Interval solution and robust validation of uncertain

Safety, Reliability, Risk and Life-Cycle Performance of Structures & Infrastructures – Deodatis, Ellingwood & Frangopol (Eds) © 2013 Taylor & Francis Group, London, ISBN 978-1-138-00086-5

Interval solution and robust validation of uncertain elastic beams S. Gabriele LaMS – Modeling and Simulation Lab, Department of Architecture, University “Roma Tre” of Rome, Italy

C. Valente Department of Engineering and Geology, University “G. d’Annunzio” of Chieti-Pescara, Italy

M. de Angelis Institute for Risk and Uncertainty, University of Liverpool, UK

ABSTRACT: In the field of structural mechanics the notion of uncertainty is employed in several contexts such as modelling, analysis, experiments, and reliability. When dealing with problems involved with uncertainty the model of the system should include an appropriate representation of the uncertain quantities. Among different formulations capable of representing uncertainties, certainly interval analysis promises to be very effective since it is not required to handle distributions as where probability is concerned. In this work the attention will be focused on the so-called direct problem where the mechanical model and the amount of uncertainty in the parameters are a priori given and the goal is to evaluate how and to what extent the uncertainty propagates and influences the response of the system. In this context, the uncertainty may come from the scarce knowledge about the materials, geometry and boundary conditions that constitute the structural model. An example of this is the uncertainty in loading, which intensity, area and location contribute to shape the mathematical equations that describe the problem. The purpose of the paper is to go through the use of the interval formulation as alternative to the probability formulation in the study of systems embodying uncertain parameters. Sample problems concerning the statics of beams will be addressed and the interval solution will be discussed and compared to analytic or Monte Carlo probability solutions. The effectiveness of the presented approach is demonstrated by means of a real case example, where a set of precast concrete beams undertaking static tests is analysed.

1

INTRODUCTION

In structural mechanics mainly two issues are concerned, which are the direct and the inverse problem (Marwala 2010). While in the inverse problem the objective consists in characterizing the structural properties by jointly using mechanical models and the available experimental data, the direct one aims at seeking the mechanical model that, given already the structural properties, better suites for a specific purpose. Both problems are affected by uncertainty (Gabriele & Valente 2009), firstly because experimental data do not suffice in characterizing the structural properties in almost every engineering application, and secondly because mechanical models represent just the attempt to reproduce the real behavior of structures. In the direct problem it is of interest assessing to what extent the uncertainty propagates and how it influences the system (Schuëller 2007). As long as the knowledge about materials, geometry, loading and boundary conditions is scarce and incomplete, structural models should be capable of providing measures of uncertainty. Interval analysis promises to be particularly effective in addressing the problem posed by the uncertainty propagation, since it is completely

distribution-free. The peculiar feature of interval analysis stems from the fact that, theoretically, every physical model can be simply reformulated within its set-based framework with straightforward extension of the conventional arithmetic. In doing so, measures of uncertainty are given, along with the model responses, leaving a little room for subjective interpretation. The introduction of interval analysis in running the direct approach (Muhanna & Mullen 2001) is motivated by more than one reason. For example, one may want to feed the mechanical model with parameters along with their respective tolerances being sure the structure performs as expected. Or alternatively, one may just desire to be given a wider range of options to choose from, in order to make the right decision at an early stage of design. In the present work interval analysis is invoked to describe the behavior of a whole set of beams made up of precast concrete. These beams are samples taken from mass production. Therefore they are nominally identical within construction tolerances, but once tested they always display different mechanical performances. This is mainly due to the uncertainties on material properties and loading conditions. In this context, interval analysis turns out to be particularly effective because allows accounting for

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uncertainties comprising inaccuracy of experimental procedures. Despite the fact that interval analysis alone constitutes a powerful tool to account for uncertainties, this does not prevent it to be extended to include probability concepts. It may be known for example, that the properties of a specific material are said to comply with a probability distribution; hence, one is able to associate a measure of probability to any interval of those properties (Beer, Zhang, Quek, & Phoon 2013). In these cases the proposed methodology based on interval analysis is still valid and may be used to predict the degree of confidence of the interval response outputting the model. 2 2.1

INTERVAL ANALYSIS Notations

Interval variables are denoted by square brackets; thus, if x is a point-wise (crisp) variable ranging within the limits xL and xU its classical interval notation reads

