Reliable Dynamic Analysis 006_Modares.ppt

Reliable Dynamic Analysis of Transportation Systems

1. RELIABLE DYNAMIC ANALYSIS OF
TRANSPORTATION SYSTEMS
Mehdi Modares, Robert L. Mullen and Dario A. Gasparini
Department of Civil Engineering
Case Western Reserve University

2. Dynamic Analysis
An essential procedure in transportation engineering to
design a structure subjected to a system of moving loads.

3. Dynamic Analysis
In conventional dynamic analysis of transportation
systems, the possible existence of any uncertainty present
in the structure’s mechanical properties and load’s
characteristics is not considered.

4. Uncertainty in Transportation Systems
Attributed to:

Structure’s Physical Imperfections

Inaccuracies in Determination of Moving Load

Modeling Complexities of Vehicle-Structure Interaction
For reliable design, the presence of uncertainty must be
included in analysis procedures

5. Objective
To introduce a method for dynamic analysis of a structure
subjected to a moving load with properties of structure and load
expressed as interval quantities.
Procedure
To enhance the conventional continuous dynamic analysis for
considering the presence of uncertainties.

6. Presentation Outline

Review of deterministic continuous dynamic analysis

Fundamentals of structural uncertainty analysis

Introduce interval continuous dynamic analysis

Example and conclusion

7. Dynamic Analysis of Continuous Systems
Considering a flexural beam subjected to a load moving with
constant velocity:

The partial differential equation of motion:
)(
),(),(
2
2
4
4
vtxP
t
txu
m
x
txu
EI 







)0(
v
L
t

8. Solution to Free Vibration
Considering the free vibration and assuming a harmonic function:

Substitute in the equation of motion, the linear eigenvalue problem:
Considering:
ti
extxu

)(),(
)(
)(
2
2
2
2
2
xm
dx
xd
EI
dx
d











EI
m
k
2
4


9. Solution to Free Vibration (Cont.)
Consider the solution:

Applying boundary conditions for simply-supported beam:
The non-trivial solution to the characteristic equation can be
obtained.
kxAkxAkxAkxAx coshsinhcossin)(
4321

0)()()0()0(  LL

10. Eigenvalues and Eigenfunctions
Natural circular frequencies (Eigenvalue):
Mass-orthonormalized mode shape (Eigenfunction):
Normalized by:
4
22
Lm
EI
n
n
)sin(
2
)(
L
xn
Lm
x
n




L
nn xmx
0
1)()(

11. Orthogonality
Considering two different mode shapes:
Orthogonality of eigenfunctions with respect to left and right
linear operators in eigenvalue problem.

 




L
mn
L
mn
xmx
xEI
dx
d
x
0
0
4
4
0)()(
0)()(


)(nm

12. Solution to Forced Vibration
The solution for the forced vibration may be expressed as:

where is the modal coordinate.
Substituting in equation of motion, decoupling by:

Premultiplying by each mode shape

Integrating over the domain

Invoking orthogonality

Adding modal damping ratio




1
)()(),(
n
nn xtytxu 
ny

13. Decoupled System
The modal equation of motion is:
or:
where is the modal participation factor.


L
nnnnnnn
dxvtxPxtytyty
0
2
)()()()(2)( 


)sin(
2
)sin()()(2)(
2
t
L
vn
Lm
Pt
L
vn
tytyty
nnnnnnn



 
LmP
n
/2



14. Updated Modal Coordinate
Considering an updated modal coordinate:

The updated modal equation:
n
n
n
ty
td


)(
)(
)sin()()(2)(
2
t
L
vn
tdtdtd
nnnnnn

 

)0(
v
L
t

15. Maximum Modal Coordinate
Response Spectrum: Function of maximum dynamic amplification
response versus the natural frequencies for an assumed damping ratio.
The maximum modal coordinate is obtained using response spectrum of
each mode for a given and .
Biot (1932)
max,nd

n

n


16. Maximum Modal Response
The maximum modal displacement response is the product of:

Maximum modal coordinate

Modal participation factor

Mode shape
)sin()
2
)(())()()((
max,max,max,
L
xn
Lm
P
dxdu
nnnnn





17. Total Response (Deterministic)
In the final step, the finite contributions (N ) all maximum modal
responses must be combined to determine the total response:

Summation of absolute values of modal responses

Square Root of Sum of Squares (SRSS) of modal maxima
Rosenbleuth (1962)



N
n
nuu
1
2
max,max )(



N
n
nuu
1
max,max

18. Engineering Uncertainty Analysis
 Formulation
Modifications on the representation of the system characteristics
due to presence of uncertainty

Computation
Development of schemes capable of considering the uncertainty
throughout the solution process

19. Uncertainty Analysis Schemes
Considerations:

Consistent with the system’s physical behavior

Computationally feasible

20. Paradigms of Uncertainty Analysis

Stochastic Analysis : Random variables

Fuzzy Analysis : Fuzzy variables

Interval Analysis : Interval variables

21. Interval Variable
A real interval is a set of the form:
Archimedes (287-212 B.C.)
}|{],[
~
ulul
zzzzzzZ 
],[
~
bax
)
7
1
3
71
10
3( 

