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Reliable Dynamic Analysis of Structures Using Imprecise Probability

Mehdi Modares and Joshua BergersonDepartment of Civil, Architectural, and Environmental Engineering,

Illinois Institute of Technology, Chicago, IL, 60616, USA,Emails: [email protected], [email protected]

Abstract: Dynamic analysis of a structure is an essential procedure in designing a structure subjected to asystem of suddenly applied loads such as wind or earthquake excitations. However, throughoutconventional dynamic analyses, the existence of uncertainties in mechanical properties of a structure hasnot been considered. One method to quantify the presence of uncertainty in a physical system is to use animprecise probability framework based on the concept of p-boxes.

In this work, a new method for reliable dynamic analysis of structures using p-box based impreciseprobability is developed. This method establishes a framework for handling incomplete information in

structural dynamics. Using this method, the reliability and robustness in dynamic analysis are achieved.Example problem that illustrates the developed algorithm is presented.

Keywords:Structural Dynamics; Uncertainty; Imprecise Probability; P-Box.

1. IntroductionThroughout conventional dynamic analysis, structural parameters in a system (including geometric,material, and load properties) are treated as deterministic values. Although the true values for structuralproperties generally have small variations from the deterministic values used in analysis, these uncertainties

affect the systems dynamic response significantly.There are two distinct types of uncertainty: aleatoric and epistemic. Aleatoric uncertainty is due to

inherent randomness in the system. This type of uncertainty is irreducible with higher precision or moredata acquisition. Epistemic uncertainty is due to lack of knowledge, modeling errors, and/or insufficient

data. This uncertainty, also known as reducible uncertainty, can be reduced with further investigations, moredata, and updated models.

A method for handling uncertainty in a system with no assumptions of a well-defined ProbabilityDensity Function (PDF) is imprecise probability. This method involves setting bounds on the CumulativeDistribution Function (CDF), between which the true CDF exists. Williamson and Downs (1990) developed

a method for combining imprecise probabilities using the four basic arithmetic operations, known asdependency bounds convolutions. These methods are enveloping methods in order to retain robustness in

calculations, while avoiding overestimation on bounds due to dependency between different imprecise

probabilities. Ferson and Donald (1998) developed a method for handling imprecise probability, referred toas Probability Bounds Analysis (PBA). Another method for handling imprecise probability is to discretizethe upper and lower CDFs into probability boxes (p-boxes), each of equal probability mass. (Ferson et al.1998)

A much older framework for handling imprecise probability is the Dempster-Shafer approach. While

5th International Conference on Reliable Engineering Computing (REC 2012)

Edited by M. Voechovsk, V. Sadlek, S. Seitl, V. Vesel, R. L. Muhanna and R. L. Mullen

Copyright 2012 BUT FCE, Institute of Structural Mechanics

ISBN 978-80-214-4507-9. Published by Ing. Vladislav Pokorn LITERA

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Mehdi Modares and Joshua Bergerson

the development of the Dempster-Shafer (DS) structures has been confirmed several times, the methodsproposed for combining multiple DS structures has shown complexities. Ferson et al. (2003) suggested thatthe concept of Dempster-Shafer structures and p-boxes are in fact highly related and outlined procedures forconverting from one to the other. They also expanded methods for combining multiple DS structures or p-

boxes, for operations including: enveloping, intersection, unweighted and weighted mixtures, averaging,and logarithmic pooling. Moreover, they outlined the various desired properties of any method ofcombining multiple p-boxes or DS structures, and summarized which methods satisfied these properties.From this, they concluded that the best method for combining is enveloping, also used in this work.

Modares et al. (2006) showed a method of dynamic analysis using finite element method formechanical uncertainties defined by interval variables, which yields exact bounds on a systems naturalfrequencies. This method utilizes the monotonic behavior of eigenvalues of symmetric matrices subject tonon-negative definite perturbations. Zhang et al. (2010) outlined a method for extending impreciseprobability methods to finite element analysis with p-box uncertainty in loading in static problems.