The central value and the radius of an interval variable are xc = (xL + xU )/2 and xr = (xU − xL )/2 respectively. By introducing the e
= [−1, 1] interval an alternative arithmetic representation of notation 1 is available, that is the so called central notation, see e.g. (Hansen & Walster 2004)

spread by means of interval operations. Because of the dependence, specific algorithmic strategies present in literature (Corsaro & Marino 2006), are to be developed in order to keep tight the interval width in the results and to successfully limiting the problem. 3

In this section two interval-based beam models are considered and compared one another. In both models the uncertainty is described by interval variables. Notice that using intervals different models may arise depending upon the strategy chosen to cope with interval extensions. The first model derives from the closed-form solution of the fourth-order linear differential equation of an Euler-Bernoulli’s beam; while the second model is obtained by considering the weak formulation of the same beam problem, whose solution is accessed via the principle of least action. This section shows how it is possible to cope with natural interval extension by redefining the mechanical formulation of the model (Muhanna & Mullen 2001, Dessombz, Thouverez, Laîné, & Jézéquel 2001). In interval analysis some mathematical expression (or extension) may be substantially better than others. Here it is shown that seeing the Euler-Bernoulli’s beam from a different mechanical perspective allows to sensibly reduce the dependence effect.

3.1 Notation 2 may turn out to be of particular convenience because allows providing the central value with the physical connotation of nominal value, while the radius quantifies the amount of uncertainty to be attributed to the interval quantity itself. Therefore, by means of this representation, it is much easier to deliver the physical meaning of uncertain quantities defined in terms of intervals. 2.2

Dependence problem

Within the framework of interval analysis, interval arithmetic allows computing with intervals in place of real numbers. Thus, the all operations can be redefined and used for interval computations, that become “interval extensions” when dealing with interval-valued functions. But, as soon as one realizes that computing with intervals is possible by complying to simple rules of common sense, it may also be noted that something might not be as expected. In fact, consider the case of subtracting the interval [x] = [xL , xU ] from itself: by applying the rule of subtraction (Moore 1969), one gets the interval [xL − xU , xU − xL ], which is not what expected [0, 0], unless xL = xU . This is what the dependence effect (Hansen & Walster 2004), also known as dependability effect, is about; i.e. the fact that the original physical uncertainty can be artificially

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INTERVAL-BASED BEAM MODELS

Closed-form solution

The point-wise solution of the Euler-Bernoulli’s beam may be obtained by solving the linear differential equation to the fourth order:

where B = EI is the bending stiffness of the beam, v is the deflection and p is the applied load. In the case of interest, the load is applied over a very small area compared to the size of the beam, and therefore may be well-described by the delta function. The delta function is characterized by the fundamental property

where x is an additional variable varying along with z, p(x) is any integrable function, and δ(.) is the delta function. In physical terms such assumption corresponds to apply a concentrated load of intensity P over the beam at location x. The point-wise solution of the above problem is unique, and can be obtained by integrating equation 3, given the boundary conditions. Integrating the delta function leads to the Heaviside step function, which introduces a singularity at x location. By splitting the beam domain up into two parts

one may still access the analytic solution, which, after nondimensionalization, reads

In the previous equation ζ and y are normalized variable with respect to the length L, which allow to better describe the solution over the beam domain; whilst the load intensity P is the integral in equation 4. Once the analytic solution 5 is found, the introduction of the interval uncertainty needs an interval extension to be chosen. As in (Moore, Kearfott, & Cloud 2009) is well presented, this choice is not unique. Different kind of interval extensions can be defined to obtain a different level of refinement in the sought solution, namely an interval in the function co-domain. It is out of the purposes of this paper to compare the solution among these different choices. For this reason we choose the straightforward way of the “natural extension”. Even if the authors are aware that this is not the best choice for numerical purposes, the natural extension maintains the structure of the original equations and allows to discuss about the mechanical relevance of the uncertain quantities B, p and x when considered as intervals. A further work extension will need to consider different interval extensions of equation 5. It can be seen from equation 5 that the variables of the problem can be treated separately with respect to the interval uncertainty. The interval solution of equation 3 when B, p, and x are uncertain is obtained by letting these quantities vary in the respective intervals [B], [p] and [x]. From a practical viewpoint, this can be achieved by using interval arithmetic to obtain the following interval extension:

where [y] is given by [y] = [x]/L, and y˜ is any value within the interval [y]. The peculiarity of the interval [y] is that it has a limited variability, because for evident reasons it may be any interval enclosed in [0, 1]; therefore it holds that 0 = min(inf ([y])) and 1 = max(sup([y])). From equation 5 it can be seen that the uncertainty concerning the beam properties and load intensity can be considered separately from the uncertain variable y. Whilst coefficient [P]/(6[B]L5 ) of equation 6 does not lead to overestimation; the function related to the load position, in both partitions of the beam domain, suffers by overestimation because of the dependence problem.

