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Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach
Accepted Manuscript
Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach Jinwen Feng , Di Wu , Wei Gao , Guoyin Li PII: DOI: Reference:
S0307-904X(17)30261-5 10.1016/j.apm.2017.03.066 APM 11707
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
23 September 2016 22 February 2017 28 March 2017
Please cite this article as: Jinwen Feng , Di Wu , Wei Gao , Guoyin Li , Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.03.066
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Highlights A unified perturbation mathematical programming (UPMP) approach is proposed. Random and interval uncertainties are simultaneously considered in this study.
Both nodal displacement and elemental force are analysed by UPMP approach.
Uncertain-but-bounded statistical characteristics are efficiently captured via NLP.
Interval dependency issue is completely eliminated from the algorithm
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Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach Jinwen Feng1, Di Wu1, Wei Gao1,*, Guoyin Li2 1
Centre for Infrastructure Engineering and Safety (CIES), School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia 2
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School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia
Abstract
A novel computational method, namely the unified perturbation mathematical programming
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(UPMP) approach, for hybrid uncertainty analysis of engineering structures is proposed in this paper. The presented study considers a mixture of random and interval system parameters which are frequently encountered in engineering applications. Within the UPMP approach, matrix perturbation theory is adopted in combination with the mathematical programming approach. The proposed computational method provides a non-simulative hybrid uncertainty
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analysis framework, which is competent to offer the extreme bounds of the statistical characteristics (i.e., mean and variance) of any concerned structural responses in
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computationally tractable fashion. In order to thoroughly explore various intricate aspects of the engineering system involving hybrid uncertainties, systematic numerical experiments have
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also been conducted. Diverse statistical analyses are implemented to identify the bounded probability profile of the uncertain structural responses. Both academic and practical
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engineering structures are investigated to justify the applicability, accuracy and efficiency of the proposed UPMP approach.
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Keywords: Random Interval Analysis; Nondeterministic Structural Responses; Hybrid Uncertainties; Mathematical Programming; Bounded Statistical Feature.
*
Corresponding author. Email addresses: [email protected] (J. Feng), [email protected] (D. Wu), [email protected] (W. Gao), [email protected] (G. Li)
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1. Introduction In engineering analysis and design, computational methods have been extensively developed by hypothetically considering models and system parameters as deterministic. Finite element method (FEM), along with the fast growth of the capability of computers, establishes a favourable analysis framework for modern engineering applications [1]. Despite the wide adoption of the deterministic analysis through FEM, the traditional framework of such
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analysis neglects nondeterministic features, or so-called uncertainties, which inevitably exist within engineering structures and modelling processes [2]. These uncertainties, either origin from the inherent variation of system parameters or the lack of knowledge or information, potentially lead to fluctuations in determinations of the structural behaviours [3, 4]. Therefore, the development of efficient computational approaches for predicting the effect of diverse
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uncertain parameters on structural behaviours becomes mandatory.
Regarding the traditional uncertainty analysis, probabilistic/stochastic approaches are frequently implemented in the uncertain structural analysis due to the solid theoretical foundation associated with uncertainty modelling. Within the probabilistic analysis framework, all uncertain parameters are modelled as either random variables or fields with
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associated statistical information [5, 6]. Consequently, by combining the FEM with the wellestablished theories on probability and statistics, the stochastic finite element method (SFEM)
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has been innovatively established with extensive application across wide range of engineering disciplines [7-9]. Within the framework of the SFEM, the well-known Monte-Carlo
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simulation (MCS) method is one of the most straightforward procedures, which has been pervasively implemented in a variety of stochastic problems [10-12]. In addition, the
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computational stochastic analysis, which has been developed basing on the theory of general matrix perturbation, offers an advantageous non-simulative strategy for uncertainty analysis
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on engineering structures involving random variables [13-16]. Furthermore, the spectral stochastic finite element method (SSFEM) was introduced to address the spatially dependent uncertain system parameters in structural engineering problems [17-19]. The core concept of SSFEM is to establish a valid surrogate model by implementing various series expansion schemes (e.g., the Karhunen-Loeve expansion) such that the statistical characteristics (i.e., mean and variance) of the structural responses can be effectively calculated. Additionally, other types of stochastic methods have been proposed as in [20, 21]. Even though the probabilistic approach of uncertainty analysis has been prevalently implemented in practical
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engineering applications, the applicability of such methodology strongly depends on the availability of the information of uncertain parameters [22]. However, accumulated experiences from engineering practice have repeatedly revealed that the requirement for the implementation of the probabilistic analysis cannot be always fulfilled due to the issue of information deficiency on the uncertain system parameters [23]. In order to achieve valid uncertainty analysis for situations where stochastic approaches are
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prohibited, the non-probabilistic approaches, such as the convex model analysis and fuzzy uncertainty analysis, have been developed as alternatives. By using convex sets to describe uncertain parameters, such non-stochastic modelling technique has been applied in uncertainty propagation analysis, reliability analysis and engineering optimizations involving uncertain system inputs [24-26]. The fuzzy analysis, stemming from the fuzzy set theory,
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incorporates membership functions to represent the possibility distributions of uncertain variables [27, 28]. In addition, interval models are gaining popularity because of the conceptual simplicity [29]. Instead of demanding the full profile of uncertain parameters, only the upper and lower bounds of uncertain parameters are required for valid uncertainty analysis. By implementing the concept of interval analysis within the framework of the FEM, various
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computational algorithms with specific emphasises have been explicitly developed for tackling intricate engineering problems with enhanced capability and accuracy [30-34].
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Furthermore, the concept of interval field was also proposed in [35], and subsequently extended to model spatially dependent uncertain parameters in a non-probabilistic fashion
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[36].
In modern engineering applications, it is frequently encountered that both probabilistic
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and non-probabilistic uncertainties exist simultaneously [37]. Therefore, a computational scheme which is capable of estimating the effect of diverse types of uncertainties on
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engineering system is inevitably necessary. A random interval perturbation method has been developed for static response analysis [38], and free vibration analysis [39] of structures with both stochastic and non-stochastic uncertainties. In these studies, the upper and lower bounds of means and variances of system responses are explicitly formulated, such that the computational efficiency of this type of approach can be well preserved. Also, the perturbation based computational method is extended to the dynamic response analysis of acoustic field and structural-acoustic interaction system with hybrid uncertain parameters [40]. Stochastic response analysis of structures with interval system parameters under random excitations has been investigated in [41]. Polynomial-Chaos-Chebyshev-Interval method is 4
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recently proposed for vehicle dynamic problems with mixed uncertainties [42]. Hybrid probabilistic and interval [43], or probabilistic and convex set models for structural reliability analysis [44] are also explicitly proposed. By thoroughly examining on the literatures regarding the perturbation theory based hybrid uncertainty analysis with random and interval variables, it is realized that either series expansion or numerical approximation has been implemented to derive explicit formulations
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for the upper and lower bounds of the statistical characteristics of the considered structural responses. The advantage of such prevalently employed technique is that explicit formulations on the bounded statistical characteristics of structural responses can be explicitly formulated. However, the drawback of such analysis is that the overestimation on the bounded statistical characteristics of system output cannot be completely exempted due to the issue of interval
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dependency. As clearly demonstrated in the previous studies [30-34], the issue of interval dependency can significantly affect the accuracy of the interval analysis, which is mainly attributed by the multiple occurrences of the same interval parameter at various locations. The general rule is that the more numbers of interval parameters appear in the system, the more likely the bounded results are suffering from overestimation.
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In the light of achieving valid hybrid uncertainty analysis with the least overestimation due to interval dependency, a novel computational method, namely the unified perturbation
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mathematical programming (UPMP) method, is proposed for static analysis of engineering structures with a mixture of random and interval uncertain parameters. Within the proposed
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computational scheme, the matrix perturbation theory and mathematical programming (MP) approach are integrated in conjunction with an alternative FEM modelling technique. Such
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novel integration reformulates the hybrid probabilistic interval static analysis into a series of nonlinear programming (NLPs) problems. Consequently, the upper and lower bounds of the
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mean and variance of the concerned structural response can be explicitly determined by solving the corresponding NLPs. Some core advantages of UPMP are including: 1. The superior computational efficiency of the non-simulative method is fully inherited by the UPMP approach, which significantly enhances the applicability of the proposed scheme in the context of analysing modern complex engineering structures. 2. The physical feasibility of all interval parameters can be rigorously maintained throughout the entire hybrid probabilistic interval analysis. Thus, the determined system outputs are fully liberated from the overestimation caused by the issue of interval dependency. 5
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3. The critical values of the interval parameters responsible for the extremities of statistical information of structural response can be effectively retrieved as byproducts of the uncertainty analysis. In order to evidently illustrate the applicability and effectiveness of the proposed UPMP approach, both academic sized and practically motivated structures have been investigated in this study. For the purpose of results verification, all computational results provided by the
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UPMP approach are rigorously testified against either analytical solution (for academic sized example) or results obtained from exhausted simulation approach (for practically motivated example). In addition to the proposition of the UPMP approach, supplementary numerical experiments have been conducted to further explore some important aspects on the hybrid uncertain static analysis with random and interval variables.
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The reminder of this paper is organised as follows. In Section 2, the formulations for static structural response analysis with hybrid uncertainties are presented. The proposed UPMP scheme is formally presented in Section 3. Furthermore, both academic sized and practically motivated numerical examples are investigated by the proposed UPMP approach and then critically compared with either analytical solution or exhausted simulation approach
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in Section 4. Finally, some concluding remarks are drawn in Section 5.
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2. Non-deterministic static analysis of structures with hybrid uncertainties For the non-deterministic analysis, the system parameters such as material properties,
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cross-sectional geometries and loading regimes are considered as uncertain variables. In general engineering practice, the availability and sufficiency of information of uncertain
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system parameters are strongly situational dependent [7-8]. Therefore, it is rational and necessary to develop a hybrid uncertainty analysis scheme which can adopt as many different
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uncertainty modelling techniques as possible in a single framework of analysis. 2.1 Generalized hybrid random interval static analysis of structures Without loss of generality, R Z () is defined as a random variable, where Z ()
represents the set of all real random variables in a probability space Ω, Α, P and denotes the set collects all real numbers; and I I () denotes an interval variable, such that
I [ , ] { | } , where I () denotes the closed set for all real intervals, and
represent the lower and upper bounds of I , respectively. For maintaining the consistency 6
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of formulations throughout this paper, () R denotes that the parameter () is a random variable/vector, () I denotes that the parameter () is an interval variable/vector, and () RI denotes that the parameter () is a variable/vector which includes both random and interval characteristics. Thus, the non-deterministic linear static response analysis of a structure with d degrees-of-freedom (DOFs) is expressed as: K(ψ R , ξ I )u RI F(ψ R , ξ I )
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such that
(1)
ψ R Ω : {ψ m | Rj ~ f R ( x), for j 1,...,m}
(2)
ξ I Φ : {ξ p | ξ k ξ kI ξ k , for k 1,..., p}
(3)
j
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where K (ψ R , ξ I ) d d denotes the non-deterministic stiffness matrix of the structure; u RI d is the vector that contains the non-deterministic displacements at all DOFs; and
F(ψ R , ξ I ) d denotes the non-deterministic externally applied forces; ψ R denotes the random vector which collects all the m probabilistic uncertain system parameters; f R (x) j
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denotes the probability density function of the random variable Rj ; ξ I denotes the interval
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vector which contains all the p interval variables presented in the system; k and k denote the lower and upper bounds of kI , respectively. It is worth to notice that due to the presence of
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hybrid uncertainties, the stiffness matrix and force vector are functions of both random and interval variables.
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The consideration of hybrid stochastic and non-stochastic uncertainties within the structural system dramatically increases the complexity of the static analysis problem. Due to
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the simultaneous existence of both random and interval variables, the behaviour of structural responses including nodal displacement and elemental force could have both probabilistic and interval characteristics. Moreover, for complex structural systems, the relation between each input variable and the structural output (i.e. nodal displacement and elemental force) cannot be analytically determined such that the exact prediction on the mercurial characteristics of structural behaviour is prohibited. Being conceptually different with the theories of imprecise probabilities [45], the random and interval uncertainties considered in this study are mutually independent rather than inclusive. 7
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2.2 Hybrid uncertainty analysis by using moment method By implementing the concept of the random moment method [7, 39] combined with the probabilistic theory, the means and standard deviations of the structural responses can be explicitly reformulated. Without loss of generality, multiple random and interval variables are respectively represented by random vector ψ R m and interval vector ξ I p . Within the framework of finite element method, the structural stiffness matrix K (ψ R , ξ I ) and the applied
expanded by implementing the random moment method as:
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load vector F(ψ R , ξ I ) , which are the functions of random and interval parameters, can be
(4)
F(ψ R , ξ I ) { R μ R } F(ψ , ξ ) F(μ ψ R , ξ ) R h h h 1 h ψ R μ R ψ 1 m m 2F(ψ R , ξ I ) { R μ }{ R μ } Re j (F) R R hR Rj h 2 h 1 j 1 h j R ψ μ ψR
(5)
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m K (ψ R , ξ I ) { R μ R } K (ψ R , ξ I ) K (μ ψ R , ξ I ) R h h h h 1 ψ R μ R ψ 1 m m K (ψ R , ξ I ) { R μ }{ R μ } Re j (K ) R R hR Rj h 2 h 1 j 1 h j R ψ μ ψR m
I
I
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R
1
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where μ ψ R [ R ,..., R ,..., R ] is the vector collects the means of all random variables; R h
m
h
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denotes the mean of the hth random variable hR ; Re () is the high-order terms of the Taylor expansion for the uncertain parameter () .