In this work, a new method for reliable dynamic analysis of structures using p-box based impreciseprobability is developed which establishes a framework for handling incomplete information in structural

dynamics. Independent uncertainties in stiffness defined by p-boxes are considered. This method applies themethod for bounding natural frequencies, developed by Modares et al. (2006), to analysis of systems withindependent interval uncertainty for each level of p-box discretization. Using this method, the reliability androbustness in dynamic analysis of an uncertain system are achieved. Although this work represents p-boxuncertainties in stiffness properties, the extension of the developed method for uncertainties in the mass

matrix is straightforward. Numerical example that illustrates the developed algorithm is presented, as wellas comparison to interval Monte-Carlo simulation.

2. Review of Deterministic Frequency AnalysisThe equation of motion for an undamped system is:

[M]{x} + [K]{x} = {0}, (1)

where is the global mass matrix, is the global stiffness matrix, x is the displacement ector, and} is the acceleration vector. Assuming a harmonic motion for the displacement

{x} { }ei tu (2)

the equation of motion becomes a generalized eigenvalue problem as:

([K] ( )[M]){ }={0}u (3)

or:

[K]{ } = ( )[M]{ },u u (4)

where is the set of circular natural frequencies of the system and {u} is the set of their correspondingmode shapes.

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selecting values of (x )UiF corresponding to zq / for every member ifor

zq ,2,1, .

This may be expressed using the inverse functions1

iF

and1

iF

as:

1 1x = ,iiqL qFz

(8)

1.x =iqU i

qF

z

(9)

These bounds can be combined for each value of qand written in interval form for member ias:

x =[x , x ]iq iqL iqU (10)

with each iqx~ having a probability mass of (1/z) for Ni ,2,1,= , whereNis the number of members in the

system.

One issue with this discretization process is that when evaluating 1

0 1iF q , many common

distributions produce infinite tails. This problem also exists when evaluating 1 1iF q z . Therefore, the

tails must be truncated. Ferson et al. (2003) details the effects of truncating infinite tails and concluded thatwhile the decision of where to truncate can have an effect on analysis, it is generally so small that it isnegligible compared with the magnitude of the p-box or compared to the effect of discretizing the p-box

even whenzis sufficiently large. Thus, the infinite tails of (x )i LF and (x )UiF

are truncated at and

(1 ) , respectively, where (1/ )z . Thus, for (x )i LF and (x )UiF

with infinite tails:

1

1x = ( )ii L F

(11)

1x = (1 )izU iF

(12)

4.2 P-BOX BASED FREQUENCY ANALYSIS

The monotonicity concept for natural frequencies proven by Modares et al. (2006) is used for boundingeach p-box based eigenvalue problem (Eqs. (6) & (7)) as follows:

1 2

21( ) 1 2( ) 2 ( )(x [K ] x [K ] x [K ]){ }=( )[M]{ }Nq U q U N q U N U u u (13)

1 2

21( ) 1 2( ) 2 ( )(x [K ] x [K ] x [K ]){ }=( )[M]{ }q L q L N q L N L

Nu u (14)

where iq are independently defined as possibilities of discretization levels for element i: [1, 2,, ]iq z .

The results are arranged and condensed back down to zvalues for each L and U

following procedure

for such condensation outlined by Williamson and Downs (1990). These z intervals are then used toconstruct p-boxes for each natural frequency.

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5.1 SOLUTION USING DEVELOPED METHOD

The above three p-boxes are discretized into ten intervals. The results are given in the Table I in the form ofthe interval of Eq. (10).

Table I. Coordinates for the discretized p-boxes of E1, E2, and E3.

1/E 2/E 3/E

x1qL

x1qR

x2qL

x2qR

x3qL

x3qR0.9000 1.0251 0.7001 1.0754 0.8001 1.0503

0.9251 1.0337 0.7754 1.1010 0.8503 1.0673

0.9337 1.0398 0.8010 1.1195 0.8673 1.0797

0.9398 1.0451 0.8195 1.1353 0.8797 1.0902

0.9451 1.0500 0.8353 1.1500 0.8902 1.1000

0.9500 1.0549 0.8500 1.1647 0.9000 1.1098

0.9549 1.0602 0.8647 1.1805 0.9098 1.1203

0.9602 1.0663 0.8805 1.1990 0.9203 1.13270.9663 1.0749 0.8990 1.2246 0.9327 1.1497

0.9749 1.1000 0.9246 1.2999 0.9497 1.1999

Figures 4a, 4b, and 4c show the discretized p-boxes for each modulus of elasticity.