Any set that contains at least all the solutions of equation 6, when variables range within respective intervals, is said to be an inclusive solution. The need to get inclusive intervals is justified by the fact that no solutions would be missed. This would allow both better describing the physical behavior and have a close match with the experimental results. The interval extension of equation 6 can be improved by arithmetically reducing the number of appearances of [y] in the second partition of the domain. For example the expression on the first partition of the beam −(1 − [y])ζ 3 − [y](1 − [y])(2 − [y])ζ may become −(1 − [y])ζ 3 + [y](([y] − 3/2)2 − 1/4)ζ. However, this version still does not provide sharp enough results, since many occurrences of the interval [y] are displayed. Despite each term of the above expression can be individually exactly bounded (for example exact bounds of [y](1 − [y]2 ) can be d obtained by studying the functions dy y(1 − y2 ) = 1 − 2

3y2 , dd2 y y(1 − y2 ) = −6y), as soon as these terms are composed back together they will no longer result in the sharpest interval, because of the dependency effect; in fact, these terms still depend on one another through the variable y. In order for equation 6 to be inclusive, extremes of the deflection v(ζ) are needed, which are obtained by looking at the minimum and maximum value of y˜ ∈ [y]. Thus the sought interval is given by

i.e. by the union of all functions given in equation 5 obtained varying y˜ ∈ [y], which is the quantity defining the two partition of the beam. The previous equation cannot be rigorously computed in an inclusive way, because the union can only be performed a finite number of times, however, good inclusion can still be obtained by numerical methods. In this sense, one may provide a set of support points, for example complying with a Latin-hypercube scheme, to pick up values from. By doing so, the more values of y˜ are drawn the closer is the approximation to the exact interval. It is worth noticing that such a numerical procedure will not lead to inclusive intervals from a rigorous viewpoint, because the exact interval is approached from the interior. In other words the extremes of the solution are sought by sweeping point by point the searching domain, according to a numerical scheme. Such a procedure allows looking at the minimum and maximum , but only among the points generated within the search; thus, it does not provide clues on how close to the edges of solution these points are, so that the outcome strongly relies on the chosen scheme for point generation, and on the number of points. 3.2 Interval solution by potential energy minimization In this section only the effects of load position are taken into consideration, since previously it was shown

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that uncertainties are separable and the dependence and inclusion problems are related only to the load position. An alternative way of finding the functions that bound v(ζ) for every point in the beam domain is to replace the above differential formulation with an integral one. Notice that such issue corresponds to looking at the intervals that enclose v(ζ). These intervals are defined as an infinite sequence; therefore they can be retrieved at any point of the beam domain as a continuous function. The principle of least action is used on purpose and the whole problem is reformulated in terms of elastic potential energy (Muhanna, Mullen, & Zhang 2005), which for the problem under consideration is the functional:

Among all the functions v(ζ) complying with the boundary conditions the sought function is the one that minimizes the energy functional, i.e. makes the variation δU = 0. This condition leads to a system of decoupled partial derivatives if a convenient basis of orthogonal functions is chosen to represent the function v(ζ). Considered the boundary conditions of a simply supported beam, a basis of sinusoidal functions is selected, thus the solution is expressed as:

where, αi is a coefficient to be determined. If one replaces this expression in equation 8, and computes the integrals considering that p(ζ) is the delta function δ(ζ − y˜ ), the functional U becomes a function of solely coefficients αi as

Figure 1. Simply supported beam and loading conditions.

