CABLE-STAYED BRIDGE SEISMIC CABLE-STAYED BRIDGE SEISMIC ANALYSIS USING ARTIFICIAL ANALYSIS USING ARTIFICIAL

ACCELEROGRAMSACCELEROGRAMS

Roman Guzeev, Ph.D.Institute Giprostroymost-Saint-Petersburg

Russian Federationhttp://www.gpsm.ru

The Eastern Bosporus bridge, Vladivostok, RussiaThe Eastern Bosporus bridge, Vladivostok, Russia1

AASHTO LFRD

Bridge Design

Specification

EUROCODE

EN 1998-1:2004

Design of structures for

earthquake resistance

Design structures in

earthquake regions

(Russian code)

Presentation of the Response spectrum in national codesPresentation of the Response spectrum in national codes2

Disadvantages of Disadvantages of

the Response Spectrum Methodthe Response Spectrum Method

it is inapplicable for structures with anti-seismic devices, it is inapplicable for structures with anti-seismic devices, which make behavior of the structures nonlinearwhich make behavior of the structures nonlinear

It does not take into account seismic wave propagation It does not take into account seismic wave propagation

It considers mode shape vibration as statistically independentIt considers mode shape vibration as statistically independent

It uses approximate relations between response spectrum It uses approximate relations between response spectrum curves with different damping.curves with different damping.

3

Time history analysis using accelerograms.Time history analysis using accelerograms.Instrumentally recorded ground acceleration.Instrumentally recorded ground acceleration.

0 5 10 15 20 25 30 35 40 45 50 55 60-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time T, sec

Sca

led

acc

eler

atio

n, m

/s2

1994, Northridge, Santa Monica, City Hall Grounds

0 5 10 15 20 25 30 35 40 45 50 55 60-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Sca

led

acc

eler

atio

n, m

/s2

Time T, sec

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

No

nd

imen

sio

nal

res

po

nse

Period T, sec

1940, El Centro Site

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

3.5

No

nd

imen

sio

nal

res

po

nse

Period T, sec

4

The main featuresThe main features

Instrumentally recorded earthquake acceleration is an Instrumentally recorded earthquake acceleration is an event of random processevent of random process

Every earthquake is unique and has its own peak Every earthquake is unique and has its own peak acceleration and spectral distributionacceleration and spectral distribution

Any earthquake Depends on ground conditionAny earthquake Depends on ground condition

Instrumentally recorded acceleration can be dangerous Instrumentally recorded acceleration can be dangerous to one type of structure and can be safe to anotherto one type of structure and can be safe to another

Instrumentally recorded ground acceleration ?Instrumentally recorded ground acceleration ?5

Artificial accelerogramsArtificial accelerograms

Artificial accelerogram should meet requirements of Artificial accelerogram should meet requirements of national codes:national codes:

1.1. There should be a peak value on accelerogram. The There should be a peak value on accelerogram. The peak value depends on the region seismic activity, peak value depends on the region seismic activity,

ground condition and period of exceedance. ground condition and period of exceedance.

2. Accelerogram response spectrum should match design 2. Accelerogram response spectrum should match design spectrum spectrum

6

Artificial accelerogram should meet physical Artificial accelerogram should meet physical requirements:requirements:

1.1. Acceleration, velocity and displacement should Acceleration, velocity and displacement should be equal to zero at the beginning and at the end be equal to zero at the beginning and at the end

of the earthquakeof the earthquake

2. Duration of the earthquake should not be less 2. Duration of the earthquake should not be less than 10 sec.than 10 sec.

Artificial accelerogramsArtificial accelerograms

7

The accelerogram generation algorithmThe accelerogram generation algorithm

Step 1. Step 1. Generating accelerogram with peak value equal to 1 Generating accelerogram with peak value equal to 1

Step 2.Step 2. Scaling accelerogram according to the design value Scaling accelerogram according to the design value

of ground acceleration.of ground acceleration.

The accelerogram to be found is presented as trigonometric series:The accelerogram to be found is presented as trigonometric series:

1

2 2( ) sin cos

N

i ii ii

a t a t b tT T

-sought coefficients,

-natural period of mode i,

,i i

i

a b

T

N -number of considered modes

We take into account the modes which contribute to effective modal mass in the earthquake direction

8

1

2 2max sin cos 1

N

i ii ii

a t b tT T

The accelerogram constraintsThe accelerogram constraints

Peak value nonlinear constraint:Peak value nonlinear constraint:

Acceleration, velocity and displacement linear constraints:Acceleration, velocity and displacement linear constraints:

At the beginning t=0 At the end t=Ts

9

1

1

2 2

2 21

2 2cos sin 0,

2 2

2 2sin cos 0,

2 2sin cos 0,

4 4

Ni i

i s i si ii

N

i s i si ii

Ni i

i s i si ii

T Ta T b T

T T

a T b TT T

T Ta T b T

T T

2

21

1

0,

0,4

0,2

N

ii

Ni

ii

Ni

ii

b

Tb

Ta

Generated accelerogram response spectrumGenerated accelerogram response spectrum

1

( ) max ( ) ( ) ( ) 0N

s сi i i i s

i

T a t a y t b y t t T

( ), ( ) 0s сi i sy t y t t T

is the solution of differential equation of motion for one DOF oscillator on sine and cosine base excitation.

2

2

2 2 2( ) 2 sin , (0) 0, (0) 0,

2 2 2( ) 2 cos , (0) 0, (0) 0

s s s s si d i i i i

c c c c ci d i i i i

y t y y t y yT T T

y t y y t y yT T T

d - damping ratio of design response spectrum

Where,

10

The coefficient of series terms to be found The coefficient of series terms to be found ссan be determined an be determined by means of the least square method with linear and nonlinear by means of the least square method with linear and nonlinear

constraintsconstraints

We minimize the sum square of differences between We minimize the sum square of differences between accelerogram response spectrum and the design response accelerogram response spectrum and the design response

spectrum spectrum

{ ( ) ( )} { ( ) ( )}Tj d j j d jF T T W T T

F – object sum square function

[W] – diagonal matrix of weight factors

{ ( ) ( )}j d jT T

accelerogram response spectrum and the design response spectrum

– vector of differences between the

11

Recommendation on analysis using artificial acelerogramRecommendation on analysis using artificial acelerogram

Terms of series should contain natural frequencies of Terms of series should contain natural frequencies of structures. It lead to resonance excitation.structures. It lead to resonance excitation.

We should take into account the modes which contribute We should take into account the modes which contribute to effective modal mass in the earthquake directionto effective modal mass in the earthquake direction

For the closest match to design response spectrum we For the closest match to design response spectrum we can add extra terms into the seriescan add extra terms into the series

We have to generate more than one design accelerogram. We have to generate more than one design accelerogram. We can do it by varying the number of terms and considered We can do it by varying the number of terms and considered modesmodes

For every strain-stress state parameter we have to built For every strain-stress state parameter we have to built an envelope caused by action generated accelerogramsan envelope caused by action generated accelerograms

12

30580 mm

33270 mm

30580 mm

33300 mm

3330

mm

331

2 m

m

Concrete deck

Steel deck

Golden Horn Bay cable-stayed bridge, Vladivostok, RussiaGolden Horn Bay cable-stayed bridge, Vladivostok, Russia

m

m mm

13

Elastic response spectrum

Seismic action input dataSeismic action input data

1( , ) ( , )d i el iS T S T K

( , )d iS T - design spectrum

( , )el iS T - elastic spectrum

1 0.25K - ductility factor

( , ) (0.08, )el i elS T S T

(0.08, )elS T - elastic spectrum

with 0.08 damping ratio

0.08

i

- dumping correction factor

i - modal damping ratio

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Natural Period T, sec

S

Peak ground acceleration

0.107gA g

Return period is 5000 years.

14

GTSTRUDL ModelGTSTRUDL Model15

Mode

Natural period /

frequency

Effective modal mass

Mode shape

1T=4.88s

f=0.205 Hz

X: 0%

Y: 0%

Z: 10.0%

lateral

2T=4.36s

f=0.229 Hz

X: 0%

Y: 6.5%

Z: 0%

vertical

vertical

3T=3.62s

f=0.276 Hz

X: 28.4%

Y: 0%

Z: 0%

longitudinal

16

ModeNatural period /

frequency

Effective modal mass

Mode shape

4T=2.84s

f=0.352 Hz

X: 45.8%

Y: 0%

Z: 0%

longitudinal

and lateral

6T=2.78s

f=0.358 Hz

X: 0%

Y: 0%

Z: 17.8%

lateral

17

Stiffness weighted average dampingStiffness weighted average damping

Structural element

Damping ratio

Steel deck 0.02

Concrete deck 0.02

Pylon 0.025

Cables 0.00096

Concrete piers 0.05

CONSTANT MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.025 GROUP 'PYLON'MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.02 GROUP 'DECK'MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.05 GROUP 'SUPP'MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.00096 GROUP 'CABLE'DYNAMIC PARAMETERS RESPONSE DAMPING STIFFNESS 1.0END OF DYNAMIC PARAMETERSCOMPUTE MODAL DAMPING RATIOS AVERAGE

18

0 4 8 12 16 20-1.1-1-0.8-0.6-0.4-0.2

00.20.40.60.8

11.1

Time t, s

acce

lera

tio

n, m

/s2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Period T, s

resp

on

se s

pec

tru

m

0 4 8 12 16 20-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time t, s

velo

sity

m/s

0 4 8 12 16 20-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Time t, s

dis

pla

cem

ent

m/s

natural period

Accelerogram generation resultsAccelerogram generation results 19

Response spectrum analysisResponse spectrum analysis.. GTSTRUDL statement GTSTRUDL statement..

