CABLE-STAYED BRIDGE SEISMIC
ANALYSIS USING ARTIFICIAL
ACCELEROGRAMS
Roman Guzeev, Ph.D.
Institute Giprostroymost-Saint-Petersburg
Russian Federation
http://www.gpsm.ru
The Eastern Bosporus bridge, Vladivostok, Russia
1
2
Presentation of the Response spectrum in national codes
AASHTO LFRD
Bridge Design
Specification
EUROCODE
EN 1998-1:2004
Design of structures for
earthquake resistance
Design structures in
earthquake regions
(Russian code)
3
Disadvantages of
the Response Spectrum Method
 it is inapplicable for structures with anti-seismic devices,
which make behavior of the structures nonlinear
 It does not take into account seismic wave propagation
 It considers mode shape vibration as statistically independent
 It uses approximate relations between response spectrum
curves with different damping.
4
Time history analysis using accelerograms.
Instrumentally recorded ground acceleration.
1994, Northridge, Santa Monica, City Hall Grounds
3
Nondimensional response
Scaled acceleration, m/s2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
10 15 20 25 30 35 40 45 50 55 60
Time T, sec
Period T, sec
1940, El Centro Site
3.5
Nondimensional response
Scaled acceleration, m/s2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
10 15 20 25 30 35 40 45 50 55 60
Time T, sec
3
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Period T, sec
5
Instrumentally recorded ground acceleration ?
The main features
 Instrumentally recorded earthquake acceleration is an
event of random process
 Every earthquake is unique and has its own peak
acceleration and spectral distribution
 Any earthquake Depends on ground condition
 Instrumentally recorded acceleration can be dangerous to
one type of structure and can be safe to another
6
Artificial accelerograms
Artificial accelerogram should meet requirements of
national codes:
1. There should be a peak value on accelerogram. The
peak value depends on the region seismic activity,
ground condition and period of exceedance.
2. Accelerogram response spectrum should match design
spectrum
7
Artificial accelerograms
Artificial accelerogram should meet physical
requirements:
1. Acceleration, velocity and displacement should
be equal to zero at the beginning and at the end
of the earthquake
2. Duration of the earthquake should not be less
than 10 sec.
The accelerogram generation algorithm
Step 1.
Generating accelerogram with peak value equal to 1
Step 2.
Scaling accelerogram according to the design value
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of ground acceleration.
The accelerogram to be found is presented as trigonometric series:
 2 
 2
a( t )   ai sin 
t  bi cos 
 Ti 
 Ti
i 1
ai , bi -sought coefficients,
N
Ti
-natural period of mode i,
N
-number of considered modes

t

We take into account the modes which contribute to effective modal mass in
the earthquake direction
The accelerogram constraints
9
Peak value nonlinear constraint:

max 


 2
 ai sin  T
 i
i 1
N

 2
t  bi cos 

 Ti

t 1
 
Acceleration, velocity and displacement linear constraints:
At the beginning t=0
N
 bi  0,
i
At the end t=Ts
 2  Ti
 2 
Ti
  2 ai cos  T Ts   2 bi sin  T Ts   0,
 i 
 i 
i 1
N
N
 2 
 2 
a
sin
T

b
cos
 i  T s  i  T Ts   0,
 i 
 i 
i 1
N
 2  Ti 2
 2 
Ti 2
a
sin
T

b
cos
 4 2 i  T s  4 2 i  T Ts   0,
 i 
 i 
i 1
Ti2
 4 2 bi  0,
i 1
Ti
 2 ai  0,
i 1
N
N
Generated accelerogram response spectrum
N


s
с
 (T )  max  a( t )   ai yi ( t )  bi yi ( t )  t  0 Ts


i

1


Where, yis ( t ),
yiс ( t ) t  0 Ts
is the solution of differential equation of motion for one DOF
oscillator on sine and cosine base excitation.
 2
s
yi ( t )  2 d 
 T
 s  2  s
 2
 yi   T  yi  sin  T





t  , yis (0)  0, yis (0)  0,

 2
c
yi ( t )  2 d 
 T
 c  2  c
 2
 yi   T  yi  cos  T





t  , yic (0)  0, yic (0)  0

2
2
 d - damping ratio of design response spectrum
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The coefficient of series terms to be found сan be determined
by means of the least square method with linear and nonlinear
constraints
We minimize the sum square of differences between
accelerogram response spectrum and the design response
spectrum
F  {  (T j )   d (T j )}T W {  (T j )   d (T j )}
F – object sum square function
[W] – diagonal matrix of weight factors
{  (T j )   d (T j )} – vector of differences between the
accelerogram response spectrum and the design response
spectrum
Recommendation on analysis using artificial acelerogram
12
 Terms of series should contain natural frequencies of
structures. It lead to resonance excitation.
 We should take into account the modes which contribute to
effective modal mass in the earthquake direction
 For the closest match to design response spectrum we can
add extra terms into the series
 We have to generate more than one design accelerogram.
We can do it by varying the number of terms and considered
modes
 For every strain-stress state parameter we have to built an
envelope caused by action generated accelerograms
13
Golden Horn Bay cable-stayed bridge, Vladivostok, Russia
Concrete deck
3330 mm
30580 mm
33270 mm
Steel deck
3312 mm
30580 mm
33300 mm
m
m
m
m
14
Seismic action input data
Sd ( i , T )  Sel ( i , T ) K 1
Elastic response spectrum
6
Sd ( i , T ) - design spectrum
5