In this expression all the possible configurations of the beam can be bounded within interval arithmetic and neither numerical techniques nor approximate methods are needed to look for the upper and lower bounds of the solution. Note that the series in equation 12 quickly converges towards the sought function, because of the integer to the forth power at the denominator. Furthermore, when the variable y˜ is replaced by the interval [y], the dependence problem will not affect much the results. In fact, as the terms of the series increase, the interval sin iπ[y] keep growing in radius until it reaches the outer interval [−1, 1], which bounds the sinusoidal function. This is because the bigger the integer i, the more sinusoidal functions of whole period can fit in the same interval π[y]. While it is understood that far terms of the series do not contribute to the point-wise outcome of equation 12, because as i approaches infinity the limit tends to zero. It is also noted that the interval radius of far terms goes to zero as i approaches infinity. This occurs because the interval sin iπ[y] cannot be however wider than [−1, 1], and the radius of the i-th term is always less than i24 . Finally, it is worth noticing that the result obtained so far is not affected anyhow by the amplitude of the interval [y], which can be taken as wider as needed within the beam length. 3.3

∂U The stationary conditions are thus, given by ∂α = 0, i which is a system of independent equations that can be analytically calculated. The integral of equation 10 is equal to 12 for i = j and is equal to zero for i = j; therefore the functional becomes

∂U = which derivatives with respect to αi are ∂α k 1 B 4 4 k π αk − P sin kπ˜y. Thus, the functional is sta2 L3

tionary for αi = now as

2PL3 B(iπ)4

sin iπ˜y and the solution displays

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Numerical results

The objective of the theoretical investigations is obtaining models capable of coherently describing the scatter in the structural performance of beams designed with identical properties. The efficacy of these models is firstly shown by comparing theoretical values from numerical analysis with the experimental results. The beams under study are made up of precast concrete, are 3.2 m long and have a t-shaped crosssection 0.09 m high and 0.12 m deep (web). Under test, the beams lie on two point-wise supports 3.0 m apart and are subjected to a vertical load applied about halfway between the supports. Neither the extent nor the shape of the contact between the load and the beam surfaces is known; nonetheless such contact is meant to be limited to a very small area. The resulting numerical model is displayed in figure 1. Keeping the uncertainties arising from the load position and from the beam properties separated one another, allows comparing clearly what model better describes the problem. In the first case, the closed-form solution (CfS) is exact but may result in too wide intervals when the load position is modeled. On the other hand the solution obtained by applying the least action principle (LAS), albeit

Table 1.

Results in terms of intervals for the middle-span deflection with uncertainty in load position.

Point-wise solution = 2.25 (mm) Interval solutions obtained with xr /xc = 0.05 Models

Interval (mm)

Central value (mm)

Radius (mm)

Direct Interval Arithmetic (CfS) Handled Interval Arithmetic (CfS) Harmonic series (LAS) Monte Carlo (CfS)

[1.8145, 2.7136] [2.1290, 2.3541] [2.2313, 2.2501] [2.2413, 2.2500]

2.2641 2.2416 2.2407 2.2457

0.4496 0.1126 0.0094 0.0044

Table 2.

Results in terms of intervals for the middle-span deflection with global uncertainty.

Point-wise solution = 2.25 (mm) Interval solutions obtained with Br /Bc = 0.15, Pr /Pc = 0.05, xr /xc = 0.05 Models

Interval (mm)

Central value (mm)

Radius (mm)

Direct Interval Arithmetic (CfS) Handled Interval Arithmetic (CfS) Harmonic series (LAS) Monte Carlo (CfS)

[1.1616, 3.9606] [1.6455, 3.0148] [1.7988, 2.7797] [1.8297, 2.7794]

2.561 2.330 2.289 2.305

1.400 0.685 0.490 0.475

Figure 2. Interval models of the beam: closed-form solution (CfS) and least action solution (LAS) with uncertain load location.

approximate, does yield much narrower intervals and therefore provides a better answer in terms of amount of uncertainty propagated. Results in terms of intervals as well as of interval radius are shown in table 3.3 where the different solutions are given when the load position ranges in the interval [x] = L2 (1 + 0.05e
), i.e. when the load is assumed to be varying in location with a ±2.5% of uncertainty with respect to the beam length L. It is worth noticing that the LAS has been obtained by rigorous interval arithmetic and therefore provides inclusive results. The significant difference in terms of radius, displayed in table, is to attribute to the dependence effect, which cannot be successfully contained for the first model by interval arithmetic. For comparison purposes, along with the interval results, an interior (not-inclusive) solution obtained

Figure 3. Interval models of the beam: closed-form solution (CfS) and least actionc solution (LAS) with uncertain bending stiffness, load intensity and load location.

by Monte Carlo numerical analysis of the closed-form model is also reported.A pictorial comparison between these solutions is displayed in figure 2. In the second case, the uncertainty arising simultaneously from the beam properties and from the load intensity (global uncertainty) is introduced. This can be quantified at once by interval arithmetic without worrying about the dependence problem. In table 3.3, results from the different models, which are obtained by considering the intervals [B] = Bc (1 + 0.15e
), [P] = Pc (1 + 0.05e
) and [x] = L2 (1 + 0.05e
), are displayed. Defining the interval bending stiffness as interval means that such a quantity can vary arbitrarily, i.e. with any law, along the beam length, within the specified interval. This is of great importance in order to provide a general answer to the beam problem. Results from the general case in which all uncertain quantities are intervals are displayed in figure 3. It can be seen that CfS and LAS

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Table 3.

