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

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

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.

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

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.

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

Dynamic Analysis of Continuous Systems Considering a flexural beam subjected to a load moving with constant velocity: The partial differential equation of motion:

Solution to Free Vibration Considering the free vibration and assuming a harmonic function: Substitute in the equation of motion, the linear eigenvalue problem: Considering:

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.

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

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

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

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

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

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)

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

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)

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

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

Paradigms of Uncertainty Analysis Stochastic Analysis : Random variables Fuzzy Analysis : Fuzzy variables Interval Analysis : Interval variables

Interval Variable A real interval is a set of the form: Archimedes ( B.C.)

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

Interval Eigenvalue Problem The solution to interval linear eigenvalue problem:

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

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.

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

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

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)

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

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

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

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:

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:

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

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

Response Spectrum for Fundamental Frequency

Results The upperbounds the mid-span displacement response for the fundamental mode Upper Bound Monte-Carlo Simulation Upper Bound Present Method e e-004

Beam Fundamental-Mode Response

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.

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