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From the theory of first-order matrix perturbation, the governing equation for linear static analysis of the structure involving various uncertain parameters can be formulated as: [K 0 1K][u 0 1u] F0 1F
(6)
K 0 K (μ ψ R , ξ I )
(7)
where
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(8)
F0 F(μ ψ R , ξ I )
(9)
2 m F(ψ R , ξ I ) { R μ R } F { R μ R } 1F h h h h h hR h 1 h 1 ψ R μ R ψ
(10)
m
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m
By neglecting the high-order terms, the random interval structural displacement based on the first-order perturbation is calculated as: u RI u 0 1u m R I F(ψ , ξ ) { R μ R } K F K R h h h 1 h ψ R μ R ψ 1 0
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1 0 0
m R I K ( ψ , ξ ) R 1 { μ R } K F h 0 0 h 1 hR h ψ R μ R ψ
(11)
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R I R I 1 F(ψ , ξ ) 1 K ( ψ , ξ ) 1 K F K 0 K0 K 0 F0 { hR μ R } R R h h h h 1 ψ R μ R ψ R μ R ψ ψ m
m
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1 0 0
K 01F0 K 01Fh K 01K hK 01F0 { hR μ R } h
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h 1
Subsequently, the mean and standard deviation of the random interval structural
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displacement responses u RI based on the first-order matrix perturbation [39] can be obtained
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by taking the mean and variance operation on Eq. (11), which leads to:
m
μ u RI K 01F0 m
(12)
σ u RI σ u RI K 01Fh K 01K h K 01F0 K 01F j K 01K j K 01F0 cov( hR , Rj )
(13)
h 1 j 1
where μ u RI d and σ u RI d denote the vectors which collect the mean and standard deviation of the displacement responses at all d degrees-of-freedom, respectively;
cov( h , j ) denotes the covariance between the hth and jth random variables h and j ; the operator ― ‖ denotes the Hadamard product of two matrices with equal dimension [47]. As 9
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indicated by Eqs. (12) and (13), the hybrid random interval problem has been transferred into two explicit interval analysis such that the statistical characteristics of the structural displacements are uncertain-but-bounded which can be further expressed as: μu RI Φ : {μu RI d | u RI u RI u RI , for v 1,..., d}
(14)
σu RI Φ : {σu RI d | u RI u RI u RI , for v 1,..., d}
(15)
v
v
v
v
v
v
v
v
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where μu RI and σ u RI denote the mean and standard deviation of the structural displacement ( u vRI ) at vth degree-of-freedom, respectively; u RI and u RI denote the upper and lower bounds of v
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μu RI , respectively; u RI and u RI denote the upper and lower bounds of σ u RI , respectively. v
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v
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In addition, the elemental force denoted by f RI w , can be explicitly formulated as: f RI D(ψ R , ξ I )u RI D(ψ R , ξ I )K (ψ R , ξ I ) 1 F(ψ R , ξ I )
where D(ψ R , ξ I ) wd can be defined as:
(16)
(17)
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m D(ψ R , ξ I ) { R μ R } D(ψ R , ξ I ) D(μ ψ R , ξ I ) R h h h h 1 ψ R μ R ψ 1 m m D(ψ R , ξ I ) { R μ }{ R μ } Re j ( D) R R hR Rj h 2 h 1 j 1 h j R ψ μ R ψ m 1 m m D0 Dh { hR μ R } Dh , j { hR μ R }{ Rj μ R } Re ( D) h h j 2 h 1 j 1 h 1
Once again, by implementing the random moment method combined with the theory of matrix perturbation, the mean and standard deviation of elemental force can be explicitly
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formulated as:
μ f RI D0 K 01F0 m
(18)
m
σ f RI σ f RI {D 0 [K 01Fh K 01K h K 01F0 ] D h K 01F0 }
(19)
h 1 j 1
{D 0 [K F j K K j K F ] D j K F }cov( , ) 1 0
1 0
1 0 0
1 0 0
R h
R j
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(20)
σf RI Φ : {σf RI w | f RI f RI f RI , for s 1,..., w}
(21)
s
s
s
s
s
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s
where μ f RI and σ f RI denotes the mean and standard deviation of f sRI which is the sth s s
component of the elemental force vector, respectively; f RI and f RI denote the upper and s
s
lower bounds of μ f RI , respectively; f RI and f RI denote the upper and lower bounds of σ f RI , s s
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s
s
respectively; w denotes the total number of elemental force in the structural system. Eqs. (14) - (15) and (20) - (21) indicate that the implementation of the random moment method meticulously transforms the hybrid random interval static analysis of linear structures
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into a series of interval problems, which provide an efficient yet systematic scheme for solving uncertainty analysis of structures with a mixture of random and interval variables. The
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explicit reformulations on the mean and standard deviation of the nodal displacement and elemental force offer preliminary information for analysing the uncertain structural behaviour under static loads. However, the calculation of the uncertain-but-bounded statistical
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characterises remains a computational challenge which is dominated by the aforementioned interval dependency issue. In this context, the interval arithmetic based approaches could lead
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to overestimation of the bounded stochastic characteristic while sampling method would be computational expensive for complex hybrid uncertainty analysis. Therefore, it is necessary to
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develop a more computational tractable approach such that the bounded means and standard deviations of the structural responses can be efficiently calculated without the interference of the issue of interval dependency. 3. The unified perturbation mathematical programming (UPMP) approach In this study, a new computational approach, namely the unified perturbation mathematical programming (UPMP) approach, is proposed to investigate the hybrid uncertain static response analysis with random and interval parameters. The UPMP approach offers a non-sampling computational strategy which can efficiently determine the mean and standard 11
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deviations of the structural responses. Furthermore, by adopting the alternative finite element (FE) formulation, the upper and lower bounds of the means and standard deviations of the structural responses can be explicitly formulated into standard nonlinear programming problems (NLPs). Consequently, the extreme bounds of all statistical characteristics (i.e. means and standard deviations) of either nodal displacements or elemental forces can be efficiently obtained by solving eight standard NLPs.
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3.1 Alternative finite element formulation As previously emphasized, the issue of interval dependency is always present in assembling global stiffness matrix with interval variables through the interval arithmetic approach. Even though the interval stiffness matrix can be easily assembled, it is highly likely that the final computational results are over-estimated due to the inappropriate manipulation
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of interval algebra. In the light of eliminating the interference on the system outputs caused by the interval dependency, an alternative FE formulation [34, 46] is implemented in interval analysis of engineering structures. Within the studies reported in [34, 46], the structural response analysis and linear bifurcation buckling analysis with interval uncertain parameters
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are transformed into mathematical programming problems. Subsequently, the interval analyses are processed directly via NLP solver by modelling the interval parameters as inequality constraints of the NLPs while the originality of the FE governing equations is
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thoroughly maintained. In this context, the interval arithmetic and the associated dependency issue are completely eliminated from the analysis such that the sharpness of results is
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improved. Therefore, such FE reformulation is meticulously adopted in this hybrid uncertainty analysis and briefly demonstrated in this subsection.
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By implementing this alternative FE modelling, the governing equations of linear elastic analysis of a structure which consist n elements, d degrees of freedom and w generalized
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stress/strain, are composed by the equilibrium condition, the compatibility condition and the elastic constitutive condition. The three governing equations can be explicitly expressed as: CT q F
(22)
Cu e
(23)
Se q
(24)
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the 2D truss and frame structures can refer to [46].
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elemental stiffness. Readers who are interested in the detailed alternative FE formulations of
When the hybrid random and interval uncertainties of system parameters are considered, the non-deterministic analysis expressed in Eq. (1) can be reformulated into: CT q RI F(ψ R , ξ I )
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Cu RI e RI
S(ψ R , ξ I )e RI q RI
(25) (26) (27)
where q RI , e RI w denote the non-deterministic generalized stress and strain vectors
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respectively. By substituting Eqs. (26) and (27) into Eq. (25), the resultant equation becomes: CT q RI CT S(ψ R , ξ I )e RI CT S(ψ R , ξ I )CuRI K(ψ R , ξ I )u RI F(ψ R , ξ I )
(28)
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It is easily recognized that Eq. (28) is exactly same as Eq. (1) with: K(ψ R , ξ I ) CT S(ψ R , ξ I )C
(29)
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stress vector as:
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Additionally, the elemental force of the structure can be obtained from the generalized
q RI S(ψ R , ξ I )e RI S(ψ R , ξ I )Cu RI
(30)
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Eqs. (25) - (27) are the general governing equations for linear elastic analyses of structures involving hybrid random and interval uncertainties, which should be satisfied regardless of element types. 3.2 Upper and lower bounds of mean of nodal displacement By implementing the alternative FE formulation as illustrated in Eqs. (28) and (29), the mean of nodal displacement as expressed in Eq.(14) can be reformulated as:
CT S(μ ψ R , ξ I )Cμ u RI F0
(31) 13
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Thus, the formulation of mean of nodal displacement can be alternative expressed by the format introduced in Eqs. (25) - (27). Mathematically, all the interval uncertain parameters can be treated as inequality constraints, and the equilibrium condition, the compatibility condition and the elastic constitutive condition are considered as equality constraints. Subsequently, the upper bound of the mean of the displacement response at vth ( v 1,...,d ) degree-of-freedom can be obtained by solving the following NLP:
μu RI maximise{ μu RI } v
v
subject to:
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C T q F0
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(P1): (32a)
(32b) (32c)
S(μ ψ R , ξ I )e q
(32d)
ξ I Φ : {ξ p | ξ k ξ kI ξ k , for k 1,..., p}
(32e)
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Cμ u RI e
Similarly, the lower bound of the mean of the displacement response at vth ( v 1,...,d )
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degree-of-freedom is calculated by solving the following problem:
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(P2):
u minimise{ μu } RI v
RI v
(33a)
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subject to: Eqs . (32 b) - (32 e)
(33b)
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Eqs. (32) and (33) indicate that by introducing the alternative finite element model into
the matrix perturbation, the upper and lower bounds of the mean of the nodal displacement can be determined without the implementation of interval arithmetic. Consequently, the extremities of the mean of the structural displacement at any degree of freedom can be rigorously established by completely eliminating the effects of interval dependency.
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3.3 Upper and lower bounds of standard deviation of nodal displacement Similarly, by adopting the alternative FE model introduced in Section 3.1, the bounds of standard deviation of the nodal displacement at the vth degree-of-freedom can be also calculated by solving two standard NPLs. To further simplify Eq. (13), additional auxiliary variables A h d , B h d , m d and n h d for h 1,2,..., m , are introduced, thus the
m
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variance of displacement can be expressed as: m
σ u RI σ u RI [ A h B h ] [ A j B j ] cov( h , j )
(34)
A h K 01Fh
(35)
h 1 j 1
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where
Bh K 01K h K 01F0 K 01K hm K 01n h
(36)
m K 01F0
(37)
n h K hm
(38)
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such that
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The introduction of these auxiliary variables enables the adoption of the alternative FE model to reformulate the standard deviation of nodal displacement. Thus, by implementing the concept expressed in Eqs. (28) and (29), Eqs. (35) – (38) can be explicitly reformulated.
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Subsequently, the determinations of the extreme bounds of standard deviation of nodal displacement could be achieved through solving two explicit NLPs. By treating interval
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uncertainties as bounded variables, the upper bound of the standard deviation of displacement response at the vth degree-of-freedom is calculated by solving the NLP defined by Eq. (39).
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Due to the non-negative property of standard deviation, ( u RI )2 σu2RI and ( u RI ) 2 σu2RI such v
v
v
v
that the validity of the proposed UPMP approach for obtaining the bounds of standard deviation through the calculation on the bounds of variance is ensured. Consequently, the upper bound of the variance of the vth ( v 1,...,d ) structural displacement can be explicitly formulated as the NLP problem (P3): (P3):
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u
2 RI v
maximise{ σ u2RI }
(39a)
v
subject to: m
m
σ u RI σ u RI [ A h B h ] [ A j B j ] cov( h , j )
(39b)
CT q A h Fh , h 1,..., m
(39c)
CAh e Ah , h 1,..., m
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h 1 j 1
(39d)
S(μ ψ R , ξ I )e A h q A h , h 1,..., m
(39e)
CT q Bh n h , h 1,..., m
(39f)
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CBh eBh , h 1,..., m
(39g) (39h)
CT q m F0
(39i)
Cm em
(39j)
S(μ ψ R , ξ I )e m q m
(39k)
CT q nh n h , h 1,..., m
(39l)
R I S(ψ , ξ ) e q , h 1,..., m nh hR m ψ R μ R ψ
(39m)
ξ I Φ : {ξ n | ξ k ξ kI ξ k , for k 1,..., p}
(39n)
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S(μ ψ R , ξ I )e Bh q Bh , h 1,..., m
Furthermore, the lower bound of the variance of displacement response at the vth
( v 1,...,d ) degree-of-freedom is calculated by solving the following NLP: (P4): u
2 RI v
minimise{ u2RI }
(40a)
v
subject to:
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(40b)
Eqs . (39 b) - (39n)
To further demonstrate the formulation of NLP problem P3 and P4, the detailed formulation of Eq. (39) is demonstrated via a proof in the Appendix A. The proposed UPMP approach transformed the computation of bounded standard deviation of nodal displacement into two NLP problems under same constraints, such that the interval arithmetic procedure is eliminated throughout the hybrid random interval analysis.
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The integration of the alternative FE formulation and mathematical programming approach advantageously accommodates the dependency of interval uncertain variables. 3.4 Upper and lower bounds of mean of elemental force
Within the framework of the proposed UPMP approach, the expression of mean of
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elemental force (Eq. (18)) is reformulated by adopting the alternative FE formulation indicated by Eq. (30) as following:
μ f RI D 0 K 01 F0
S(μ ψ R , ξ I )Cμ u RI
(41)
M
q
where q has been defined as in Eqs. (37b) – (37d) such that
ED
D 0 S(μ ψ R , ξ I )C
(42)
PT
Therefore, the upper bound of mean of the sth ( s 1,...,w ) force response can be obtained by
AC
CE
solving the NLP problem (P5) expressed as Eq. (43): (P5):
f maximise{ qs } RI s
(43a)
subject to: Eqs. (32b) - (32e)
(43b)
and the calculation of the lower bound of mean of the sth ( s 1,...,w ) force response is summarised as the following NLP (P6): (P6):
17
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RI s
minimise{ qs }
(44a)
subject to: (44b)
Eqs. (32b) - (32e)
The Eqs. (43) and (44) indicate that the computation of upper and lower bounds of the mean of elemental force are based on solving two NLP problems with same constraints as in
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the calculation of bounded mean of nodal displacement which saves the computational effort in the hybrid random interval uncertainty analysis.
3.5 Upper and lower bounds of standard deviation of elemental force
The expression of the standard deviation of elemental force for the hybrid uncertainty
m
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analysis can be reformulated by utilising the proposed UPMP approach as Eq. (45): m
σ f RI σ f RI {D 0 [ A h B h ] D h m} {D 0 [ A j B j ] D j m} cov( hR , Rj ) h 1 j 1 m
m
{S(μ ψ R , ξ I )C[ A h B h ] D h m} h 1 j 1
(45)
{S(μ ψ R , ξ )C[ A j B j ] D j m} cov( , ) m
m
h
j
R h
M
I
R j
ED
{q A h q B h q n h } {q A j q B j q n j } cov( h , j ) where A h d , B h d , m d are the auxiliary variables and q A h w , q B h w ,
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q n h w are the pseudo-vectors introduced in the reformulation for standard deviation of nodal displacement. By adopting the same concept of numerical decomposition for the
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calculation of the bounds of the standard deviation of nodal displacement, the general computational formulation of UPMP for the calculation of the upper bound of the standard
AC
deviation of the sth ( s 1,...,w ) elemental force response can be explicitly formulated into a NLP, such that: (P7):
f
2 RI s
maximise{ 2f RI }
(46a)
s
subject to:
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m
m
h
j
σ f RI σ f RI {q A h q B h q n h } {q A j q B j q n j } cov( h , j )
(46b) (46c)
Eqs. (39c) - (39n)
and the calculation of the lower bound of the standard deviation of the sth ( s 1,...,w ) elemental force response can be explicitly expressed as:
f
2 RI s
minimise{ 2f RI } s
subject to: Eqs. (50b)
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Eqs. (39c) - (39n)
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(P8): (47a)
(47b) (47c)
Eqs. (46) and (47) comprise the general formulations of the proposed UPMP approach for calculating the upper and lower bounds of standard deviation of elemental force of linear structures. The proposed computational scheme rigorously transforms a complicated hybrid
M
random interval analysis of elemental force into four NLP problems (P5 – P8) which could be solved by any available NLP solver. Additionally, the constraints for the calculation of
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standard deviation of nodal displacement and elemental force are identical such that only the objective functions of NLP problems are necessarily to be modified.
PT
3.6 The implementation of the UPMP approach The proposed UPMP approach transforms the uncertain structural responses analysis with
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hybrid random and interval parameters into mathematical programming problems. In this context, the means and standard deviations of random variables become deterministic inputs.
AC
Furthermore, by modelling the interval uncertain parameters as inequality constraints, the extreme bounds of the stochastic characteristic of structural responses are directly obtained by respectively solving the NLP problems P1 – P8. The solution algorithm of the proposed UPMP methods is summarised by a flowchart shown in Fig. 1.
