RELIABLE DYNAMIC ANALYSIS OF

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 structures
  mechanical properties and loads characteristics
  is not considered.

4

• Uncertainty in Transportation Systems
• Attributed to
• Structures 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

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

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.

10

• Eigenvalues and Eigenfunctions
• Natural circular frequencies (Eigenvalue)
• Mass-orthonormalized mode shape (Eigenfunction)
  
• Normalized by

11

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

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

13

• Decoupled System
• The modal equation of motion is
• or
• where is the modal
  participation factor.

14

• Updated Modal Coordinate
• Considering an updated modal coordinate
• The updated modal equation

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)

16

• Maximum Modal Response
• The maximum modal displacement response is the
  product of
• Maximum modal coordinate
• Modal participation factor
• Mode shape

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)

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 systems 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.)

22

• Interval Dynamic Analysis
• Considering a beam with uncertain modulus of
  elasticity subjected to an uncertain load
  ,
• The partial differential equation of motion

23

• Interval Eigenvalue Problem
• The solution to interval linear eigenvalue
  problem

24

• Solution
• Interval natural circular frequencies (Interval
  Eigenvalue)
• Mode shape (Eigenfunction)

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.

26

• Interval Eigenvalue Problem in Discrete Systems
• Interval eigenvalue problem using the interval
  global stiffness matrix
• Rayleigh quotient (ratio of quadratics)

27

• Bounds on Natural Frequencies
• The first eigenvalue Minimum
• The next eigenvalues Maximin Characterization

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)

29

• Maximum Modal Coordinate
• Having the interval natural frequency, the
  interval modal coordinate is determined using
  modal response spectrum as
• The maximum modal coordinate

30

• Maximum Modal Participation Factor
• The interval modal participation factor
• The maximum modal participation factor

31

• Maximum Modal Response
• The maximum modal displacement response is the
  product of
• Maximum modal coordinate
• Maximum modal participation factor
• Mode shape

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

33

• Example
• A continuous flexural simply-supported beam with
  interval uncertainty in the modulus of elasticity
  and magnitude of moving load.
• Structures Properties
• Load Properties

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)

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

37

• Results
• The upperbounds the mid-span displacement
  response for the fundamental mode

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 problems
  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
•