Limit State Imprecise Interval Analysis in
Geotechnical Engineering
Sónia H. MARQUES a,b , Michael BEER b , António T. GOMES a and António A. HENRIQUES a a
Department of Civil Engineering, Faculty of Engineering, University of Porto, Portugal b
Institute for Risk and Uncertainty, University of Liverpool, United Kingdom
Abstract. The first applications of nonprobabilistic interval analysis in geotechnical engineering have been recently explored. In
fact, the solution of practical problems in this field often requires judgement based on limited data and interval analysis has become a research area formally motivated by input information characterised by imprecision. Whenever no information apart from bounds is available, intervals may be considered in the form of a model noted that they are among the most widely used analytical tools to describe uncertainty by using nonprobabilistic approaches. However, the application of a pure interval analysis to engineering problems may lead to large overestimation due to the dependency phenomenon and meaningful results are only obtained when a decision is based on threshold values. A mixed approach that admits imprecise information as well as probabilistic information is therefore desirable. In this paper the conventional probabilistic approach to uncertainty is extended to include imprecise information in the form of intervals. For demonstration, results are provided for a strip spread foundation designed by the Eurocode 7 methodology, wherein the shear strength parameters of the foundation soil are implemented as intervals and then combined with other uncertain parameters in the form of bounded random variables under dependence. A limit state imprecise interval analysis for safety assessment is then presented in the format of a sensitivity analysis.
Keywords. Uncertainty; imprecise interval analysis; geotechnical engineering
1. Introduction
Real engineering systems are characterised by multiple uncertainties which affect their design and performance. These uncertainties are commonly described within a probabilistic framework as random variables or random fields. However, the accuracy of a probabilistic analysis relies on the availability of data to describe the distributions of basic input variables, particularly in the tails, as reliability estimates are sensitive to variations in probabilistic models.
In the last decades, several nonprobabilistic approaches have been developed to include uncertainties described by scarce information. If reliability is seen as a probability related to the satisfactory performance of a system under given circunstances, a nonprobabilistic concept of reliability holds on the acceptable range of performance fluctuations. According to the discussion of Haim and Elishakoff (1995), when considered only input interval variables, the proposed nonprobabilistic concept of reliability is approached only by bounds as the safety margin is as well expressed in the interval format.
The capabilities of nontraditional models for engineering computation under uncertainty have been under continuous review, see for instance Möller and Beer (2008). In particular, intervals represent an appropriate model to describe uncertainty in cases when a possible range between bounds is known and no other information concerning frequencies is available.
Among nonprobabilistic approaches, the ordinary interval analysis involves the mapping of interval input to interval output quantities. The main advantage is the evaluation of the analytical enclosure of the true solution. However, the application to real engineering systems is quite complex due to the assignment of threshold values under a binary treatment of information rather often characterised by large conservatism.
In recent years, a considerable number of papers devoted to interval analysis have been published, namely the study of Su and Wen (2013) regarding the interval risk analysis for gravity dam instability or a number of advanced works dedicated to various structures, see for instance Impollonia and Muscolino (2011), Muscolino and Sofi ( 2012) and Muscolino et al. © 2015 The authors and IOS Press.
This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License.
(2013). These latter authors focused on novel algorithms to account for issues as the dependency phenomenon or interval fields.
Interval quantities may as well be included in computation based on other uncertainty models. This procedure is particularly convenient for sensitivity analysis, a very efficient tool to identify important design parameters and to possibly improve the performance level of real engineering systems in the context of optimisation procedures. Thus, whenever some probabilistic information is available, a mixed approach gathering imprecise information in the form of intervals may be explored. Beer et al. (2013) provided an overview about imprecise probabilities in engineering analysis, and a number of practical applications have been presented, see for instance Zhang et al. (2010) in a comparison of uncertainty models for reliability analysis. Then, interval probabilities emerge from the consideration of a set of plausible probabilistic models in order to find the lower and upper probability bounds. In addition, unrealistic model assumptions may be revealed by a spare interval-based sensitivity analysis wherein the random variables are bounded by using interval quantities in order to express different levels of probability. Accordingly, the safety margin is as well expressed in the interval format and a limit state imprecise interval analysis for safety assessment may be pursued.