19
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Fig. 1 Flowchart of the solution algorithm of UPMP approach As a combination of perturbation method and mathematical programming approach, the
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proposed UPMP method offers a computational framework which contains the advantages of both approaches. By employing the perturbation-based random moment method with respect to random variables, the explicit expressions of first two moments of structural responses are respectively formulated such that only the means and standard deviations of random variables are enclosed. Thus, the achievement of means and standard deviations of system outputs is no longer rely on generating a large number of samples for random variables. Furthermore, the adoption of the alternative FE modelling technique conveniently reformulated the expressions of the statistical characteristics of structural responses into the NLP format. Subsequently, the
20
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hybrid uncertain structural response analysis with random and interval variables can be efficiently processed by any available NLP solver. Consequently, the interval arithmetic is completely eliminated throughout the proposed computational strategy. 4. Numerical examples In this section, the applicability, accuracy and efficiency of the proposed UPMP approach are critically justified through investigations on both academic sized and practically motivated
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engineering structures. For the simple academic sized examples, the accuracy of the proposed UPMP approach can be rigorously justified by comparing with some analytical solutions, whereas the applicability of the proposed approach can be illustrated by analysing the practically motivated examples.
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Within the context of this study, all NLPs involved in the proposed UPMP approach are solved by a commercial NLP solver named CONOPT [48], which is operated within an advanced mathematical modelling environment, namely General Algebraic Modelling System (GAMS) [49]. For all the numerical examples investigated in this study, uncertain parameters from the same category of different elements are treated as independent to each other. The
M
accuracy of the results is rigorously verified through a number of different approaches. For academic sized examples, the results calculated by the proposed UPMP approach are
ED
compared with the analytical solutions where possible. For the circumstance that analytical solution cannot be determined, the computationally exhaustive Monte-Carlo Simulation combined with Quasi-Monte-Carlo Simulation (MCS-QMCS) method is implemented to
PT
verify the uncertain-but-bounded statistical characteristics obtained by UPMP approach into certain extent. The realizations for both random and interval variables used in the MCS-
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QMCS approach are generated by the Statistics toolbox of MATLAB R2015b [50]. For the QMCS scheme, the Halton low discrepancy sequence has been implemented with a dynamic
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skip and leap scheme combined with a ‗‗RR2‖ scramble scheme [51]. The presented numerical results are obtained by using a workstation with CPU of Intel Core i7-4770, 32 GB of memory, and 1 TB of hard drive. 4.1 Stepped cantilever beam The first investigation considers a two-stepped cantilever Euler-Bernoulli beam subjected to an external load acting at the right end, as depicted in Fig. 2. The Young‘s modulus and cross-sectional geometries of the two elements (element 1 is between nodes 1 and 2, and
21
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element 2 is between nodes 2 and 3) are different. For this particularly example, the vertical displacements at nodes 2 ( u 2, y ) and 3 ( u3, y ) have been analysed by employing the UPMP approach considering three cases of analysis with distinctive mixture of random and interval variables. The information of uncertain parameters of the considered three cases is presented
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in Table 1.
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Fig. 2 Two-stepped cantilever beam
The upper and lower bounds of the stochastic characteristics of u 2, y and u3, y of Case 1 obtained by the proposed UPMP approach are presented in Table 2. In order to verify the accuracy of the proposed method, the analytical expressions of u 2, y and u3, y are respectively obtained as Eqs. (48) and (49) based on the Euler-Bernoulli beam theory. Subsequently, the
M
analytical solutions for the upper and lower bounds of the means and standard deviations of
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u 2, y and u3, y for Case 1 can be calculated as demonstrated in Table 2. In such way, the accuracy of the UPMP approach has been evidently justified for this particular case of
4l 3 6l 2l u 2, y 1 3 1 2 3 F E1b1h1 E1b1h1
(48)
4l 3 12l12l2 12l1l22 4l23 u3, y 1 F 3 3 E b h E b h 1 1 1 2 2 2
(49)
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CE
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analysis.
For the investigations conducted in Cases 2 and 3, the analytical solutions of bounded
statistical characteristics of the nodal displacement with hybrid uncertainties cannot be explicitly determined, thus the MCS-QMCS method has been implemented to verify the results obtained from the proposed UPMP approach. Regarding the investigations for Cases 2 and 3, 1,000 realizations have been generated for all the interval variables within the scheme of QMCS, and 10,000 random samples have been explored at each realization of the interval
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variable. Therefore, the bounds of stochastic characteristics of u 2, y and u3, y obtained by both approaches for Cases 2 and 3 of the cantilever beam are summarised in Table 3. Table 1 Information of uncertain parameters of Example 1
E1 ( GPa )
Case 3
Interval
Normal E1 200 , E1 20
Same as Case 1
Interval
E
62.1 E2 75.9
2
Interval 0.054 b1 0.066 Interval
Same as Case 1 Same as Case 1
0.036 b2 0.044 Interval 0.0736 h1 0.0864
h2 ( m )
Interval 0.0552 h2 0.0648
F ( kN )
Lognormal F 50 , F 7.5
Lognormal E2 69 , E2 6.9 Same as Case 1
Normal b2 0.04 , b2 0.002
Same as Case 1
Same as Case 1
Same as Case 1
Normal h2 0.06 , h2 0.003
Interval 42.5 F 57.5
Normal F 50 , F 7.5
M
h1 ( m )
Normal 69 , E2 6.9
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b2 ( m )
Case 2
180 E1 220
E2 ( GPa ) b1 ( m )
Case 1
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Parameters
2, y
Analytical solution ( m )
7.019 103
7.019 103
u
2, y
1.053 103
1.053 103
u
2, y
2.904 103
2.904 103
u
2, y
4.357 104
4.357 104
u
3, y
3.067 102
3.067 102
u
3, y
4.600 103
4.600 103
u
3, y
1.269 102
1.269 102
u
3, y
1.904 103
1.904 103
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UPMP ( m )
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u
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Table 2 The statistical characteristics of Case 1 of cantilever beam
By thoroughly examining Table 3, all computational results provided by the proposed UPMP approach are fully enclosing the results reported from the dually simulative MCSQMCS approach. The reason for such observed incompetence of the simulative approach is 23
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that even though extensive amount of sampling points with considerable amount of uniformity have been explored for the interval counterpart, it still remains quite challenging for the global search based method to sample the points at the boundaries of the interval parameters (i.e., the upper and lower bounds of the interval parameters). However, from Eqs. (48) and (49), monotonic relationship can be observed for the considered interval parameters in Cases 2 and 3 with the concerned structural responses. Consequently, once explicit formulations for the mean and standard deviation for the structural responses are established,
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the UPMP approach can utilize the computational benefit offered by the mathematical programming approach to search for the extreme bounds of the statistical characteristics in a more appropriate manner.
Table 3 The statistical characteristics of Case 2 and 3 of cantilever beam Case 3
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Case 2 MCS-QMCS
UPMP
MCS-QMCS
u ( m )
7.265 103
6.711 103
7.019 103
6.442 103
u ( m )
7.265 104
6.833 104
1.053 103
9.621 104
u ( m )
2.716 103
2.982 103
2.904 103
3.142 103
u ( m )
2.716 104
3.112 104
4.357 104
4.688 104
u ( m )
3.174 102
2.840 102
2.537 102
2.432 102
u ( m )
2.249 103
2.054 103
4.165 103
4.076 103
u ( m )
1.187 102
1.372 102
1.581 102
1.660 102
u ( m ) 8.408 104
9.966 104
2.914 103
3.060 103
230.76
1.47
253.64
2, y
2, y
2, y
3, y
3, y
3, y
PT
3, y
ED
2, y
1.52
CE
t com (s)
M
UPMP
AC
In addition, the computational superiority of the proposed method is also highlighted in Table 3. That is, by consuming on average 0.62% of the computational effort required by the dual simulative method, the proposed UPMP approach is competent to deliver more robust bounds (i.e., larger upper bound and smaller lower bound) on the statistical characteristics of the concerned structural responses for Cases 2 and 3. Overall, by comparing with both analytical solutions in Case 1 and simulative results provided by the MCS-QMCS approach in Cases 2 and 3, the proposed UPMP approach offers several superiorities over the simulative approach for all the investigated scenarios.
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4.2 Three-span truss bridge In the second example, the hybrid uncertain static analysis of a three-span truss bridge involving both random and interval uncertainties is conducted. The general layout of the structure with the loading regime at reference configuration is shown in Fig. 3. The truss bridge consists of 113 elements with 108 degrees-of-freedom. Moreover, the elements of the truss bridge are categorised into four groups with different cross-sectional areas such that A1
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denotes the cross-sectional areas of all the vertical elements, A2 denotes the cross-sectional areas of all the diagonal elements, A3 denotes the cross-sectional areas of the elements between nodes 1 and 29, and A4 denotes the cross-sectional areas of the elements between
30
F2
F1 2
F1 3
F1 4
F1 5
F1 6
F1 7
F1 8
31
32
33
34
35
36
37 38
9
F2 10
F2 11
39
40
F2 12
F2 13
41
y
F2 14
42
F2
F2
F2
F2
F2
F2
F2
F3 F3 F3 F3 F3 F3 F3 F3 22 23 24 25 26 27 28 29
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F1 1
43
15
44
16
17
45
46
18
47
19
48
20
21
49 50
51
7.316m
nodes 30 and 58.
53
52
54
55
56
57
58
28x6.1m=170.8m
M
x
Fig. 3 Three-span truss bridge
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The investigation of structural response of the truss structure involves two distinctive cases with various uncertain situations. The details of the two cases are outlined as follows:
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Case 1. The Young‘s modulus ( E R , unit: GPa) is lognormal random variable and the four different cross-sectional areas ( A1R , A2R , A3R and A4R , unit: m 2 ) are normally distributed
CE
random variables. The statistical information of random variables are: E 200 GPa,
E 20 GPa; A 1.232 102 m 2 , A 6.160 104 m 2 ; A 1.361 102 m 2 , 1
1
2
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A 6.805 104 m 2 ; A 1.171 102 m 2 , A 8.550 104 m 2 ; A 1.671 102 m 2 , 2
3
3
4
A 8.355 104 m 2 . The externally applied loads ( F1I , F2I and F3I , units: kN) at reference 4
configuration are interval parameters which have: F1I ΦF1 : {F1 8 | 288 F1, 352, for 1,...,8}
(50)
F2I ΦF2 : {F2 13 | 360 F2 , 440, for 1,...,13}
(51)
F3I ΦF3 : {F3 8 | 288 F3, 352, for 1,...,8}
(52) 25
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Case 2. The effect of varying degree of uncertainty of system parameters against the structural responses is analysed. The change range (CR) of interval variables is varying between 0.04 CR 0.3 and the coefficient of variation (COV) of random variables is varying between 0.04 COV 0.3 . In this case, Young‘s modulus ( E I , unit: GPa) and crosssectional areas ( A1I , A 2I , A 3I and A 4I , unit: m 2 ) are interval variables which have the nominal C values as: E 200 GPa, A1C 1.232 102 m 2 , A2C 1.361 102 m 2 , A3C 1.171 102 m 2 ,
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and A4C 1.671 102 m 2 . Thus, the interval parameters can be expressed as:
(53)
A1I Φ A1 : {A1 29 | A1C (1 CR ) A1, A1C (1 CR ), for 1,...,29}
(54)
A2I ΦA 2 : {A2 28 | A2C (1 CR ) A2, A2C (1 CR ), for 1,...,28}
(55)
A3I ΦA3 : {A3 28 | A3C (1 CR ) A3, A3C (1 CR ), for 1,...,28}
(56)
A4I ΦA 4 : {A4 28 | A4C (1 CR ) A4, A4C (1 CR ), for 1,...,28}
(57)
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E I Φ E : {E 113 | E C (1 CR ) E E C (1 CR ), for 1,...,113}
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The random variables considered in this case are the normally distributed externally applied loads at reference configuration. The means of the externally applied loads are: F1 320 kN, 2
3
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F 400 kN, and F 320 kN.
For Case 1, both UPMP approach and dual simulative MCS-QMCS method are
PT
implemented to investigate the structural responses of the truss bridge with hybrid uncertainties specified as above. For the purpose of demonstration, the bounds of statistical
CE
characteristics of the horizontal displacement at node 10 ( u10, x ), the vertical displacement at
AC
node 44 ( u 44, y ) and the internal force of element 69 ( f 69 ) obtained by both approaches are reported in Table 7. For the MCS-QMCS method, various sample sizes are adopted in calculating the upper and lower bounds of the statistical characteristics. It is evidently illustrated that all the results of MCS-QMCS approach are enclosed within the results obtained by the proposed UPMP approach. From Table 4, it can be observed that increasing the initial sample size in MCS-QMCS approach could slightly improve the computational results. However, the computational effort increases dramatically with the increase of sample size. Therefore, the proposed UPMP approach surpassed the performance of MCS-QMCS method in both accuracy and computational efficiency. 26
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Table 4 The bounds of statistical characteristics of Case 1 of truss bridge
UPMP
MCS-QMCS
MCS-QMCS
MCS-QMCS
MCS-QMCS
100 interval
500 interval
500 interval
1000 interval
5000 random
5000 random
10,000 random
10,000 random
(m)
1.128 102
1.048 10 2
1.048 10 2
1.046 10 2
1.069 10 2
u
(m)
1.209 103
1.134 10 3
1.134 10 3
1.129 10 3
1.153 10 3
u
(m)
7.897 103
9.328 10 3
9.14110 3
9.12110 3
9.12110 3
u
(m)
8.419 104
9.867 10 4
9.634 10 4
9.66110 4
9.66110 4
u
44, y
(m)
8.305 102
7.913 10 2
7.93110 2
7.934 10 2
8.042 10 2
u
44, y
(m)
8.655 103
8.418 10 3
8.418 10 3
8.292 10 3
8.375 10 3
u
44, y
(m)
6.567 102
7.150 10 2
7.133 10 2
7.124 10 2
7.124 10 2
u
44, y
(m)
6.849 103
7.474 10 3
7.425 10 3
7.415 10 3
7.364 10 3
f (kN)
1.731103
1.613 103
1.613 103
1.614 103
1.637 103
f (kN)
22.580
21.344
21.439
21.282
21.590
f (kN)
1.277 103
1.464 103
1.442 103
1.442 103
1.442 103
f (kN)
17.844
19.386
19.211
19.177
19.177
t com (s)
3.109
679.906
2.856 103
5.487 103
1.346 10 4
1 0, x
1 0, x
69
69
69
AC
CE
PT
ED
69
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1 0, x
M
1 0, x
CR IP T
u
(a)
(b)
Fig. 4 Variations of upper and lower bounds of (a) mean and (b) standard deviation of u10, x of Case 2 of truss bridge against alternations of degree of uncertainty
27
CR IP T
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(a)
(b)
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PT
ED
M
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Fig. 5 Variations of upper and lower bounds of (a) mean and (b) standard deviation of u44, y of Case 2 of truss bridge against alternations of degree of uncertainty
(a)
(b)
AC
Fig. 6 Variations of upper and lower bounds of (a) mean and (b) standard deviation of f 69 of Case 2 of truss bridge against alternations of degree of uncertainty
28
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Fig. 7 Symmetric two-bar truss with interval uncertain Young‘s modulus and cross-sectional area
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For Case 2, the response surfaces of the upper and lower bounds of the stochastic characteristics of u10, x , u 44, y and f 69 against the variations of the random forces, interval Young‘s modulus and cross-sectional areas are constructed by the proposed UPMP approach and shown as the 3D plots through Figs. 4 to 6. Additionally, the MCS-QMCS method with
M
100 interval samples and 5000 random samples against selected combinations of change ratio (CR, which equals to the interval width divided by the midpoint) of an interval variable and
ED
coefficient of variation (COV, which equals to the standard deviation divided by the mean value) of a random variable are also implemented. Thus, for each combination of CR and COV, 100 samples of mean and standard deviation are obtained and shown in the 3D plots.