STORE RESPONSE SPECTRA ACCELERATION LINEAR vs - NATURAL PERIOD LINEAR 'SEYSM‘ DAMPING RATIO 0.02 FACTOR 0.26242……………………………………………………………………………………………………………………………………………………………………………………END OF RESPONSE SPECTRA RESPONSE SPECTRA LOADING 'RSP' 'response'SUPPORT ACCELERATION TRANS X FILE 'SEYSM'END RESPONSE SPECTRUM LOADLOAD LIST 'RSP'ACTIVE MODES ALLPERFORM MODE SUPERPOSITION ANALYSISCOMPUTE RESPONSE SPECTRA FORCES MODAL COMBINATION RMS MEM ALLCOMPUTE RESPONSE SPECTRA DISPL MODAL COMBINATION RMS JOINTS ALL

20

STORE TIME HISTORY ACCELERATION TRANSLATION – 'EARTHQ' FACTOR 0.262310.0000000 0.0000000-0.0441006 0.0100000-0.0805970 0.0200000……………………………………………………………………………………………………………………………………………………………………………………TRANSIENT LOADING 1SUPPORT ACCELERATION TRANSLATION X FILE 'EARTHQ'INTEGRATION FROM 0.0 TO 25.0 AT 0.01ACTIVE MODES ALLDYNAMIC ANALYSIS MODAL

STORE TIME HISTORY ACCELERATION TRANSLATION – 'EARTHQ' FACTOR 0.262310.0000000 0.0000000-0.0441006 0.0100000-0.0805970 0.0200000……………………………………………………………………………………………………………………………………………………………………………………TRANSIENT LOADING 1SUPPORT ACCELERATION TRANSLATION X FILE 'EARTHQ'INTEGRATION FROM 0.0 TO 25.0 AT 0.01ACTIVE MODES ALLDYNAMIC ANALYSIS MODAL

Time history analysisTime history analysis.. GTSTRUDL statement GTSTRUDL statement..21

0 5 10 15 20 25-6

-4

-2

0

2

4

6x 10

4

Time t, s

Pyl

on

leg

mo

me

nt,

mto

n x

m

0 5 10 15 20 25-6000

-4000

-2000

0

2000

4000

6000

Time t, s

Pie

re m

om

ent,

mto

n x

m

0 5 10 15 20 25-400-300-200-100

0100200300400

Time t, s

ST

U f

orc

e, m

ton

0 5 10 15 20 25-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time t, s

Pyl

on

to

p d

isp

lace

men

t, m

The time history analysis resultsThe time history analysis results 22

Time history analysis recordTime history analysis record23

The Result comparisonThe Result comparison

Parameter Response spectrum Time history

Pylon leg bending moment

49990 mton x m 57300 mton x m

Pier bending moment

4767 mton x m 5074 mton x m

Shock-transmitter

unit force302 mton 363 mton

Pylon top displacement

0.127 m 0.116 m

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ConclusionConclusion1.1. Time history analysis using artificial accelerograms Time history analysis using artificial accelerograms

overcome weaknesses of the response spectrum method:overcome weaknesses of the response spectrum method:

a)a) this analysis is applicable for structures with anti-seismic this analysis is applicable for structures with anti-seismic devices, which make behavior of the structures nonlinear;devices, which make behavior of the structures nonlinear;

b)b) this analysis can take into account seismic wave this analysis can take into account seismic wave propagation;propagation;

c)c) this analysis does not consider mode shape vibration as this analysis does not consider mode shape vibration as statistically independent;statistically independent;

d)d) this analysis uses exact methods of taking into account this analysis uses exact methods of taking into account structural damping. structural damping.

2. Time history analysis using artificial acelerograms does not 2. Time history analysis using artificial acelerograms does not contradict with national codes.contradict with national codes.

3. Time history analysis using artificial acelerograms usually 3. Time history analysis using artificial acelerograms usually gives higher value of forces and displacements.gives higher value of forces and displacements.

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Thank you for your attentionThank you for your attention