At last, a special remark about geotechnical materials, among the most variable of all engineering materials. Indeed, geotechnical design parameters enclose a physical meaning and are typically limited in a range. As the solution of practical problems in geotechnical engineering often requires judgement based on limited information, the assumption of interval-valued parameters in a mixed approach that admits nonprobabilistic as well as probabilistic information verily complies with the imprecision from the input scenario. For demonstration, results are provided for a strip spread foundation designed by the Eurocode 7 methodology, wherein the shear strength parameters of the foundation soil are implemented as intervals and then combined with other uncertain parameters in the form of bounded random variables under dependence. A limit state imprecise interval analysis for safety assessment is then presented in the format of a sensitivity analysis.
2. Optimisation Algorithm
In this section, a brief explanation of the considered methodology for minimisation and maximisation which considers dependence while building the limit state charts for safety assessment is summarised as follows:
[Step 1] Express the distributions and the statistics of basic input variables; [Step 2] Express the equivalent standard
normal correlation matrix;
[Step 3] Express the outcome matrix from the Cholesky decomposition of the equivalent standard normal correlation matrix;
[Step 4] Express the set of random and interval variables vector in the standard normal space of uncorrelated random and interval variables;
[Step 5] Express the performance function and the limit state in the standard normal space of uncorrelated random and interval variables;
[Step 6] Express the set of constraints and whenever required the set of guess values for initialisation of the suboptimisation algorithm;
[Step 7] Select the suboptimisation algorithm properly and run the iterative process in a multistart approach;
[Step 8] Estimate the pair minimum-maximum and the corresponding coordinates in the standard normal space of uncorrelated as well as in the original space of correlated random and interval variables.
Some further remarks might be helpful, the first about the equivalent standard normal correlation matrix. Considered the next design example with correlated normal and lognormal distributions, the values of the correlation coefficients between the basic input variables may be transformed by exact relationship according to the expression in Table 1, which may be verified in Haldar and Mahadevan (2000). Thus, the transformation F may be used for the computation of the resultant equivalent standard normal correlation coefficient, when the variables are normal and lognormal.
Table 1. Transformation F by exact relationship for
representation of the correlation coefficient between two standard normal variables xi and xj as = F × .
Variable xi Variable xj Transformation F
Normal Lognormal cv ln1 + cv cv-coefficient of variation.
Then, the performance function in the standard normal space of uncorrelated random and interval variables is expressed by the following equivalent transformations =f( )
and f()=derived by Eq. (1) in compensative
probability:
= F
[()] [F()] = (1)
if =variable in the original space; =variable
in the standard normal space; F=cumulative
nonnormal distribution function; F=inverse
cumulative nonnormal distribution function;
=cumulative standard normal distribution function; and =inverse cumulative standard
normal distribution function.
Considered the next design example with correlated normal and lognormal distributions, the described transformations for representation of variables with different distributions as a
function of standard normal variables are
detailed as follows in Table 2.
Table 2. Transformations for representation of variables
with different distributions as a function of standard normal variables . Distributions =f() Normal = + Lognormal = e( ) Distributions f()= Normal ! = Lognormal ln( ) ! l l =
-mean; -standard deviation; "-log mean; "-log standard deviation.
The correlation assignment in Eq. (2) is as well added to this standard normal space script:
[#] = [$%][&] (2)
if #=set of variables vector in the standard
normal space of correlated variables; &=set of
variables vector in the standard normal space of uncorrelated variables; and Ch =outcome matrix from the Cholesky decomposition of the equivalent standard normal correlation matrix.
3. Design Example
The design example is referred to the strip spread foundation on a relatively homogeneous c-' soil shown in Figure 1, wherein groundwater level is away. Considered the vertical noneccentric loading problem and the calculation model for bearing capacity, the performance function may be described by the simplified Eq. (3):
M = f(B, D, *., c/, '/, */, P, Q) (3)
if B is the foundation width; D is the soil height above the foundation base; *sis the unit weight of the soil above the foundation base; c/ is the
cohesion of the foundation soil; '/ is the friction
angle of the foundation soil; */is the unit weight
of the foundation soil; P is the dead load; and Q is the live load.
Table 3 summarises the description of basic input variables, with different distributions. The considered coefficients of correlation between the basic input variables are either presented in the correlation matrix shown in Table 4.
The strip spread foundation is designed by the Eurocode 7 methodology, Design Approach DA.2*, noted that the imprecise interval analysis is performed in a set of scenarios wherein the shear strength parameters of the foundation soil are implemented as intervals, but separately in a bounded imprecise probabilistic approach.
4. Results and Discussion
The limit state charts for safety assessment are presented in Figure 2 and Figure 3, considered respectively the cases cohesion and friction angle interval scenario, separately in a sensitivity analysis wherein the random variables are bounded on different levels of probability.