PT
Once again, the results provided by the UPMP approach enclose all the results obtained from the dual sampling method. Comparing with Case 1, the relative difference between the results
CE
obtained by both approaches are relatively large. The reason for such noticeable difference is because of the implementation of insufficient amount of sample for situations involving
AC
relatively large number of interval variables. The quality of results obtained from the QMCSMCS approach can be improved if more sampling points are explored. However, the increasing of the sample size would also burden the computational cost of such analysis. By further examining Figs. 4 - 6, several additional interesting yet important points can be noticed, which includes: 1. When the COV of random variables varies and CR of interval variables remains constant, both the upper and lower bounds of mean of the selected structural
29
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responses remain unchanged. On the other hand, the bounds of the standard deviations of the selected structural responses vary monotonically with the change of COV. 2. Under the circumstance that the COV is constant, a monotonic increase of the upper bounds and a monotonic decrease of the lower bounds of means and standard deviations of the selected structural responses can be observed when the CRs of Young‘s modulus and cross-sectional areas increases. Additionally, nonlinear relationship between the variations of the bounds of statistical characteristics and the
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changes of CRs are observed from Figs. 4 and 5. Therefore, it can be concluded from Case 2 that the influence on the bounds of the stochastic characteristics of structural responses due to the variations of the interval Young‘s modulus and cross-sectional areas cannot be predicted by simple linear scaling. Moreover, it can be observed from
u
44, y
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Fig. 5 that the increase of u 44, y is relatively faster in comparison with the decrease of .
3. By examining Fig. 4a, the u10, x obtained by the proposed UPMP approach decreases from positive to negative when the CRs of Young‘s modulus and cross-sectional area
M
increase, while the simulated samples from MCS-QMCS remain positive. The physical plausibility of such phenomenon can be demonstrated based on a symmetric
ED
two-bar truss structure, as shown in Figure 7, (element 1 is between nodes 1 and 2, and element 2 is between nodes 1 and 3) with interval Young‘s modulus ( E1 and E 2 )
PT
and cross-sectional area ( A1 and A2 ) of both elements subjected to vertical load EI ΦE : {E 2 | E E E, for 1,2}
(58)
A I ΦA : {A 2 | A A A, for 1,2}
(59)
AC
CE
applied at node 1. The bounds of Young‘s modulus and cross-sectional area are:
By simply applying the interval arithmetic with structural analysis, a positive u x can be easily obtained when E1 E , A1 A , E2 E and A2 A while the u x is negative when E2 E , A2 A , E1 E and A1 A . Thus, if the externally applied load is a random variable, the mean of u x will varies from negative to positive. Therefore, the proposed UPMP approach has the superior advantage for situations involving the positive-to-negative variations of bounds of mean of the structural responses.
30
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By implementing the UPMP approach, the random interval behaviors of the structural responses of the three-span truss bridge have been investigated against two different cases with various uncertain conditions. The accuracy and computational efficiency of the proposed method has been clearly illustrated in the study of Example 2. Additionally, the effects of varying interval Young‘s modulus and cross-sectional area and random externally applied loads on the structural responses have been investigated. From this investigation, it is necessary for engineering application to explicitly analyze the influence of the fluctuation of
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random and interval uncertain system parameters acting upon the structural responses of structures. 4.3 Multi-bay multi-storey frame
To further investigate the performance of the proposed UPMP approach, a practical-sized
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multi-bay multi-storey frame subjected to diverse vertical and horizontal loadings at various locations is considered in the last numerical investigation. In this particular example, the applicability, accuracy and computational efficiency of the UPMP approach are thoroughly demonstrated through three distinctive investigations which are including a probabilistic
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analysis, an interval analysis, and a hybrid random and interval analysis of structural responses. The general layout and loading regime of the frame is illustrated in Fig. 8. The plane frame model consists of 753 degrees-of-freedom and 376 elements. The considered
ED
uncertain parameters are the Young‘s modulus ( E , unit: GPa), cross-sectional areas of beams
PT
( AB , unit: m 2 ) and columns ( AC , unit: m 2 ), and externally applied loads ( F1 , F2 and F3 , units: kN) at reference configurations. For this frame structure, 310UC118 has been used for all columns and 310UB32 has been employed for all beams [52]. In order to maintain the
CE
compatibility between cross-sectional area and the corresponding second moment of area for both column and beam, the following compatibility functions are introduced such that the
AC
second moment of area of each element is expressed as a function of cross-sectional area. For each 310UC118 column the compatibility function is expressed as Eq. (60), whereas the compatibility function of each 310UB32 beam is expressed as Eq. (61).
I C ( AC ) 0.5238 AC2 0.0114 AC 3 10 5
(60)
I B ( AB ) 0.8876 AB2 0.0288 AB 4 105
(61)
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F2
F2
F2
F2
F2 F2 F2F2
11 F1
10 F1
9F1
8F1
7F1
6F1
5F1
4F1 3F1
2F1
1F1
180
28
193
42
206
56 70
13
179
27
192
41
205
55 69
12
178
26
191
40
204
54 68
11
177
25
190
39
203
53 67
10
176
24
189
38
202
52 66
9
175
23
188
37
201
51 65
8
174
22
187
36
200
50 64
7
173
21
186
35
199
49 63
6
172
20
185
34
198
48 62
5
171
19
184
33
197
47 61
4
170
18
183
32
196
46 60
3
169
17
182
31
195
45 59
2
168
16
181
30
194
44 58
1
15
29
F2
43 57
219
111
232
125
82 96
218
110
231
124
81 95
217
109
230
80 94
216
108
79 93
215
78 92
214
F3 F3 262 139
F3
F3
F3
F3
245
153
258
167
138
244
152
257
166
123
137
243
151
256
165
229
122
136
242
150
255
164
107
228
121
135
241
149
254
163
106
227
120
134
240
148
253
162
133
239
147
252
161
132
238
146
251
160
261
77 91
213
105
226
119
76 90
212
104
225
118
75 89
211
103
224
117
131
237
145
250
159
74 88
210
102
223
116
130
236
144
249
158
73 87
209
101
222
115
129
235
143
248
157
72 86
208
100
221
114
128
234
142
247
156
71 85
207
99
220
113
127
233
141
246
155
112
126
259
98
140
154
1m
10@3 m
M
2.25 m
1m
F3
260
y 6@3 m
F3
83 97
84
x
F2
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12 F1
14
13@3 m
F2
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F2 13 F1
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Fig. 8 The multi-bay multi-storey frame For this particular example, three cases with various uncertain scenarios are investigated.
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The details of the three distinct investigations are outlined as follows: Case 1: All system parameters are considered as random variables whose statistical
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information is collectively summarized in Table 5 [53, 54].
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Table 5 Statistical information of all random variables considered in Case 1 Parameter Distribution
Mean
COV
E (GPa)
lognormal
200
0.06
AB ( m 2 )
normal
40.8 104
0.05
AC ( m 2 )
normal
150 104
0.05
F1 (kN)
normal
5.5
0.1
F2 (kN)
normal
60
0.1
F3 (kN)
normal
80
0.1
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Case 2: The Young‘s modulus, cross-sectional areas of each beam and column, as well as all the applied loads at reference configuration are considered as interval uncertain parameters. The detailed information regarding all considered interval parameters is summarized through Eqs. (62) - (67). (62)
A IB ΦAB : {A B 151 | 36.72 104 AB, 44.88 104 , for 1,...,151}
(63)
ACI ΦAC : {AC 130 | 135 104 AC , 165 104 , for 1,...,130}
(64)
F1I ΦF1 : {F1 8 | 4.95 F1, 6.05, for 1,...,13}
(65)
F2I ΦF2 : {F2 156 | 54 F2, 66, for 1,...,156}
(66)
F3I ΦF3 : {F3 95 | 72 F3, 88, for 1,...,95}
(67)
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EI ΦE : {E 281 | 180 E 220, for 1,...,281}
Case 3: Two scenarios are thoroughly investigated to demonstrate the performance of the
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proposed UPMP approach in hybrid random and interval analysis. For both scenarios, the interval cross-sectional areas of beams and columns are expressed as Eqs. (68) and (69), and
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the details of random uncertain parameters are listed in Table 6. (68)
ACI ΦA C : {AC 130 | 141 104 AC , 159 104 , for 1,...,130}
(69)
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A IB ΦA B : {A B 151 | 38.352 104 AB, 43.248 104 , for 1,...,151}
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Table 6 Random uncertain parameters of frame structure Distribution
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Parameter
Mean COV Scenario 1 Scenario 2
E (GPa)
lognormal
lognormal
200
0.045
F1 (kN)
normal
gumbel
5.5
0.1
F2 (kN)
lognormal
lognormal
60
0.1
F3 (kN)
normal
normal
80
0.1
Table 7 The statistical characteristics of frame structure of Case 1
33
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MCS
MCS
10,000
100,000
UPMP
R1 (%) R2 (%)
(m)
9.194 102
9.263 102
9.260 102
-0.748 -0.714
u
(m)
1.190 102
1.194 102
1.205 102
-0.339 -1.208
u (m)
1.689 102
1.700 102
1.699 102
-0.628 -0.616
u (m)
1.931 103
1.953 103
1.946 103
-1.111 -0.793
M (kNm)
92.652
92.757
92.790
-0.113 -0.148
M (kNm)
9.655
9.613
9.685
0.440
-0.311
t com (s)
71.765
9.470 102
1.767 104
N/A
N/A
167, x
167, x
70, y
70, y
1
1
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u
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For Case 1 of Example 3, both UPMP and Monte-Carlo Simulation (MCS) method are implemented to investigate the random behaviour of the structural responses of this complex frame. For the MCS approach, two simulations with 10,000 and 100,000 samples have been adopted to verify the results obtained by the proposed method. The means and standard deviations of the horizontal displacement at node 167 ( u167, x ), vertical displacement at node
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70 ( u70, y ), and the bending moment at node 1 ( M 1 ) calculated by both approaches are
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outlined in Table 7, where R1 and R2 denote the relative differences between the results obtained by UPMP approach and MCS method with 10,000 and 100,000 samples,
R
UPMP MCS % MCS
(74)
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respectively. The relative difference in this case is defined as:
It is indicated in Table 7 that the results calculated by the UPMP approach agree well with the statistical information obtained from the MCS approach with various sample sizes,
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but of course in a much more computational friendly manner. Therefore, the applicability, accuracy and computational efficiency of the proposed UPMP approach for pure stochastic analysis have been rigorously justified through this investigation.
34
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(a)
(b)
Fig. 9 Bounds of (a) u70, x and (b) M 1 of frame structure of Case 2.
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For Case 2, interval analyses of the displacement and force responses of the frame structure have been conducted by both UPMP and Quasi-Monte-Carlo Simulation (QMCS) approaches with various sample sizes. The reason for adopting QMCS in this investigation is due to the advantage of QMCS approach in generating more uniform samples than the MCS method, which could potentially enhance the effectiveness of the QMCS method. For this
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particular case, 100, 1000, 10,000 and 100,000 simulations have been implemented through the QMCS scheme. For the demonstration purpose, the upper and lower bounds of u70, y and
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M 1 obtained by both approaches are shown in Fig. 9, where the colour scale in the plots indicates the computational time consumed by the QMCS approach. It can be easily observed
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that the UPMP approach surpasses the capability of the QMCS method by providing a larger upper bound and smaller lower bound within 20 seconds. Indeed, limited improvements of
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results obtained from the QMCS approach can be observed by exponentially increasing the sample size, but the possibility of engineering application for such global search based
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computational approach is quite challengeable due to the inevitably substantial consumption of computational effort.
Table 8 The bounds of statistical characteristics of the frame structure in Case 3. MCS-QMCS UPMP
u
167, x
(m )
1.016 101
Scenario 1
Scenario 2
9.351102
9.351102 35
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(m )
1.127 102
1.041 102
1.048 102
u
(m )
8.350 102
9.105 102
9.105 102
(m )
9.640 103
1.019 102
1.027 102
(m )
1.815 10 2
1.734 102
1.734 102
(m )
1.764 10 3
1.691103
1.702 103
(m )
1.578 102
1.654 102
1.654 102
(m )
1.550 103
1.616 103
1.627 103
M (kNm)
110.832
101.412
M (kNm)
10.568
9.581
M (kNm)
76.804
85.679
M (kNm)
7.140
7.919
t com (s)
2.031103
167, x
u
167, x
u
70, y
u
70, y
u
70, y
u
70, y
1
1
1
1
101.416 9.657
85.682 7.983
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167, x
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u
6.399 104
6.366 104
For Case 3, the uncertain-but-bounded statistical characteristic of selected displacement and force responses obtained by both approaches are listed in Table 8. For the MCS-QMCS
M
method, 100 simulations have been implemented for generating realizations for interval variables through the QMCS scheme, and subsequently, 10,000 simulations have been
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conducted at each interval realization for the stochastic analysis. By inspecting the results reported in Table 8, the upper and lower bounds of statistical information regarding concerned
PT
structural responses obtained by MCS-QMCS method are enclosed by the results determined from the UPMP approach for both Scenarios 1 and 2. Since that the proposed approach
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simply requires the mean and standard deviations of the random variables, it leads to the same UPMP formulations and subsequently the results for the structural responses for both
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Scenarios 1 and 2. Comparing with the adopted sampling method, the proposed UPMP approach is capable of offering sharper bounds on the means and standard deviations of all the considered structural responses with dramatically less computational cost. Therefore, the accuracy and computational efficiency of the proposed approach in hybrid uncertainty analysis have been evidently demonstrated. In addition to the illustration on the applicability and effectiveness of the proposed UPMP approach, both parametric and non-parametric statistical inference analyse methods which are widely adopted [55 - 59] in structural engineering for identifying the probability distributions of random variables are selected to construct the probability density functions 36
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(PDFs) and cumulative distribution functions (CDFs) of the bounds of the concerned structural responses demonstrated in Table 8. The selected parametric methods include loglikelihood test, Akaike information criterion (AIC) test and Bayesian information criterion (BIC) test. The non-parametric analyses adopted involve Kolmogorov–Smirnov (KS) test and Anderson–Darling (AD) test. These methods are implemented on the simulated results of
u167, x , u10, y and M 1 at each realisation of interval parameters from MCS-QMCS for both
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Scenarios 1 and 2. According to the test statistics, the types of probability distribution which are closest fit for describing the selected structural responses are summarised in Table 9. For demonstration purpose, the test statistics for the samples corresponding to the 50th realisation of interval parameters for Scenarios 1 and 2 are respectively summarised as in Table B1 and B2 in Appendix B.