Nominal values, tolerances and uncertainties to be inputting the model.

Model parameters Bending stiffness (Nm) Span length (m) Load intensity (N) Load location (m) 1st load history (N) 2nd load history (N) 3rd load history (N)

Symbol B L P x

Central value

Relative radius

218376 0.050 3.0 0.001 (see load history) 0.025 1.5 0.050

CASE STUDY

The efficacy of the beam models based on the interval formulation is now analyzed against experimental data. A lab work has been carried out to evaluate the deflections of a set of precast concrete beams. The beams to undergoing test are attached a load that steps in intensity from zero to a certain value to get three loading cycles all within the elastic range. Even though the test conditions are accurately set up, errors may come out as a result of many concomitant factors such as the experimental setup and the partial knowledge of the beam properties and geometry. All this sources of errors are usually accounted for by means of a separate error analysis. However, by doing so, one assumes firstly that the numerical model allows describing closely the specimen behavior, and secondly that the error has nothing to do with the specimen properties, which in this case it may reveal to be false. Here is where the method previously developed may turn out to be useful. In fact, the proposed method has been tailored to embody the uncertainties in the numerical model, which derive both from the experimental setup and from the specimen properties. By this method it is possible to assess to what extent the nominal material properties are reasonable as well as whether the experimental setup influences the results. 4.1

[206490, 230260] [2.9924, 3.0076] – [1.4249, 1.5751]

0, 100.59, 200.48, 299.38, 395.76 0, 100.59, 200.48, 299.38, 395.76, 494.17, 593.83, 690.53, 787.52 0, 100.59, 200.48, 299.38, 395.76, 494.17, 593.83, 690.53, 787.52, 885.35, 985.68, 1082.5, 1179.1

do not differ much; this is because, in the case under study, the uncertainty on load position is very small in magnitude, compared to the others, as well as their effect on the numerical results. 4

Interval

Comparison between numerical and experimental results

A set of four beams, namely beam T1, T3, T4 and T6, originating form the same production, underwent a number of static tests. Out of these tests the deflections were measured using a series of indicators placed along the beams. In this section only results for the middle-span point are displayed. Any test has been running three times loading the beam from zero up to a certain value and then back to zero, by means of constant load’s steps. Time histories corresponding to each load cycle are displayed in table 3.3. Figure 4 clearly shows the variegate behaviors of the tested

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Table 4. Middle-span deflection: comparison between experimental data and numerical solutions. Values in mm. Experimental

CfS

LAS

min

max

interval

interval

0.475 0.715 0.940 1.170 1.410 1.645 1.865 2.110 2.335 2.580 2.785

0.560 0.820 1.075 1.325 1.580 1.820 2.070 2.315 2.570 2.810 3.025

[0.452, [0.675, [0.892, [1.114, [1.338, [1.556, [1.775, [1.995, [2.222, [2.440, [2.657,

0.586] 0.875] 1.156] 1.444] 1.735] 2.017] 2.301] 2.586] 2.879] 3.162] 3.444]

[0.473, [0.707, [0.935, [1.167, [1.403, [1.631, [1.860, [2.091, [2.328, [2.557, [2.785,

0.560] 0.836] 1.105] 1.380] 1.658] 1.928] 2.200] 2.472] 2.752] 3.022] 3.292]

beams, despite they all originate from the same production. Because of the errors and uncertainties said above, each beam of the tested set displays a somehow different response in terms of deflection. In figure 5 are also plotted two theoretical sets of responses (CfS and LAS models), which are obtained setting the beam parameters as shown in table 3. The intervals of table 3 are obtained by making simple considerations over the errors arising from the testing procedure. The interval associated to the span length L derives from the tolerances of the gauges, which elementary units are the millimeters. The uncertainty in the load location comprises the above tolerance, the uncertainty in placing the load (partially related to the gauging tolerances) and the uncertainty in the loading area, which may be more or less spread over the beam’s axis. The uncertainty in the load intensities includes the errors in converting values from unit of mass (as known at the time of the experiment) to unit of force, and the tolerances in weighing the other parts that composed the loading apparatus. For the bending stiffness one shall be using the tolerances provided by the producer. In this case, however, such tolerances were not available thus a rough interval was established using good judgment. In table 4 the interval quantities inputting the model are given. The model’s response is shown in figure 5 where it can be seen that all the provided

Figure 4. Normalized deflections against load intensity for some beams of the same production.