Distinct levels of credibility are used to sketch the lines which express the limit state bounds. It is possible to search for a credibility level which ensures no failure regardless of the parameter value on the horizontal axis, see safe level for arrow in Figure 3 against unsafe level for arrow in Figure 2. Conversely, it is possible to find the threshold parameter which ensures no
failure for a given credibility level and then to proceed with proper ground investigation and testing or improvement, see in both charts the circles crossing the zero limit state boundary and the 0.9900 credibility level line. In decision making, this valuable approach may be extended by numerical analysis for high dimensional cases with several indecision variables in simultaneous.
Figure 1. Strip spread foundation. Table 3. Summary description of basic input variables.
Basic input variables Distributions Mean value Coefficient of variation Statistics
B (m) Deterministic - - 1.30 D (m) Deterministic - - 1.00 *. (kN/m3) Normal 16.80 0.05 - c/ (kN/m2) Lognormal 14.00 0.40 "02.5648 "00.3853 Interval - - [0.00,35.00]a '/ (º) Lognormal 32.00 0.10 "03.4608 "00.0998 Interval - - [25.00,35.00]b */ (kN/m3) Normal 17.80 0.05 - P (kN/m) Normal 370.00 0.10 - Q (kN/m) Normal 70.00 0.25 -
"-log mean; "-log standard deviation. acase cohesion interval scenario. bcase friction angle interval scenario.
Table 4. Coefficients of correlation between the basic input variables.
Correlation matrix 2 3 4 5 2 3 4 5 2 2 22 23 24 25 3 3 32 33 34 35 4 4 42 43 44 45 5 5 52 53 54 55 = 1.0 0.0 0.5 0.9 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.5 0.0 1.0 0.5 0.0 0.0 0.9 0.0 0.5 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 -coefficient of correlation; x1-*.; x2-c/; x3-'/; x4-*/; x5-P; x6-Q.
Figure 2. Limit state chart for safety assessment for the case cohesion interval scenario.
The first remark is devoted to dependence, important feature in geotechnical engineering practice wherein design parameters encompass a physical meaning. In a brief analysis, the dependence between the cohesion and the friction angle of the foundation soil relies particularly on the material, but the considered hypothesis of independence is possibly cautious, further noted that the characteristic values of soil properties are often calculated separately, neglecting the effects of dependence. However, a Monte Carlo simulation obtained from 5e6 steps in the considered probabilistic scenario returns reliability indexes of 3.4388 and 3.6041, respectively under dependence and independence assumptions. Like this, optimisation algorithms should consider any important dependence relationships and if interval variables are involved, see the case friction angle interval scenario, proper constraints are required.
The Eurocode 7 Design Approach DA.2* corresponds to a particular form of load and resistance factor design wherein partial factors are applied to the resulting effects of actions and to the ground resistance. According to the imprecise probabilistic approach and considered the cases cohesion and friction angle interval scenario, the threshold values which satisfy the Eurocodes 3.8 target ultimate limit state reliability index are respectively 12 kN/m2 and 26º-27º. Thereby, satisfactory levels of reliability estimates are attained by a probability level of 0.9900. In fact, unrealistic model assumptions are evinced by the charts when higher levels of probability are considered, see the case cohesion interval scenario wherein the limit state upper bounds are obtained from unrealistic coordinates. In addition, the interval uncertainty comprehends a probabilistic meaning for every combinatory possibility, see the case cohesion interval scenario wherein nonsatisfactory levels of reliability estimates are attained in a significant part of the cohesion interval. Therefore, a critical evaluation from interval scenario may influence the safety-based decision in ground investigation and testing or improvement. Another important issue is the position of the median trend which separates the fifty percent chance cases, here sketched on the safe side, noted that the linear and nonlinear trends evinced by the charts express the influence of the shear strength parameters in the calculation model for bearing
capacity. At last, in the multivariate case the considered probability level prescribes a credible region in the hyperspace, then the imprecise interval analysis complies with a mixed set of probabilistic and nonprobabilistic interval models wherein different bounding measures may be applied in order to find the limit state lower and upper bounds in different scenarios.
5. Conclusion
A geotechnical engineering design example is detailed. Practical experience suggests that important dependence relationships should be considered in optimisation algorithms. The imprecise probabilistic approach is crossed with a limit state imprecise interval analysis for safety assessment. This sensitivity analysis may reveal unrealistic model assumptions and may provide meaningful results for safety-based decision in ground investigation and testing or improvement.
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