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Table 9 Probability distribution types identified by statistical inference analyses in Case 3. Probability Distribution Scenario 1
Scenario 2
Normal
Gumbel
u167, x
Normal
Gumbel
ED
M1
Lognormal Lognormal
M
u70, y
Subsequently, the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the demonstrated displacement and force responses of Scenarios 1 and 2
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are shown in Figs. 10 and 11, respectively. It can be observed from Figs. 10 and 11 that the PDFs of the MCS-QMCS results are bounded by the upper bound of PDF (PDF-UB) and
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lower bound of PDF (PDF-LB) constructed based on the results of UPMP approach. Moreover, the upper bound of CDF (CDF-UB) and lower bound of CDF (CDF-LB) of the
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selected nodal displacement and elemental force are rigorously enclosing all CDFs constructed basing on the results of the MCS-QMCS approach.
37
AC
CE
PT
ED
M
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Fig. 10 PDFs and CDFs of selected structural responses of Scenario 1 of Case 3 of the frame structure.
38
AC
CE
PT
ED
M
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Fig. 11 PDFs and CDFs of selected structural responses of Scenario 2 of Case 3 of the frame structure.
39
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5. Conclusions This paper presents a computational scheme named as unified perturbation mathematical programming (UPMP) approach for structural analysis involving random and interval parameters. The proposed approach offers a potent non-sampling strategy for investigating the static response for various engineering structures. The integration of matrix perturbation with mathematical programming within the proposed computational approach transforms the
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intricate hybrid random interval analysis into a series of NLP problems. By treating interval parameters as inequality constraints in the mathematical programming, the interval dependency is eliminated from the algorithm.
The applicability of the proposed UPMP approach is critically validated by investigating various types of engineering structures through numbers of uncertain circumstances. By
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comparing with analytical solution and results obtained from sampling method, the accuracy and computational efficiency of the proposed uncertainty analysis scheme are evidently illuminated. Additionally, hypothesis test is implemented in this study for identifying the probability distribution of the structural responses.
M
From the computational benefits associated with the proposed UPMP approach, it is possible to further extend the proposed computational scheme for other types of engineering analyses, such as hybrid uncertain linear and nonlinear buckling analysis, dynamic analysis of
ED
structures involving random and interval uncertainties, etc. Furthermore, the concept of the proposed approach can be also extended to other areas of study where the effects of
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uncertainties must be evaluated.
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6. Acknowledgement
The work presented in this paper has been supported by Australian Research Council
AC
through projects DP140101887 and DP160103919. The authors would like to sincerely express our gratitude to the anonymous reviewers for
their valuable comments and constructive suggestions.
40
ACCEPTED MANUSCRIPT Appendix A. Details of reformulation of Eqs. (35) – (38) The details of the reformulation of Eqs. (35) – (38) are illustrated by the following proof, which provides a reference for the construction of the two NLP problems indicated by Eqs. (39) and (40). Proof.
reformulation of Eq. (35) is demonstrated as:
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By implementing the alternative FE formulation presented in Section 3.1, the
(A1)
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CT q F Ah h 1 T I A h K 0 Fh C S(μ ψ R , ξ )CA h Fh CA h e A h I S(μ ψ R , ξ )e A h q A h
where q A h w and e A h w denote the two pseudo-vectors corresponding to A h . The two pseudo-vectors are introduced here for numerical decomposition purpose. By adopting the same concept, Eq. (36) can be decomposed as:
(A2)
ED
M
CT q n Bh h 1 T I B h K 0 n h C S(μ ψ R , ξ )CB h n h CB h e B h S(μ ψ R , ξ I )e B h q B h
where q B h w and e B h w denotes the two pseudo-vectors corresponding to B h .
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Similarly, Eq. (37) is reformulated as Eq. (A3) by adding two pseudo-vectors q m w
AC
CE
and e m w . That is,
CT q F m 0 1 T I m K 0 F0 C S(μ ψ R , ξ )Cm F0 Cm e m I S(μ ψ R , ξ )e m q m
(A3)
Then, Eq. (38) could be decomposed into Eq. (A4) by introducing a pseudo-vector
q n h w , such that: CT q n h n h n h K h m CT S h Cm n h Cm e m S e q nh h m
(A4a)
41
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where
S(ψ R , ξ I ) Sh hR ψ R μ
(A4b) ψR
The combination of Eqs. (A1) – (A4) constructs the equality constraints listed as Eqs. (39c) – □
(39m). This completes the proof.
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Appendix B. Results of parametric and non-parametric analysis for u167, x of Case 3 in Example 3
The test statistics of the implemented parametric and non-parametric statistical inference methods for u167, x Case 3 in Example 3 are summarised in Tables B1 and B2. The maximum
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test statistics obtained by the log likelihood approach indicates the best fitted probability distribution for the given samples, while the distribution with minimum test statistics obtained by other methods is inferred as the most suitable.
Table B1 Parametric and non-parametric test statistics for u167, x for Scenario 1 of Case 3.
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Scenario 1- u167, x Parametric Method
AC
CE
PT
ED
Distribution Log Likelihood 10 4 Normal 3.0879 Gamma 3.0862 Lognormal 3.0821 Gumbel 3.0195 Logistic 3.0769 Log logistic 3.0742 Weibull 3.0461 Rayleigh 1.9963
AIC 10 4 -6.1755 -6.1720 -6.1638 -6.0386 -6.1533 -6.1480 -6.0918 -3.9925
Non-parametric Method BIC 10 4 -6.1740 -6.1706 -6.1624 -6.0372 -6.1519 -6.1465 -6.0903 -3.9918
AD Test
KS Test
0.9730 3.1464 8.3085 97.6199 10.4052 13.6244 59.0456
0.0114 0.0135 0.0191 0.0581 0.0225 0.0229 0.0428 0.4340
2.8050 103
Table B2 Parametric and non-parametric test statistics for u167, x for Scenario 2 of Case 3. Case 2- u167, x Parametric Method Distribution Log Likelihood 10 Normal 3.1477 Gamma 3.1754 Lognormal 3.1859
4
AIC 10 -6.2949 -6.3503 -6.3714
4
Non-parametric Method BIC 10 -6.2935 -6.3489 -6.3700
4
AD Test
KS Test
83.7019 46.1837 31.6121
0.0623 0.0475 0.0399 42
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Gumbel Logistic Log logistic Weibull Rayleigh
3.2039 3.1645 3.1866 3.0184 2.0533
-6.4074 -6.3286 -6.3727 -6.0363 -4.1064
-6.4060 -6.3272 -6.3713 -6.0349 -4.1057
4.8649 48.6919 22.4190 296.2260
2.8695 103
0.0157 0.0381 0.0243 0.1078 0.4648
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Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach Jinwen Feng , Di Wu , Wei Gao , Guoyin Li PII: DOI: Reference:
S0307-904X(17)30261-5 10.1016/j.apm.2017.03.066 APM 11707
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
23 September 2016 22 February 2017 28 March 2017
Please cite this article as: Jinwen Feng , Di Wu , Wei Gao , Guoyin Li , Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.03.066
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Highlights A unified perturbation mathematical programming (UPMP) approach is proposed. Random and interval uncertainties are simultaneously considered in this study.
Both nodal displacement and elemental force are analysed by UPMP approach.
Uncertain-but-bounded statistical characteristics are efficiently captured via NLP.
Interval dependency issue is completely eliminated from the algorithm
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Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach Jinwen Feng1, Di Wu1, Wei Gao1,*, Guoyin Li2 1
Centre for Infrastructure Engineering and Safety (CIES), School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia 2
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School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia
Abstract
A novel computational method, namely the unified perturbation mathematical programming
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(UPMP) approach, for hybrid uncertainty analysis of engineering structures is proposed in this paper. The presented study considers a mixture of random and interval system parameters which are frequently encountered in engineering applications. Within the UPMP approach, matrix perturbation theory is adopted in combination with the mathematical programming approach. The proposed computational method provides a non-simulative hybrid uncertainty
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analysis framework, which is competent to offer the extreme bounds of the statistical characteristics (i.e., mean and variance) of any concerned structural responses in
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computationally tractable fashion. In order to thoroughly explore various intricate aspects of the engineering system involving hybrid uncertainties, systematic numerical experiments have
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also been conducted. Diverse statistical analyses are implemented to identify the bounded probability profile of the uncertain structural responses. Both academic and practical
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engineering structures are investigated to justify the applicability, accuracy and efficiency of the proposed UPMP approach.
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Keywords: Random Interval Analysis; Nondeterministic Structural Responses; Hybrid Uncertainties; Mathematical Programming; Bounded Statistical Feature.
*
Corresponding author. Email addresses: [email protected] (J. Feng), [email protected] (D. Wu), [email protected] (W. Gao), [email protected] (G. Li)
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1. Introduction In engineering analysis and design, computational methods have been extensively developed by hypothetically considering models and system parameters as deterministic. Finite element method (FEM), along with the fast growth of the capability of computers, establishes a favourable analysis framework for modern engineering applications [1]. Despite the wide adoption of the deterministic analysis through FEM, the traditional framework of such
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analysis neglects nondeterministic features, or so-called uncertainties, which inevitably exist within engineering structures and modelling processes [2]. These uncertainties, either origin from the inherent variation of system parameters or the lack of knowledge or information, potentially lead to fluctuations in determinations of the structural behaviours [3, 4]. Therefore, the development of efficient computational approaches for predicting the effect of diverse
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uncertain parameters on structural behaviours becomes mandatory.
Regarding the traditional uncertainty analysis, probabilistic/stochastic approaches are frequently implemented in the uncertain structural analysis due to the solid theoretical foundation associated with uncertainty modelling. Within the probabilistic analysis framework, all uncertain parameters are modelled as either random variables or fields with
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associated statistical information [5, 6]. Consequently, by combining the FEM with the wellestablished theories on probability and statistics, the stochastic finite element method (SFEM)
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has been innovatively established with extensive application across wide range of engineering disciplines [7-9]. Within the framework of the SFEM, the well-known Monte-Carlo
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simulation (MCS) method is one of the most straightforward procedures, which has been pervasively implemented in a variety of stochastic problems [10-12]. In addition, the
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computational stochastic analysis, which has been developed basing on the theory of general matrix perturbation, offers an advantageous non-simulative strategy for uncertainty analysis
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on engineering structures involving random variables [13-16]. Furthermore, the spectral stochastic finite element method (SSFEM) was introduced to address the spatially dependent uncertain system parameters in structural engineering problems [17-19]. The core concept of SSFEM is to establish a valid surrogate model by implementing various series expansion schemes (e.g., the Karhunen-Loeve expansion) such that the statistical characteristics (i.e., mean and variance) of the structural responses can be effectively calculated. Additionally, other types of stochastic methods have been proposed as in [20, 21]. Even though the probabilistic approach of uncertainty analysis has been prevalently implemented in practical
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engineering applications, the applicability of such methodology strongly depends on the availability of the information of uncertain parameters [22]. However, accumulated experiences from engineering practice have repeatedly revealed that the requirement for the implementation of the probabilistic analysis cannot be always fulfilled due to the issue of information deficiency on the uncertain system parameters [23]. In order to achieve valid uncertainty analysis for situations where stochastic approaches are
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prohibited, the non-probabilistic approaches, such as the convex model analysis and fuzzy uncertainty analysis, have been developed as alternatives. By using convex sets to describe uncertain parameters, such non-stochastic modelling technique has been applied in uncertainty propagation analysis, reliability analysis and engineering optimizations involving uncertain system inputs [24-26]. The fuzzy analysis, stemming from the fuzzy set theory,
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incorporates membership functions to represent the possibility distributions of uncertain variables [27, 28]. In addition, interval models are gaining popularity because of the conceptual simplicity [29]. Instead of demanding the full profile of uncertain parameters, only the upper and lower bounds of uncertain parameters are required for valid uncertainty analysis. By implementing the concept of interval analysis within the framework of the FEM, various
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computational algorithms with specific emphasises have been explicitly developed for tackling intricate engineering problems with enhanced capability and accuracy [30-34].
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Furthermore, the concept of interval field was also proposed in [35], and subsequently extended to model spatially dependent uncertain parameters in a non-probabilistic fashion
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[36].
In modern engineering applications, it is frequently encountered that both probabilistic
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and non-probabilistic uncertainties exist simultaneously [37]. Therefore, a computational scheme which is capable of estimating the effect of diverse types of uncertainties on
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engineering system is inevitably necessary. A random interval perturbation method has been developed for static response analysis [38], and free vibration analysis [39] of structures with both stochastic and non-stochastic uncertainties. In these studies, the upper and lower bounds of means and variances of system responses are explicitly formulated, such that the computational efficiency of this type of approach can be well preserved. Also, the perturbation based computational method is extended to the dynamic response analysis of acoustic field and structural-acoustic interaction system with hybrid uncertain parameters [40]. Stochastic response analysis of structures with interval system parameters under random excitations has been investigated in [41]. Polynomial-Chaos-Chebyshev-Interval method is 4
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recently proposed for vehicle dynamic problems with mixed uncertainties [42]. Hybrid probabilistic and interval [43], or probabilistic and convex set models for structural reliability analysis [44] are also explicitly proposed. By thoroughly examining on the literatures regarding the perturbation theory based hybrid uncertainty analysis with random and interval variables, it is realized that either series expansion or numerical approximation has been implemented to derive explicit formulations
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for the upper and lower bounds of the statistical characteristics of the considered structural responses. The advantage of such prevalently employed technique is that explicit formulations on the bounded statistical characteristics of structural responses can be explicitly formulated. However, the drawback of such analysis is that the overestimation on the bounded statistical characteristics of system output cannot be completely exempted due to the issue of interval
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dependency. As clearly demonstrated in the previous studies [30-34], the issue of interval dependency can significantly affect the accuracy of the interval analysis, which is mainly attributed by the multiple occurrences of the same interval parameter at various locations. The general rule is that the more numbers of interval parameters appear in the system, the more likely the bounded results are suffering from overestimation.
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In the light of achieving valid hybrid uncertainty analysis with the least overestimation due to interval dependency, a novel computational method, namely the unified perturbation
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mathematical programming (UPMP) method, is proposed for static analysis of engineering structures with a mixture of random and interval uncertain parameters. Within the proposed
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computational scheme, the matrix perturbation theory and mathematical programming (MP) approach are integrated in conjunction with an alternative FEM modelling technique. Such
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novel integration reformulates the hybrid probabilistic interval static analysis into a series of nonlinear programming (NLPs) problems. Consequently, the upper and lower bounds of the
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mean and variance of the concerned structural response can be explicitly determined by solving the corresponding NLPs. Some core advantages of UPMP are including: 1. The superior computational efficiency of the non-simulative method is fully inherited by the UPMP approach, which significantly enhances the applicability of the proposed scheme in the context of analysing modern complex engineering structures. 2. The physical feasibility of all interval parameters can be rigorously maintained throughout the entire hybrid probabilistic interval analysis. Thus, the determined system outputs are fully liberated from the overestimation caused by the issue of interval dependency. 5
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3. The critical values of the interval parameters responsible for the extremities of statistical information of structural response can be effectively retrieved as byproducts of the uncertainty analysis. In order to evidently illustrate the applicability and effectiveness of the proposed UPMP approach, both academic sized and practically motivated structures have been investigated in this study. For the purpose of results verification, all computational results provided by the
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UPMP approach are rigorously testified against either analytical solution (for academic sized example) or results obtained from exhausted simulation approach (for practically motivated example). In addition to the proposition of the UPMP approach, supplementary numerical experiments have been conducted to further explore some important aspects on the hybrid uncertain static analysis with random and interval variables.