4.5
4
3.5
S
Sel ( i , T ) - elastic spectrum
5.5
3
2.5
K 1  0.25 - ductility factor
Sel ( i , T )  Sel (0.08, T )
Sel (0.08, T ) - elastic spectrum
with 0.08 damping ratio

0.08
i

2
1.5
1
0.5
0
0

1
2
3
4
5
6
Natural Period T, sec
Peak ground acceleration
Ag  0.107 g
Return period is 5000 years.
- dumping
correction factor
 i - modal damping ratio
GTSTRUDL Model
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Mode
Natural
period /
frequency
Effective
modal mass
Mode shape
lateral
1
T=4.88s
f=0.205 Hz
X: 0%
Y: 0%
Z: 10.0%
vertical
2
T=4.36s
f=0.229 Hz
X: 0%
Y: 6.5%
Z: 0%
vertical
longitudinal
3
T=3.62s
f=0.276 Hz
X: 28.4%
Y: 0%
Z: 0%
17
Mode
Natural
period /
frequency
Effective modal
mass
Mode shape
longitudinal
and lateral
4
T=2.84s
f=0.352 Hz
X: 45.8%
Y: 0%
Z: 0%
lateral
6
T=2.78s
f=0.358 Hz
X: 0%
Y: 0%
Z: 17.8%
Stiffness weighted average damping
Structural
element
Damping ratio
Steel deck
0.02
Concrete deck
0.02
Pylon
Cables
0.025
0.00096
Concrete piers
0.05
CONSTANT
MODAL DAMPING PROPORTIONAL TO STIFFNESS
MODAL DAMPING PROPORTIONAL TO STIFFNESS
MODAL DAMPING PROPORTIONAL TO STIFFNESS
MODAL DAMPING PROPORTIONAL TO STIFFNESS
DYNAMIC PARAMETERS
RESPONSE DAMPING STIFFNESS 1.0
END OF DYNAMIC PARAMETERS
COMPUTE MODAL DAMPING RATIOS AVERAGE
0.025 GROUP 'PYLON'
0.02 GROUP 'DECK'
0.05 GROUP 'SUPP'
0.00096 GROUP 'CABLE'
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19
3
response spectrum
1.1
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.1
0
4
8
12
Time t, s
16
2.5
2
1.5
1
0.5
0.06
0.03
0.04
0.02
0.02
0
-0.02
-0.04
-0.06
0
4
8
12
Time t, s
16
20
natural period
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Period T, s
20
displacement m/s
velosity m/s
acceleration, m/s2
Accelerogram generation results
0.01
0
-0.01
-0.02
-0.03
0
4
8
12
Time t, s
16
20
Response spectrum analysis. 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 LOAD
LOAD LIST 'RSP'
ACTIVE MODES ALL
PERFORM MODE SUPERPOSITION ANALYSIS
COMPUTE RESPONSE SPECTRA FORCES MODAL COMBINATION
RMS MEM ALLCOMPUTE RESPONSE SPECTRA DISPL MODAL
COMBINATION RMS JOINTS ALL
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Time history analysis. GTSTRUDL statement.
STORE TIME HISTORY ACCELERATION TRANSLATION –
'EARTHQ' FACTOR 0.26231
0.0000000 0.0000000
-0.0441006 0.0100000
-0.0805970 0.0200000
…………………………………………………………………………………………
…………………………………………………………………………………………
TRANSIENT LOADING 1
SUPPORT ACCELERATION
TRANSLATION X FILE 'EARTHQ'
INTEGRATION FROM 0.0 TO 25.0 AT 0.01
ACTIVE MODES ALL
DYNAMIC ANALYSIS MODAL
21
22
4
4
2
0
-2
-4
-6
0
400
300
200
100
0
-100
-200
-300
-400
0
5
5
10
15
Time t, s
10
15
Time t, s
20
20
25
25
Piere moment, mton x m
6
x 10
Pylon top displacement, m
STU force, mton
Pylon leg moment, mton x m
The time history analysis results
6000
4000
2000
0
-2000
-4000
-6000
0
5
10
15
Time t, s
20
25
5
10
15
Time t, s
20
25
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0
Time history analysis record
23
24
The Result comparison
Parameter
Response spectrum
Time history
Pylon leg bending
moment
49990 mton x m
57300 mton x m
4767 mton x m
5074 mton x m
302 mton
363 mton
0.127 m
0.116 m
Pier bending
moment
Shock-transmitter
unit force
Pylon top
displacement
Conclusion
1. Time history analysis using artificial accelerograms
overcome weaknesses of the response spectrum method:
a) this analysis is applicable for structures with anti-seismic
devices, which make behavior of the structures nonlinear;
b) this analysis can take into account seismic wave
propagation;
c) this analysis does not consider mode shape vibration as
statistically independent;
d) this analysis uses exact methods of taking into account
structural damping.
2. Time history analysis using artificial acelerograms does not
contradict with national codes.
3. Time history analysis using artificial acelerograms usually
gives higher value of forces and displacements.
25
Thank you for your attention