Figure 4a, 4b, & 4c.Discretized p-box for E1, E2, & E3, respectively.

Using Eqs. (13) and (14), lower and upper bounds on each natural frequency of the system are obtained,and the data is condensed tozintervals. Results are given in Table II and depicted in Figures 5a, 5b, and 5c.

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lower and upper natural frequencies. The results for lower and upper natural frequencies are sorted andcondensed to ten discretizations. The results are summarized below in Table III.

Table III. Condensed p-box coordinates from Monte-Carlo simulation for all natural frequencies

1MCL/(E/)

2MCL/(E/)

3MCL/(E/)

1LMCL/(E/)

1UMCL/(E/)

2LMCL/(E/)

2UMCL/(E/)

3LMCL/(E/)

3UMCL/(E/)

0.5719 0.6595 0.8969 1.0299 1.2303 1.4035

0.5923 0.6623 0.9262 1.0361 1.2668 1.4107

0.5962 0.6644 0.9335 1.0409 1.2745 1.4163

0.5990 0.6663 0.9391 1.0447 1.2802 1.4208

0.6015 0.6680 0.9437 1.0484 1.2850 1.4249

0.6038 0.6697 0.9482 1.0517 1.2894 1.4293

0.6060 0.6715 0.9521 1.0555 1.2938 1.4335

0.6083 0.6736 0.9566 1.0597 1.2984 1.4388

0.6110 0.6765 0.9617 1.0655 1.3040 1.4461

0.6145 0.6902 0.9687 1.0885 1.3118 1.4777

Figures 6a, 6b, and 6c show the results from both the developed method (solid lines) and the Monte-

Carlo simulation (dashed lines). As observed, the bounds on natural frequency from Monte-Carlosimulation are inner-bounds of the developed method.

Figure 6a,6b, & 6c. Condensed p-boxes for 1, 2, & 3, respectively, with Monte-Carlo results represented by dashed lines.

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Reliable Dynamic Analysis of Structures Using Imprecise Probability

6. ConclusionA new method for dynamic analysis of structures with uncertain properties defined by independent p-boxesis developed. The developed method allows for uncertainty in the stiffness matrix, but can be shown to

handle uncertainty in the mass matrix under the same framework. The developed method also allows for anuncertain dynamic analysis, yielding best-possible, discrete p-boxes for all natural frequencies of thedynamic system. The results obtained by this method are upperbounds of the results obtained by intervalMonte-Carlo simulation procedures.

References

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Berleant, D., and Goodman-Strauss, C. Bounding the Results of Arithmetic Operations on Random Variables of UnknownDependency Using Intervals,Reliable Computing, 4(2), 147-165, 1998.

Ferson, S., and Donald, S. Probability Bounds Analysis,Probabilistic Safety Assessment and Management, Mosleh, A., and Bari,R.A. Eds., Springer-Verlag, New York, NY, 1203-1208, 1998.

Ferson, S., Kreinovich, V., Ginzburg, L., Myers, D., S., Sentz, K. Construction probability boxes and Dempster-Shafer structures,

Tech. Rep. SAND2002-4015, Sandia National Laboratories 2003.Modares, M., Mullen, R., L., Muhanna, R., L. Natural Frequencies of a Structure with Bounded Uncertainty,Journal of

Engineering Mechanics, ASCE, December ed., 1363-1371, 2006.Muhanna, R. L., and Mullen, R. L. Uncertainty in Mechanics Problems-Interval-Based Approach. Journal of Engineering

Mechanics, June-2001, 557-566, 2001.Oberkampf, W., Helton, J. and Sentz, K. athematical Representations of Uncertainty,AIAA Non-Deterministic Approaches

Forum, AIAA 2001-1645, Seattle, WA, April 16-19, 2001.Williamson, R., and Downs, T. Probabilistic Arithmetic I: Numerical Methods for Calculating Convolutions and Dependency

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Zhang, H., Mullen, R., L., Muhanna, R., L. Finite Element Structural Analysis using Imprecise Probabilities Based on P-BoxRepresentation, 4thInternational Workshop on Reliable Engineering Computing, 211-225, 2010.


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