Figure 5. Comparison between theoretical and experimental results: deflections against load intensity.

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measurements are included in the interval solutions provided by the methods. LAS definitely provides the best answer in terms of interval solution, because includes the experimental data with the tightest intervals. A comprehensive comparison between the models’ solutions and the experimental data is given in table 4.1, where it may be appreciated to what extent the intervals include the experimental deflection. The LAS model provides the narrower inclusion of the experimental results. This means that if such model were used in place of the CfS, stricter requirements in terms of material heterogeneity might be applied to the production process. 5

CONCLUSIONS

In this paper a novel method to account for tolerances, inaccuracies and variability of material properties and loading conditions based on interval analysis is presented.The efficacy of the method has been demonstrated according to a proper formulation of beam models, whose capability in providing bounded sets of all the feasible solutions has been checked against numerical and real-case examples. In this context rigorous comparisons against the experimental results without needs to perform a separate error analysis can be made. The strength of the method stems from its feature of including every source of uncertainty in the same framework. Nonetheless, the proposed methodology has been derived on a particular case and its generalization might not be straightforward because of the dependency (or dependability) problem, which makes naive interval analysis not available for built-in implementations in a computer program. Therefore, tailored strategies are needed to deliver the solution of the problem. These strategies can be developed both from an algorithmic viewpoint, i.e. within the framework of interval analysis and from a mechanical perspective looking at different mathematical solutions of the same problem.The union of the algorithmic and mechanical approaches makes interval analysis a fruitful and powerful tool for treating the uncertainties in cases of engineering interest.The advantage of using interval analysis is to ascribe to its rigorousness, which makes it easy to judge what model best describes the problem under study. With interval analysis, room for

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approximations is only allowed in the sense of increasing the width of the solution, which might make the model uninformative but never wrong. The intervalbased model has shown to be particularly effective in providing the full set of feasible solutions of a set of precast concrete beams, sampled from the same set of cast elements, although presenting different responses because of the tolerances inherent to the mass production. Thanks to the proposed method and its inclusive property, it is possible to define the interval that bounds the observed scatter of the response. This aspect is very promising for design purposes; furthermore, in conjunction with identification methods it can allow to pick out the parameter that is mainly responsible to affect the spread of the results. REFERENCES Beer, M., Y. Zhang, S. T. Quek, & K. K. Phoon (2013). Reliability analysis with scarce information: Comparing alternative approaches in a geotechnical engineering context. Structural Safety 41, 1–10. Corsaro, S. & M. Marino (2006). Interval linear systems: the state of the art. Computational Statistics 21(2), 365–384. Dessombz, O., F. Thouverez, J.-P. Laîné, & L. Jézéquel (2001). Analysis of mechanical systems using interval computations applied to finite element methods. Journal of Sound and Vibration 239(5), 949–968. Gabriele, S. & C. Valente (2009). An interval-based technique for fe model updating. International Journal of Reliability and Safety 3(1), 79–103. Hansen, E. R. & G. W. Walster (2004). Global optimization using interval analysis, Volume 264. CRC Press. Marwala, T. (2010). Finite-element-model Updating Using Computational Intelligence Techniques: Applications to Structural Dynamics. Springer. Moore, R. E., R. B. Kearfott, & M. J. Cloud (2009). Introduction to interval analysis. Siam. Moore, R. (1969). Interval analysis. 1966. Prince-Hall, Englewood Cliffs, NJ . Muhanna, R. L., R. L. Mullen, & H. Zhang (2005). Penaltybased solution for the interval finite-element methods. Journal of engineering mechanics 131(10), 1102–1111. Muhanna, R. L. & R. L. Mullen (2001). Uncertainty in mechanics problems-interval-based approach. Journal of Engineering Mechanics 127(6), 557–566. Schuëller, G. (2007). On the treatment of uncertainties in structural mechanics and analysis. Computers & structures 85(5), 235–243.