.05 9.656

9 0.5307E+04 0.13 0.04 3.98 92.83 62.45 86.03 10.042

10 0.1038E+04 0.06 0.01 0.04 92.90 62.46 86.08 11.795

11 0.1405E+04 0.13 0.01 0.00 93.02 62.47 86.08 11.830

12 0.1671E+04 0.74 0.01 0.03 93.77 62.48 86.10 13.265

13 0.4010E+03 1.74 0.00 0.04 95.51 62.49 86.14 13.321

14 0.8892E+03 0.00 0.43 0.05 95.51 62.92 86.20 13.890

15 0.5452E+04 0.01 0.03 0.25 95.52 62.95 86.45 14.077

16 0.1986E+04 0.08 0.03 0.87 95.59 62.97 87.32 16.719

17 0.6586E+03 0.03 5.91 0.03 95.63 68.88 87.35 16.936

18 0.6484E+03 0.09 3.54 0.00 95.72 72.42 87.35 16.961

19 0.1086E+04 0.00 7.02 0.00 95.72 79.44 87.35 17.275

20 0.1866E+04 0.02 0.01 0.00 95.74 79.45 87.35 18.408

21 0.1310E+04 0.11 0.00 3.47 95.85 79.45 90.82 21.221

22 0.2060E+04 0.06 0.00 0.00 95.91 79.45 90.82 22.277

23 0.1474E+04 0.06 0.00 0.00 95.97 79.45 90.83 24.414

24 0.2324E+04 0.04 0.00 0.00 96.00 79.45 90.83 24.983

25 0.1613E+04 0.00 0.00 0.00 96.01 79.45 90.83 26.843


Response SpectrumModal Decomposition


Response Spectrum

• Combination of natural modes

• One mass oscillator

• Oscillating loads

• Intensity factor

• Single contribution

• Synchronization by Stochastic Calculation Rules: ABS,SRSS,CQC, etc

Spectral Response Acceleration


AASHTO Definition

Solution in Frequency Domain

• Solution by combining the contributions of the eigenvectors

• Superposition of eigenvectors– Loading has lost information about correlation during

conversion– Solution has no information on phase differences

between the contributions of different eigenvectorsUse Stochastic methodology

• Use Stochastic methodology

Combination Rules

• Max/Min results with different rules available:

• ABS – Rule (Sum of absolute values)

• SRSS – Rule (Square root of sum of sqaures)

• DSC – Rule (Newmark/Rosenblueth)

• CQC – Rule (Complete quadratic combination)

• GENERAL : a lot of other rules exist

Earthquake Load

Response Spectrum in RM Bridge

Time-HistoryTime Integration


Time History

• Direct Time Integration

• Linear and Non-Linear analysis

• Standard event is defined: time-histories of ground acceleration are site specific

• Probability of bearable damage

• Most accurate method to evaluate structure response under earthquake event.

tttttttt PuKuCuM

What Can Be Non-Linear in RM Bridge?

Structure-stiffness- Springs- Connections- Materials- Interaction between the substructure and bridge- Large deformations- Cables

Mass of structure- Moving vehicle traffic

Structure-damping- Raleigh damping effect- Viscous damping

Load dependent on time- Change of position, intensity or direction- Time delay of structural elements

Comparison

• Solution of uncoupled differential equations

• Each eigenmode as single mass oscillator

• Coupled system of differential equations

• Time domain approximated

• Static starting condition

• Analysis of secondary systems: vehicles, equipment, extra bridge features

• All Non-Linearities possible

MODAL ANALYSIS

TIME-HISTORY

Element 105

40 m 60 m 40 m

Element 110

Element 118

Element 125 Element 131

Application Example

Bentley RM Bridge Seismic AnalysisConclusions


Kobe, Japan (1995)


Akashi-Kaikyo – “Pearl Bridge”

RM Bridge Benefits

• Bentley BrIM vision

• Bentley portfolio

• Intuitive step-by-step calculation

• One tool for all: static, modal, time-history

• Integrated reports and drawings

Bentley RM Bridge Seismic Design and AnalysisQuestions


Thank you for your [email protected]

bentley rm bridge seismic design and analysis alexander mabrich, pe, msc

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