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The reminder of this paper is organised as follows. In Section 2, the formulations for static structural response analysis with hybrid uncertainties are presented. The proposed UPMP scheme is formally presented in Section 3. Furthermore, both academic sized and practically motivated numerical examples are investigated by the proposed UPMP approach and then critically compared with either analytical solution or exhausted simulation approach
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in Section 4. Finally, some concluding remarks are drawn in Section 5.
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2. Non-deterministic static analysis of structures with hybrid uncertainties For the non-deterministic analysis, the system parameters such as material properties,
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cross-sectional geometries and loading regimes are considered as uncertain variables. In general engineering practice, the availability and sufficiency of information of uncertain
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system parameters are strongly situational dependent [7-8]. Therefore, it is rational and necessary to develop a hybrid uncertainty analysis scheme which can adopt as many different
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uncertainty modelling techniques as possible in a single framework of analysis. 2.1 Generalized hybrid random interval static analysis of structures Without loss of generality, R Z () is defined as a random variable, where Z ()
represents the set of all real random variables in a probability space Ω, Α, P and denotes the set collects all real numbers; and I I () denotes an interval variable, such that
I [ , ] { | } , where I () denotes the closed set for all real intervals, and
represent the lower and upper bounds of I , respectively. For maintaining the consistency 6
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of formulations throughout this paper, () R denotes that the parameter () is a random variable/vector, () I denotes that the parameter () is an interval variable/vector, and () RI denotes that the parameter () is a variable/vector which includes both random and interval characteristics. Thus, the non-deterministic linear static response analysis of a structure with d degrees-of-freedom (DOFs) is expressed as: K(ψ R , ξ I )u RI F(ψ R , ξ I )
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such that
(1)
ψ R Ω : {ψ m | Rj ~ f R ( x), for j 1,...,m}
(2)
ξ I Φ : {ξ p | ξ k ξ kI ξ k , for k 1,..., p}
(3)
j
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where K (ψ R , ξ I ) d d denotes the non-deterministic stiffness matrix of the structure; u RI d is the vector that contains the non-deterministic displacements at all DOFs; and
F(ψ R , ξ I ) d denotes the non-deterministic externally applied forces; ψ R denotes the random vector which collects all the m probabilistic uncertain system parameters; f R (x) j
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denotes the probability density function of the random variable Rj ; ξ I denotes the interval
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vector which contains all the p interval variables presented in the system; k and k denote the lower and upper bounds of kI , respectively. It is worth to notice that due to the presence of
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hybrid uncertainties, the stiffness matrix and force vector are functions of both random and interval variables.
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The consideration of hybrid stochastic and non-stochastic uncertainties within the structural system dramatically increases the complexity of the static analysis problem. Due to
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the simultaneous existence of both random and interval variables, the behaviour of structural responses including nodal displacement and elemental force could have both probabilistic and interval characteristics. Moreover, for complex structural systems, the relation between each input variable and the structural output (i.e. nodal displacement and elemental force) cannot be analytically determined such that the exact prediction on the mercurial characteristics of structural behaviour is prohibited. Being conceptually different with the theories of imprecise probabilities [45], the random and interval uncertainties considered in this study are mutually independent rather than inclusive. 7
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2.2 Hybrid uncertainty analysis by using moment method By implementing the concept of the random moment method [7, 39] combined with the probabilistic theory, the means and standard deviations of the structural responses can be explicitly reformulated. Without loss of generality, multiple random and interval variables are respectively represented by random vector ψ R m and interval vector ξ I p . Within the framework of finite element method, the structural stiffness matrix K (ψ R , ξ I ) and the applied
expanded by implementing the random moment method as:
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load vector F(ψ R , ξ I ) , which are the functions of random and interval parameters, can be
(4)
F(ψ R , ξ I ) { R μ R } F(ψ , ξ ) F(μ ψ R , ξ ) R h h h 1 h ψ R μ R ψ 1 m m 2F(ψ R , ξ I ) { R μ }{ R μ } Re j (F) R R hR Rj h 2 h 1 j 1 h j R ψ μ ψR
(5)
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m K (ψ R , ξ I ) { R μ R } K (ψ R , ξ I ) K (μ ψ R , ξ I ) R h h h h 1 ψ R μ R ψ 1 m m K (ψ R , ξ I ) { R μ }{ R μ } Re j (K ) R R hR Rj h 2 h 1 j 1 h j R ψ μ ψR m
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R
1
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where μ ψ R [ R ,..., R ,..., R ] is the vector collects the means of all random variables; R h
m
h
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denotes the mean of the hth random variable hR ; Re () is the high-order terms of the Taylor expansion for the uncertain parameter () .
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From the theory of first-order matrix perturbation, the governing equation for linear static analysis of the structure involving various uncertain parameters can be formulated as: [K 0 1K][u 0 1u] F0 1F
(6)
K 0 K (μ ψ R , ξ I )
(7)
where
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(8)
F0 F(μ ψ R , ξ I )
(9)
2 m F(ψ R , ξ I ) { R μ R } F { R μ R } 1F h h h h h hR h 1 h 1 ψ R μ R ψ
(10)
m
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m
By neglecting the high-order terms, the random interval structural displacement based on the first-order perturbation is calculated as: u RI u 0 1u m R I F(ψ , ξ ) { R μ R } K F K R h h h 1 h ψ R μ R ψ 1 0
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1 0 0
m R I K ( ψ , ξ ) R 1 { μ R } K F h 0 0 h 1 hR h ψ R μ R ψ
(11)
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R I R I 1 F(ψ , ξ ) 1 K ( ψ , ξ ) 1 K F K 0 K0 K 0 F0 { hR μ R } R R h h h h 1 ψ R μ R ψ R μ R ψ ψ m
m
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1 0 0
K 01F0 K 01Fh K 01K hK 01F0 { hR μ R } h
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h 1
Subsequently, the mean and standard deviation of the random interval structural
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displacement responses u RI based on the first-order matrix perturbation [39] can be obtained
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by taking the mean and variance operation on Eq. (11), which leads to:
m
μ u RI K 01F0 m
(12)
σ u RI σ u RI K 01Fh K 01K h K 01F0 K 01F j K 01K j K 01F0 cov( hR , Rj )
(13)
h 1 j 1
where μ u RI d and σ u RI d denote the vectors which collect the mean and standard deviation of the displacement responses at all d degrees-of-freedom, respectively;
cov( h , j ) denotes the covariance between the hth and jth random variables h and j ; the operator ― ‖ denotes the Hadamard product of two matrices with equal dimension [47]. As 9
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indicated by Eqs. (12) and (13), the hybrid random interval problem has been transferred into two explicit interval analysis such that the statistical characteristics of the structural displacements are uncertain-but-bounded which can be further expressed as: μu RI Φ : {μu RI d | u RI u RI u RI , for v 1,..., d}
(14)
σu RI Φ : {σu RI d | u RI u RI u RI , for v 1,..., d}
(15)
v
v
v
v
v
v
v
v
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where μu RI and σ u RI denote the mean and standard deviation of the structural displacement ( u vRI ) at vth degree-of-freedom, respectively; u RI and u RI denote the upper and lower bounds of v
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μu RI , respectively; u RI and u RI denote the upper and lower bounds of σ u RI , respectively. v
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In addition, the elemental force denoted by f RI w , can be explicitly formulated as: f RI D(ψ R , ξ I )u RI D(ψ R , ξ I )K (ψ R , ξ I ) 1 F(ψ R , ξ I )
where D(ψ R , ξ I ) wd can be defined as:
(16)
(17)
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m D(ψ R , ξ I ) { R μ R } D(ψ R , ξ I ) D(μ ψ R , ξ I ) R h h h h 1 ψ R μ R ψ 1 m m D(ψ R , ξ I ) { R μ }{ R μ } Re j ( D) R R hR Rj h 2 h 1 j 1 h j R ψ μ R ψ m 1 m m D0 Dh { hR μ R } Dh , j { hR μ R }{ Rj μ R } Re ( D) h h j 2 h 1 j 1 h 1
Once again, by implementing the random moment method combined with the theory of matrix perturbation, the mean and standard deviation of elemental force can be explicitly
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formulated as:
μ f RI D0 K 01F0 m
(18)
m
σ f RI σ f RI {D 0 [K 01Fh K 01K h K 01F0 ] D h K 01F0 }
(19)
h 1 j 1
{D 0 [K F j K K j K F ] D j K F }cov( , ) 1 0
1 0
1 0 0
1 0 0
R h
R j
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(20)
σf RI Φ : {σf RI w | f RI f RI f RI , for s 1,..., w}
(21)
s
s
s
s
s
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s
where μ f RI and σ f RI denotes the mean and standard deviation of f sRI which is the sth s s
component of the elemental force vector, respectively; f RI and f RI denote the upper and s
s
lower bounds of μ f RI , respectively; f RI and f RI denote the upper and lower bounds of σ f RI , s s
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s
s
respectively; w denotes the total number of elemental force in the structural system. Eqs. (14) - (15) and (20) - (21) indicate that the implementation of the random moment method meticulously transforms the hybrid random interval static analysis of linear structures
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into a series of interval problems, which provide an efficient yet systematic scheme for solving uncertainty analysis of structures with a mixture of random and interval variables. The
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explicit reformulations on the mean and standard deviation of the nodal displacement and elemental force offer preliminary information for analysing the uncertain structural behaviour under static loads. However, the calculation of the uncertain-but-bounded statistical
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characterises remains a computational challenge which is dominated by the aforementioned interval dependency issue. In this context, the interval arithmetic based approaches could lead
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to overestimation of the bounded stochastic characteristic while sampling method would be computational expensive for complex hybrid uncertainty analysis. Therefore, it is necessary to
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develop a more computational tractable approach such that the bounded means and standard deviations of the structural responses can be efficiently calculated without the interference of the issue of interval dependency. 3. The unified perturbation mathematical programming (UPMP) approach In this study, a new computational approach, namely the unified perturbation mathematical programming (UPMP) approach, is proposed to investigate the hybrid uncertain static response analysis with random and interval parameters. The UPMP approach offers a non-sampling computational strategy which can efficiently determine the mean and standard 11
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deviations of the structural responses. Furthermore, by adopting the alternative finite element (FE) formulation, the upper and lower bounds of the means and standard deviations of the structural responses can be explicitly formulated into standard nonlinear programming problems (NLPs). Consequently, the extreme bounds of all statistical characteristics (i.e. means and standard deviations) of either nodal displacements or elemental forces can be efficiently obtained by solving eight standard NLPs.
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3.1 Alternative finite element formulation As previously emphasized, the issue of interval dependency is always present in assembling global stiffness matrix with interval variables through the interval arithmetic approach. Even though the interval stiffness matrix can be easily assembled, it is highly likely that the final computational results are over-estimated due to the inappropriate manipulation
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of interval algebra. In the light of eliminating the interference on the system outputs caused by the interval dependency, an alternative FE formulation [34, 46] is implemented in interval analysis of engineering structures. Within the studies reported in [34, 46], the structural response analysis and linear bifurcation buckling analysis with interval uncertain parameters
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are transformed into mathematical programming problems. Subsequently, the interval analyses are processed directly via NLP solver by modelling the interval parameters as inequality constraints of the NLPs while the originality of the FE governing equations is
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thoroughly maintained. In this context, the interval arithmetic and the associated dependency issue are completely eliminated from the analysis such that the sharpness of results is
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improved. Therefore, such FE reformulation is meticulously adopted in this hybrid uncertainty analysis and briefly demonstrated in this subsection.
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By implementing this alternative FE modelling, the governing equations of linear elastic analysis of a structure which consist n elements, d degrees of freedom and w generalized
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stress/strain, are composed by the equilibrium condition, the compatibility condition and the elastic constitutive condition. The three governing equations can be explicitly expressed as: CT q F
(22)
Cu e
(23)
Se q
(24)
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the 2D truss and frame structures can refer to [46].
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elemental stiffness. Readers who are interested in the detailed alternative FE formulations of
When the hybrid random and interval uncertainties of system parameters are considered, the non-deterministic analysis expressed in Eq. (1) can be reformulated into: CT q RI F(ψ R , ξ I )
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Cu RI e RI
S(ψ R , ξ I )e RI q RI
(25) (26) (27)
where q RI , e RI w denote the non-deterministic generalized stress and strain vectors
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respectively. By substituting Eqs. (26) and (27) into Eq. (25), the resultant equation becomes: CT q RI CT S(ψ R , ξ I )e RI CT S(ψ R , ξ I )CuRI K(ψ R , ξ I )u RI F(ψ R , ξ I )
(28)
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It is easily recognized that Eq. (28) is exactly same as Eq. (1) with: K(ψ R , ξ I ) CT S(ψ R , ξ I )C
(29)
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stress vector as:
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Additionally, the elemental force of the structure can be obtained from the generalized
q RI S(ψ R , ξ I )e RI S(ψ R , ξ I )Cu RI
(30)
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Eqs. (25) - (27) are the general governing equations for linear elastic analyses of structures involving hybrid random and interval uncertainties, which should be satisfied regardless of element types. 3.2 Upper and lower bounds of mean of nodal displacement By implementing the alternative FE formulation as illustrated in Eqs. (28) and (29), the mean of nodal displacement as expressed in Eq.(14) can be reformulated as:
CT S(μ ψ R , ξ I )Cμ u RI F0
(31) 13
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Thus, the formulation of mean of nodal displacement can be alternative expressed by the format introduced in Eqs. (25) - (27). Mathematically, all the interval uncertain parameters can be treated as inequality constraints, and the equilibrium condition, the compatibility condition and the elastic constitutive condition are considered as equality constraints. Subsequently, the upper bound of the mean of the displacement response at vth ( v 1,...,d ) degree-of-freedom can be obtained by solving the following NLP:
μu RI maximise{ μu RI } v
v
subject to:
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C T q F0
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(P1): (32a)
(32b) (32c)
S(μ ψ R , ξ I )e q
(32d)
ξ I Φ : {ξ p | ξ k ξ kI ξ k , for k 1,..., p}
(32e)
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Cμ u RI e
Similarly, the lower bound of the mean of the displacement response at vth ( v 1,...,d )
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degree-of-freedom is calculated by solving the following problem:
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(P2):
u minimise{ μu } RI v
RI v
(33a)
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subject to: Eqs . (32 b) - (32 e)
(33b)
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Eqs. (32) and (33) indicate that by introducing the alternative finite element model into
the matrix perturbation, the upper and lower bounds of the mean of the nodal displacement can be determined without the implementation of interval arithmetic. Consequently, the extremities of the mean of the structural displacement at any degree of freedom can be rigorously established by completely eliminating the effects of interval dependency.
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3.3 Upper and lower bounds of standard deviation of nodal displacement Similarly, by adopting the alternative FE model introduced in Section 3.1, the bounds of standard deviation of the nodal displacement at the vth degree-of-freedom can be also calculated by solving two standard NPLs. To further simplify Eq. (13), additional auxiliary variables A h d , B h d , m d and n h d for h 1,2,..., m , are introduced, thus the
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variance of displacement can be expressed as: m
σ u RI σ u RI [ A h B h ] [ A j B j ] cov( h , j )
(34)
A h K 01Fh
(35)
h 1 j 1
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where
Bh K 01K h K 01F0 K 01K hm K 01n h
(36)
m K 01F0
(37)
n h K hm
(38)
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such that
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The introduction of these auxiliary variables enables the adoption of the alternative FE model to reformulate the standard deviation of nodal displacement. Thus, by implementing the concept expressed in Eqs. (28) and (29), Eqs. (35) – (38) can be explicitly reformulated.