22. Interval Dynamic Analysis
Considering a beam with uncertain modulus of elasticity
subjected to an uncertain load ,

The partial differential equation of motion:
)0(
v
L
t
],[
~
ul
EEE
)(
~),(),(~
2
2
4
4
vtxP
t
txu
m
x
txu
IE 







],[
~
ul
PPP


23. Interval Eigenvalue Problem
The solution to interval linear eigenvalue problem:
















 )(
)(
:
~
|
~ 2
2
2
2
2
xm
dx
xd
EI
dx
d
EE 



24. Solution
Interval natural circular frequencies (Interval Eigenvalue):
Mode shape (Eigenfunction):

4
22~
Lm
EI
n
n
)sin(
2
)(
L
xn
Lm
x
n



25. Monotonic Behavior of Frequencies
Re-writing interval natural circular frequencies:
In continuous dynamic system, it is self-evident that the variation
in stiffness properties causes a monotonic change in values of
frequencies.

4
22
]),([
~
Lm
I
EEn
ul
n

26. Interval Eigenvalue Problem in Discrete Systems
Interval eigenvalue problem using the interval global stiffness matrix:
Rayleigh quotient (ratio of quadratics):
}]{)[
~
(}){]])[,([(
2
1
 MKul
n
i
iii 

xx
Axx
xR
T
T
)(

27. Bounds on Natural Frequencies
The first eigenvalue – Minimum:
The next eigenvalues – Maximin Characterization:
)
][
]),([(min]
)]])[,([(
[min
~
1
1
1
Mxx
xKx
ul
Mxx
xKulx
T
i
Tn
i
ii
Rx
T
n
i
iii
T
Rx
nn 






]
][
]),([minmax[]
)]])[,([(
minmax[
~
1
1,...,1,0.
1
1,...,1,0. Mxx
xKx
ul
Mxx
xKulx
T
i
Tn
i
ii
kizx
T
n
i
iii
T
kizx
k
ii








28. Bounding Deterministic Eigenvalue Problems
Solution to interval eigenvalue problem correspond to the maximum
and minimum natural frequencies:
Two deterministic problems capable of bounding all natural
frequencies of the interval system
(Modares and Mullen 2004)
}]{)[(}){])[((
2
max
1
 MKu
n
i
ii 

}]{)[(}){])[((
2
min
1
 MKl
n
i
ii 


29. Maximum Modal Coordinate
Having the interval natural frequency, the interval modal coordinate is
determined using modal response spectrum as:
The maximum modal coordinate:
)
~
max(
max, nn
dd

~
nd
~

30. Maximum Modal Participation Factor
The interval modal participation factor:
The maximum modal participation factor:
Lm
P
n
2~~

Lm
P
u
nn
2
)()
~
max(
max, 

31. Maximum Modal Response
The maximum modal displacement response is the product of:

Maximum modal coordinate

Maximum modal participation factor

Mode shape
)sin()
2
)(())()()((
max,max,max,max,
L
xn
Lm
P
dxdu
u
nnnnn





32. Total Response
In the final step, the finite contributions all maximum modal responses is
combined using Square Root of Sum of Squares (SRSS) of modal
maxima:



N
n
n
uu
1
2
max,max
)(

33. Example
A continuous flexural simply-supported beam with interval uncertainty
in the modulus of elasticity and magnitude of moving load.
Structure’s Properties:
Load Properties:
ftL200
2
/576000])1.1,9.0([ ftkipsE%1gkipsm /11
4
700ftI
 PP ]1.1,9.0[
~
mphv55

34. Solution
The problem is solved by:

The present interval method

Monte-Carlo simulation
(using bounded uniformly distributed random variables in
10000 simulations)

35. Results
Bounds on the fundamental natural frequency (first mode)
1

Lower Bound
Present
Method
Lower Bound
Monte-Carlo
Simulation
Upper Bound
Monte-Carlo
Simulation
Upper Bound
Present
Method
1.41717 1.41718 1.56673 1.56675

36. Response Spectrum for Fundamental Frequency
)sin()()(2)(
2
t
L
vn
tdtdtd
nnnnnn

 


37. Results
The upperbounds the mid-span displacement response for the
fundamental mode
P
u
1
Upper Bound
Monte-Carlo
Simulation
Upper Bound
Present Method
8.06557e-004 8.12128e-004

38. Beam Fundamental-Mode Response

39. Conclusion

A new method for continuous dynamic analysis of transportation
systems with uncertainty in the mechanical characteristics of the
system as well as the properties of the moving load is developed.

This computationally efficient method shows that implementation of
interval analysis in a continuous dynamic system preserves the
problem’s physics and the yields sharp and robust results. This may
be attributed to nature of the closed-form solution in continuous
dynamic systems.

The results show that obtaining bounds does not require expensive
stochastic procedures such as Monte-Carlo simulations.

The simplicity of the proposed method makes it attractive to
introduce uncertainty in analysis of continuous dynamic systems.

40. Questions