Bentley RM Bridge Seismic Design and Analysis Alexander Mabrich, PE, Msc.
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Transcript of Bentley RM Bridge Seismic Design and Analysis Alexander Mabrich, PE, Msc.
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
8 | WWW.BENTLEY.COM
Equivalent Static Analysis
• Uniform Load Analysis
• Orthogonal Displacements
• Simultaneously
• Fundamental mode
12 | WWW.BENTLEY.COM
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
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
27 | WWW.BENTLEY.COM
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
29 | WWW.BENTLEY.COM
Response Spectrum
• Combination of natural modes
• One mass oscillator
• Oscillating loads
• Intensity factor
• Single contribution
• Synchronization by Stochastic Calculation Rules: ABS,SRSS,CQC, etc
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
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
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
Thank you for your [email protected]