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Subsequently, the determinations of the extreme bounds of standard deviation of nodal displacement could be achieved through solving two explicit NLPs. By treating interval
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uncertainties as bounded variables, the upper bound of the standard deviation of displacement response at the vth degree-of-freedom is calculated by solving the NLP defined by Eq. (39).
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Due to the non-negative property of standard deviation, ( u RI )2 σu2RI and ( u RI ) 2 σu2RI such v
v
v
v
that the validity of the proposed UPMP approach for obtaining the bounds of standard deviation through the calculation on the bounds of variance is ensured. Consequently, the upper bound of the variance of the vth ( v 1,...,d ) structural displacement can be explicitly formulated as the NLP problem (P3): (P3):
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u
2 RI v
maximise{ σ u2RI }
(39a)
v
subject to: m
m
σ u RI σ u RI [ A h B h ] [ A j B j ] cov( h , j )
(39b)
CT q A h Fh , h 1,..., m
(39c)
CAh e Ah , h 1,..., m
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h 1 j 1
(39d)
S(μ ψ R , ξ I )e A h q A h , h 1,..., m
(39e)
CT q Bh n h , h 1,..., m
(39f)
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CBh eBh , h 1,..., m
(39g) (39h)
CT q m F0
(39i)
Cm em
(39j)
S(μ ψ R , ξ I )e m q m
(39k)
CT q nh n h , h 1,..., m
(39l)
R I S(ψ , ξ ) e q , h 1,..., m nh hR m ψ R μ R ψ
(39m)
ξ I Φ : {ξ n | ξ k ξ kI ξ k , for k 1,..., p}
(39n)
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S(μ ψ R , ξ I )e Bh q Bh , h 1,..., m
Furthermore, the lower bound of the variance of displacement response at the vth
( v 1,...,d ) degree-of-freedom is calculated by solving the following NLP: (P4): u
2 RI v
minimise{ u2RI }
(40a)
v
subject to:
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(40b)
Eqs . (39 b) - (39n)
To further demonstrate the formulation of NLP problem P3 and P4, the detailed formulation of Eq. (39) is demonstrated via a proof in the Appendix A. The proposed UPMP approach transformed the computation of bounded standard deviation of nodal displacement into two NLP problems under same constraints, such that the interval arithmetic procedure is eliminated throughout the hybrid random interval analysis.
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The integration of the alternative FE formulation and mathematical programming approach advantageously accommodates the dependency of interval uncertain variables. 3.4 Upper and lower bounds of mean of elemental force
Within the framework of the proposed UPMP approach, the expression of mean of
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elemental force (Eq. (18)) is reformulated by adopting the alternative FE formulation indicated by Eq. (30) as following:
μ f RI D 0 K 01 F0
S(μ ψ R , ξ I )Cμ u RI
(41)
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q
where q has been defined as in Eqs. (37b) – (37d) such that
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D 0 S(μ ψ R , ξ I )C
(42)
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Therefore, the upper bound of mean of the sth ( s 1,...,w ) force response can be obtained by
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solving the NLP problem (P5) expressed as Eq. (43): (P5):
f maximise{ qs } RI s
(43a)
subject to: Eqs. (32b) - (32e)
(43b)
and the calculation of the lower bound of mean of the sth ( s 1,...,w ) force response is summarised as the following NLP (P6): (P6):
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RI s
minimise{ qs }
(44a)
subject to: (44b)
Eqs. (32b) - (32e)
The Eqs. (43) and (44) indicate that the computation of upper and lower bounds of the mean of elemental force are based on solving two NLP problems with same constraints as in
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the calculation of bounded mean of nodal displacement which saves the computational effort in the hybrid random interval uncertainty analysis.
3.5 Upper and lower bounds of standard deviation of elemental force
The expression of the standard deviation of elemental force for the hybrid uncertainty
m
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analysis can be reformulated by utilising the proposed UPMP approach as Eq. (45): m
σ f RI σ f RI {D 0 [ A h B h ] D h m} {D 0 [ A j B j ] D j m} cov( hR , Rj ) h 1 j 1 m
m
{S(μ ψ R , ξ I )C[ A h B h ] D h m} h 1 j 1
(45)
{S(μ ψ R , ξ )C[ A j B j ] D j m} cov( , ) m
m
h
j
R h
M
I
R j
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{q A h q B h q n h } {q A j q B j q n j } cov( h , j ) where A h d , B h d , m d are the auxiliary variables and q A h w , q B h w ,
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q n h w are the pseudo-vectors introduced in the reformulation for standard deviation of nodal displacement. By adopting the same concept of numerical decomposition for the
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calculation of the bounds of the standard deviation of nodal displacement, the general computational formulation of UPMP for the calculation of the upper bound of the standard
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deviation of the sth ( s 1,...,w ) elemental force response can be explicitly formulated into a NLP, such that: (P7):
f
2 RI s
maximise{ 2f RI }
(46a)
s
subject to:
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m
m
h
j
σ f RI σ f RI {q A h q B h q n h } {q A j q B j q n j } cov( h , j )
(46b) (46c)
Eqs. (39c) - (39n)
and the calculation of the lower bound of the standard deviation of the sth ( s 1,...,w ) elemental force response can be explicitly expressed as:
f
2 RI s
minimise{ 2f RI } s
subject to: Eqs. (50b)
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Eqs. (39c) - (39n)
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(P8): (47a)
(47b) (47c)
Eqs. (46) and (47) comprise the general formulations of the proposed UPMP approach for calculating the upper and lower bounds of standard deviation of elemental force of linear structures. The proposed computational scheme rigorously transforms a complicated hybrid
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random interval analysis of elemental force into four NLP problems (P5 – P8) which could be solved by any available NLP solver. Additionally, the constraints for the calculation of
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standard deviation of nodal displacement and elemental force are identical such that only the objective functions of NLP problems are necessarily to be modified.
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3.6 The implementation of the UPMP approach The proposed UPMP approach transforms the uncertain structural responses analysis with
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hybrid random and interval parameters into mathematical programming problems. In this context, the means and standard deviations of random variables become deterministic inputs.
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Furthermore, by modelling the interval uncertain parameters as inequality constraints, the extreme bounds of the stochastic characteristic of structural responses are directly obtained by respectively solving the NLP problems P1 – P8. The solution algorithm of the proposed UPMP methods is summarised by a flowchart shown in Fig. 1.
19
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Fig. 1 Flowchart of the solution algorithm of UPMP approach As a combination of perturbation method and mathematical programming approach, the
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proposed UPMP method offers a computational framework which contains the advantages of both approaches. By employing the perturbation-based random moment method with respect to random variables, the explicit expressions of first two moments of structural responses are respectively formulated such that only the means and standard deviations of random variables are enclosed. Thus, the achievement of means and standard deviations of system outputs is no longer rely on generating a large number of samples for random variables. Furthermore, the adoption of the alternative FE modelling technique conveniently reformulated the expressions of the statistical characteristics of structural responses into the NLP format. Subsequently, the
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hybrid uncertain structural response analysis with random and interval variables can be efficiently processed by any available NLP solver. Consequently, the interval arithmetic is completely eliminated throughout the proposed computational strategy. 4. Numerical examples In this section, the applicability, accuracy and efficiency of the proposed UPMP approach are critically justified through investigations on both academic sized and practically motivated
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engineering structures. For the simple academic sized examples, the accuracy of the proposed UPMP approach can be rigorously justified by comparing with some analytical solutions, whereas the applicability of the proposed approach can be illustrated by analysing the practically motivated examples.
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Within the context of this study, all NLPs involved in the proposed UPMP approach are solved by a commercial NLP solver named CONOPT [48], which is operated within an advanced mathematical modelling environment, namely General Algebraic Modelling System (GAMS) [49]. For all the numerical examples investigated in this study, uncertain parameters from the same category of different elements are treated as independent to each other. The
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accuracy of the results is rigorously verified through a number of different approaches. For academic sized examples, the results calculated by the proposed UPMP approach are
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compared with the analytical solutions where possible. For the circumstance that analytical solution cannot be determined, the computationally exhaustive Monte-Carlo Simulation combined with Quasi-Monte-Carlo Simulation (MCS-QMCS) method is implemented to
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verify the uncertain-but-bounded statistical characteristics obtained by UPMP approach into certain extent. The realizations for both random and interval variables used in the MCS-
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QMCS approach are generated by the Statistics toolbox of MATLAB R2015b [50]. For the QMCS scheme, the Halton low discrepancy sequence has been implemented with a dynamic
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skip and leap scheme combined with a ‗‗RR2‖ scramble scheme [51]. The presented numerical results are obtained by using a workstation with CPU of Intel Core i7-4770, 32 GB of memory, and 1 TB of hard drive. 4.1 Stepped cantilever beam The first investigation considers a two-stepped cantilever Euler-Bernoulli beam subjected to an external load acting at the right end, as depicted in Fig. 2. The Young‘s modulus and cross-sectional geometries of the two elements (element 1 is between nodes 1 and 2, and
21
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element 2 is between nodes 2 and 3) are different. For this particularly example, the vertical displacements at nodes 2 ( u 2, y ) and 3 ( u3, y ) have been analysed by employing the UPMP approach considering three cases of analysis with distinctive mixture of random and interval variables. The information of uncertain parameters of the considered three cases is presented
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in Table 1.
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Fig. 2 Two-stepped cantilever beam
The upper and lower bounds of the stochastic characteristics of u 2, y and u3, y of Case 1 obtained by the proposed UPMP approach are presented in Table 2. In order to verify the accuracy of the proposed method, the analytical expressions of u 2, y and u3, y are respectively obtained as Eqs. (48) and (49) based on the Euler-Bernoulli beam theory. Subsequently, the
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analytical solutions for the upper and lower bounds of the means and standard deviations of
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u 2, y and u3, y for Case 1 can be calculated as demonstrated in Table 2. In such way, the accuracy of the UPMP approach has been evidently justified for this particular case of
4l 3 6l 2l u 2, y 1 3 1 2 3 F E1b1h1 E1b1h1
(48)
4l 3 12l12l2 12l1l22 4l23 u3, y 1 F 3 3 E b h E b h 1 1 1 2 2 2
(49)
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CE
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analysis.
For the investigations conducted in Cases 2 and 3, the analytical solutions of bounded
statistical characteristics of the nodal displacement with hybrid uncertainties cannot be explicitly determined, thus the MCS-QMCS method has been implemented to verify the results obtained from the proposed UPMP approach. Regarding the investigations for Cases 2 and 3, 1,000 realizations have been generated for all the interval variables within the scheme of QMCS, and 10,000 random samples have been explored at each realization of the interval
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variable. Therefore, the bounds of stochastic characteristics of u 2, y and u3, y obtained by both approaches for Cases 2 and 3 of the cantilever beam are summarised in Table 3. Table 1 Information of uncertain parameters of Example 1
E1 ( GPa )
Case 3
Interval
Normal E1 200 , E1 20
Same as Case 1
Interval
E
62.1 E2 75.9
2
Interval 0.054 b1 0.066 Interval
Same as Case 1 Same as Case 1
0.036 b2 0.044 Interval 0.0736 h1 0.0864
h2 ( m )
Interval 0.0552 h2 0.0648
F ( kN )
Lognormal F 50 , F 7.5
Lognormal E2 69 , E2 6.9 Same as Case 1
Normal b2 0.04 , b2 0.002
Same as Case 1
Same as Case 1
Same as Case 1
Normal h2 0.06 , h2 0.003
Interval 42.5 F 57.5
Normal F 50 , F 7.5
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h1 ( m )
Normal 69 , E2 6.9
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b2 ( m )
Case 2
180 E1 220
E2 ( GPa ) b1 ( m )
Case 1
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Parameters
2, y
Analytical solution ( m )
7.019 103
7.019 103
u
2, y
1.053 103
1.053 103
u
2, y
2.904 103
2.904 103
u
2, y
4.357 104
4.357 104
u
3, y
3.067 102
3.067 102
u
3, y
4.600 103
4.600 103
u
3, y
1.269 102
1.269 102
u
3, y
1.904 103
1.904 103
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UPMP ( m )
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u
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Table 2 The statistical characteristics of Case 1 of cantilever beam
By thoroughly examining Table 3, all computational results provided by the proposed UPMP approach are fully enclosing the results reported from the dually simulative MCSQMCS approach. The reason for such observed incompetence of the simulative approach is 23
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that even though extensive amount of sampling points with considerable amount of uniformity have been explored for the interval counterpart, it still remains quite challenging for the global search based method to sample the points at the boundaries of the interval parameters (i.e., the upper and lower bounds of the interval parameters). However, from Eqs. (48) and (49), monotonic relationship can be observed for the considered interval parameters in Cases 2 and 3 with the concerned structural responses. Consequently, once explicit formulations for the mean and standard deviation for the structural responses are established,
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the UPMP approach can utilize the computational benefit offered by the mathematical programming approach to search for the extreme bounds of the statistical characteristics in a more appropriate manner.
Table 3 The statistical characteristics of Case 2 and 3 of cantilever beam Case 3
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Case 2 MCS-QMCS
UPMP
MCS-QMCS
u ( m )
7.265 103
6.711 103
7.019 103
6.442 103
u ( m )
7.265 104
6.833 104
1.053 103
9.621 104
u ( m )
2.716 103
2.982 103
2.904 103
3.142 103
u ( m )
2.716 104
3.112 104
4.357 104
4.688 104
u ( m )
3.174 102
2.840 102
2.537 102
2.432 102
u ( m )
2.249 103
2.054 103
4.165 103
4.076 103
u ( m )
1.187 102
1.372 102
1.581 102
1.660 102
u ( m ) 8.408 104
9.966 104
2.914 103
3.060 103
230.76
1.47
253.64
2, y
2, y
2, y
3, y
3, y
3, y
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3, y
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2, y
1.52
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t com (s)
M
UPMP
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In addition, the computational superiority of the proposed method is also highlighted in Table 3. That is, by consuming on average 0.62% of the computational effort required by the dual simulative method, the proposed UPMP approach is competent to deliver more robust bounds (i.e., larger upper bound and smaller lower bound) on the statistical characteristics of the concerned structural responses for Cases 2 and 3. Overall, by comparing with both analytical solutions in Case 1 and simulative results provided by the MCS-QMCS approach in Cases 2 and 3, the proposed UPMP approach offers several superiorities over the simulative approach for all the investigated scenarios.
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4.2 Three-span truss bridge In the second example, the hybrid uncertain static analysis of a three-span truss bridge involving both random and interval uncertainties is conducted. The general layout of the structure with the loading regime at reference configuration is shown in Fig. 3. The truss bridge consists of 113 elements with 108 degrees-of-freedom. Moreover, the elements of the truss bridge are categorised into four groups with different cross-sectional areas such that A1
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denotes the cross-sectional areas of all the vertical elements, A2 denotes the cross-sectional areas of all the diagonal elements, A3 denotes the cross-sectional areas of the elements between nodes 1 and 29, and A4 denotes the cross-sectional areas of the elements between
30
F2
F1 2
F1 3
F1 4
F1 5
F1 6
F1 7
F1 8
31
32
33
34
35
36
37 38
9
F2 10
F2 11
39
40
F2 12
F2 13
41
y
F2 14
42
F2
F2
F2
F2
F2
F2
F2
F3 F3 F3 F3 F3 F3 F3 F3 22 23 24 25 26 27 28 29
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F1 1
43
15
44
16
17
45
46
18
47
19
48
20
21
49 50
51
7.316m
nodes 30 and 58.
53
52
54
55
56
57
58
28x6.1m=170.8m
M
x
Fig. 3 Three-span truss bridge
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The investigation of structural response of the truss structure involves two distinctive cases with various uncertain situations. The details of the two cases are outlined as follows:
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Case 1. The Young‘s modulus ( E R , unit: GPa) is lognormal random variable and the four different cross-sectional areas ( A1R , A2R , A3R and A4R , unit: m 2 ) are normally distributed
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random variables. The statistical information of random variables are: E 200 GPa,
E 20 GPa; A 1.232 102 m 2 , A 6.160 104 m 2 ; A 1.361 102 m 2 , 1
1
2
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A 6.805 104 m 2 ; A 1.171 102 m 2 , A 8.550 104 m 2 ; A 1.671 102 m 2 , 2
3
3
4
A 8.355 104 m 2 . The externally applied loads ( F1I , F2I and F3I , units: kN) at reference 4
configuration are interval parameters which have: F1I ΦF1 : {F1 8 | 288 F1, 352, for 1,...,8}
(50)
F2I ΦF2 : {F2 13 | 360 F2 , 440, for 1,...,13}
(51)
F3I ΦF3 : {F3 8 | 288 F3, 352, for 1,...,8}
(52) 25
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Case 2. The effect of varying degree of uncertainty of system parameters against the structural responses is analysed. The change range (CR) of interval variables is varying between 0.04 CR 0.3 and the coefficient of variation (COV) of random variables is varying between 0.04 COV 0.3 . In this case, Young‘s modulus ( E I , unit: GPa) and crosssectional areas ( A1I , A 2I , A 3I and A 4I , unit: m 2 ) are interval variables which have the nominal C values as: E 200 GPa, A1C 1.232 102 m 2 , A2C 1.361 102 m 2 , A3C 1.171 102 m 2 ,
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and A4C 1.671 102 m 2 . Thus, the interval parameters can be expressed as:
(53)
A1I Φ A1 : {A1 29 | A1C (1 CR ) A1, A1C (1 CR ), for 1,...,29}
(54)
A2I ΦA 2 : {A2 28 | A2C (1 CR ) A2, A2C (1 CR ), for 1,...,28}
(55)
A3I ΦA3 : {A3 28 | A3C (1 CR ) A3, A3C (1 CR ), for 1,...,28}
(56)
A4I ΦA 4 : {A4 28 | A4C (1 CR ) A4, A4C (1 CR ), for 1,...,28}
(57)
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E I Φ E : {E 113 | E C (1 CR ) E E C (1 CR ), for 1,...,113}
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The random variables considered in this case are the normally distributed externally applied loads at reference configuration. The means of the externally applied loads are: F1 320 kN, 2
3
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F 400 kN, and F 320 kN.
For Case 1, both UPMP approach and dual simulative MCS-QMCS method are
PT
implemented to investigate the structural responses of the truss bridge with hybrid uncertainties specified as above. For the purpose of demonstration, the bounds of statistical
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characteristics of the horizontal displacement at node 10 ( u10, x ), the vertical displacement at
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node 44 ( u 44, y ) and the internal force of element 69 ( f 69 ) obtained by both approaches are reported in Table 7. For the MCS-QMCS method, various sample sizes are adopted in calculating the upper and lower bounds of the statistical characteristics. It is evidently illustrated that all the results of MCS-QMCS approach are enclosed within the results obtained by the proposed UPMP approach. From Table 4, it can be observed that increasing the initial sample size in MCS-QMCS approach could slightly improve the computational results. However, the computational effort increases dramatically with the increase of sample size. Therefore, the proposed UPMP approach surpassed the performance of MCS-QMCS method in both accuracy and computational efficiency. 26
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Table 4 The bounds of statistical characteristics of Case 1 of truss bridge
UPMP
MCS-QMCS
MCS-QMCS
MCS-QMCS
MCS-QMCS
100 interval
500 interval
500 interval
1000 interval
5000 random
5000 random
10,000 random
10,000 random
(m)
1.128 102
1.048 10 2
1.048 10 2
1.046 10 2
1.069 10 2
u
(m)
1.209 103
1.134 10 3
1.134 10 3
1.129 10 3
1.153 10 3
u
(m)
7.897 103
9.328 10 3
9.14110 3
9.12110 3
9.12110 3
u
(m)
8.419 104
9.867 10 4
9.634 10 4
9.66110 4
9.66110 4
u
44, y
(m)
8.305 102
7.913 10 2
7.93110 2
7.934 10 2
8.042 10 2
u
44, y
(m)
8.655 103
8.418 10 3
8.418 10 3
8.292 10 3
8.375 10 3
u
44, y
(m)
6.567 102
7.150 10 2
7.133 10 2
7.124 10 2
7.124 10 2
u
44, y
(m)
6.849 103
7.474 10 3
7.425 10 3
7.415 10 3
7.364 10 3
f (kN)
1.731103
1.613 103
1.613 103
1.614 103
1.637 103
f (kN)
22.580
21.344
21.439
21.282
21.590
f (kN)
1.277 103
1.464 103
1.442 103
1.442 103
1.442 103
f (kN)
17.844
19.386
19.211
19.177
19.177
t com (s)
3.109
679.906
2.856 103
5.487 103
1.346 10 4
1 0, x
1 0, x
69
69
69
AC
CE
PT
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69
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1 0, x
M
1 0, x
CR IP T
u
(a)
(b)
Fig. 4 Variations of upper and lower bounds of (a) mean and (b) standard deviation of u10, x of Case 2 of truss bridge against alternations of degree of uncertainty
27
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(a)
(b)
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M
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Fig. 5 Variations of upper and lower bounds of (a) mean and (b) standard deviation of u44, y of Case 2 of truss bridge against alternations of degree of uncertainty
(a)
(b)
AC
Fig. 6 Variations of upper and lower bounds of (a) mean and (b) standard deviation of f 69 of Case 2 of truss bridge against alternations of degree of uncertainty
28
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Fig. 7 Symmetric two-bar truss with interval uncertain Young‘s modulus and cross-sectional area
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For Case 2, the response surfaces of the upper and lower bounds of the stochastic characteristics of u10, x , u 44, y and f 69 against the variations of the random forces, interval Young‘s modulus and cross-sectional areas are constructed by the proposed UPMP approach and shown as the 3D plots through Figs. 4 to 6. Additionally, the MCS-QMCS method with
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100 interval samples and 5000 random samples against selected combinations of change ratio (CR, which equals to the interval width divided by the midpoint) of an interval variable and
ED
coefficient of variation (COV, which equals to the standard deviation divided by the mean value) of a random variable are also implemented. Thus, for each combination of CR and COV, 100 samples of mean and standard deviation are obtained and shown in the 3D plots.
PT
Once again, the results provided by the UPMP approach enclose all the results obtained from the dual sampling method. Comparing with Case 1, the relative difference between the results
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obtained by both approaches are relatively large. The reason for such noticeable difference is because of the implementation of insufficient amount of sample for situations involving
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relatively large number of interval variables. The quality of results obtained from the QMCSMCS approach can be improved if more sampling points are explored. However, the increasing of the sample size would also burden the computational cost of such analysis. By further examining Figs. 4 - 6, several additional interesting yet important points can be noticed, which includes: 1. When the COV of random variables varies and CR of interval variables remains constant, both the upper and lower bounds of mean of the selected structural
29
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responses remain unchanged. On the other hand, the bounds of the standard deviations of the selected structural responses vary monotonically with the change of COV. 2. Under the circumstance that the COV is constant, a monotonic increase of the upper bounds and a monotonic decrease of the lower bounds of means and standard deviations of the selected structural responses can be observed when the CRs of Young‘s modulus and cross-sectional areas increases. Additionally, nonlinear relationship between the variations of the bounds of statistical characteristics and the
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changes of CRs are observed from Figs. 4 and 5. Therefore, it can be concluded from Case 2 that the influence on the bounds of the stochastic characteristics of structural responses due to the variations of the interval Young‘s modulus and cross-sectional areas cannot be predicted by simple linear scaling. Moreover, it can be observed from
u
44, y
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Fig. 5 that the increase of u 44, y is relatively faster in comparison with the decrease of .
3. By examining Fig. 4a, the u10, x obtained by the proposed UPMP approach decreases from positive to negative when the CRs of Young‘s modulus and cross-sectional area
M
increase, while the simulated samples from MCS-QMCS remain positive. The physical plausibility of such phenomenon can be demonstrated based on a symmetric
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two-bar truss structure, as shown in Figure 7, (element 1 is between nodes 1 and 2, and element 2 is between nodes 1 and 3) with interval Young‘s modulus ( E1 and E 2 )
PT
and cross-sectional area ( A1 and A2 ) of both elements subjected to vertical load EI ΦE : {E 2 | E E E, for 1,2}
(58)
A I ΦA : {A 2 | A A A, for 1,2}
(59)
AC
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applied at node 1. The bounds of Young‘s modulus and cross-sectional area are:
By simply applying the interval arithmetic with structural analysis, a positive u x can be easily obtained when E1 E , A1 A , E2 E and A2 A while the u x is negative when E2 E , A2 A , E1 E and A1 A . Thus, if the externally applied load is a random variable, the mean of u x will varies from negative to positive. Therefore, the proposed UPMP approach has the superior advantage for situations involving the positive-to-negative variations of bounds of mean of the structural responses.
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By implementing the UPMP approach, the random interval behaviors of the structural responses of the three-span truss bridge have been investigated against two different cases with various uncertain conditions. The accuracy and computational efficiency of the proposed method has been clearly illustrated in the study of Example 2. Additionally, the effects of varying interval Young‘s modulus and cross-sectional area and random externally applied loads on the structural responses have been investigated. From this investigation, it is necessary for engineering application to explicitly analyze the influence of the fluctuation of
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random and interval uncertain system parameters acting upon the structural responses of structures. 4.3 Multi-bay multi-storey frame
To further investigate the performance of the proposed UPMP approach, a practical-sized
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multi-bay multi-storey frame subjected to diverse vertical and horizontal loadings at various locations is considered in the last numerical investigation. In this particular example, the applicability, accuracy and computational efficiency of the UPMP approach are thoroughly demonstrated through three distinctive investigations which are including a probabilistic
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analysis, an interval analysis, and a hybrid random and interval analysis of structural responses. The general layout and loading regime of the frame is illustrated in Fig. 8. The plane frame model consists of 753 degrees-of-freedom and 376 elements. The considered
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uncertain parameters are the Young‘s modulus ( E , unit: GPa), cross-sectional areas of beams
PT
( AB , unit: m 2 ) and columns ( AC , unit: m 2 ), and externally applied loads ( F1 , F2 and F3 , units: kN) at reference configurations. For this frame structure, 310UC118 has been used for all columns and 310UB32 has been employed for all beams [52]. In order to maintain the
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compatibility between cross-sectional area and the corresponding second moment of area for both column and beam, the following compatibility functions are introduced such that the
AC
second moment of area of each element is expressed as a function of cross-sectional area. For each 310UC118 column the compatibility function is expressed as Eq. (60), whereas the compatibility function of each 310UB32 beam is expressed as Eq. (61).
I C ( AC ) 0.5238 AC2 0.0114 AC 3 10 5
(60)
I B ( AB ) 0.8876 AB2 0.0288 AB 4 105
(61)
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F2
F2
F2
F2
F2 F2 F2F2
11 F1
10 F1
9F1
8F1
7F1
6F1
5F1
4F1 3F1
2F1
1F1
180
28
193
42
206
56 70
13
179
27
192
41
205
55 69
12
178
26
191
40
204
54 68
11
177
25
190
39
203
53 67
10
176
24
189
38
202
52 66
9
175
23
188
37
201
51 65
8
174
22
187
36
200
50 64
7
173
21
186
35
199
49 63
6
172
20
185
34
198
48 62
5
171
19
184
33
197
47 61
4
170
18
183
32
196
46 60
3
169
17
182
31
195
45 59
2
168
16
181
30
194
44 58
1
15
29
F2
43 57
219
111
232
125
82 96
218
110
231
124
81 95
217
109
230
80 94
216
108
79 93
215
78 92
214
F3 F3 262 139
F3
F3
F3
F3
245
153
258
167
138
244
152
257
166
123
137
243
151
256
165
229
122
136
242
150
255
164
107
228
121
135
241
149
254
163
106
227
120
134
240
148
253
162
133
239
147
252
161
132
238
146
251
160
261
77 91
213
105
226
119
76 90
212
104
225
118
75 89
211
103
224
117
131
237
145
250
159
74 88
210
102
223
116
130
236
144
249
158
73 87
209
101
222
115
129
235
143
248
157
72 86
208
100
221
114
128
234
142
247
156
71 85
207
99
220
113
127
233
141
246
155
112
126
259
98
140
154
1m
10@3 m
M
2.25 m
1m
F3
260
y 6@3 m
F3
83 97
84
x
F2
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12 F1
14
13@3 m
F2
CR IP T
F2 13 F1
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Fig. 8 The multi-bay multi-storey frame For this particular example, three cases with various uncertain scenarios are investigated.
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The details of the three distinct investigations are outlined as follows: Case 1: All system parameters are considered as random variables whose statistical
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information is collectively summarized in Table 5 [53, 54].
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Table 5 Statistical information of all random variables considered in Case 1 Parameter Distribution
Mean
COV
E (GPa)
lognormal
200
0.06
AB ( m 2 )
normal
40.8 104
0.05
AC ( m 2 )
normal
150 104
0.05
F1 (kN)
normal
5.5
0.1
F2 (kN)
normal
60
0.1
F3 (kN)
normal
80
0.1
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Case 2: The Young‘s modulus, cross-sectional areas of each beam and column, as well as all the applied loads at reference configuration are considered as interval uncertain parameters. The detailed information regarding all considered interval parameters is summarized through Eqs. (62) - (67). (62)
A IB ΦAB : {A B 151 | 36.72 104 AB, 44.88 104 , for 1,...,151}
(63)
ACI ΦAC : {AC 130 | 135 104 AC , 165 104 , for 1,...,130}
(64)
F1I ΦF1 : {F1 8 | 4.95 F1, 6.05, for 1,...,13}
(65)
F2I ΦF2 : {F2 156 | 54 F2, 66, for 1,...,156}
(66)
F3I ΦF3 : {F3 95 | 72 F3, 88, for 1,...,95}
(67)
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EI ΦE : {E 281 | 180 E 220, for 1,...,281}
Case 3: Two scenarios are thoroughly investigated to demonstrate the performance of the
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proposed UPMP approach in hybrid random and interval analysis. For both scenarios, the interval cross-sectional areas of beams and columns are expressed as Eqs. (68) and (69), and
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the details of random uncertain parameters are listed in Table 6. (68)
ACI ΦA C : {AC 130 | 141 104 AC , 159 104 , for 1,...,130}
(69)
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A IB ΦA B : {A B 151 | 38.352 104 AB, 43.248 104 , for 1,...,151}
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Table 6 Random uncertain parameters of frame structure Distribution
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Parameter
Mean COV Scenario 1 Scenario 2
E (GPa)
lognormal
lognormal
200
0.045
F1 (kN)
normal
gumbel
5.5
0.1
F2 (kN)
lognormal
lognormal
60
0.1
F3 (kN)
normal
normal
80
0.1
Table 7 The statistical characteristics of frame structure of Case 1
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MCS
MCS
10,000
100,000
UPMP
R1 (%) R2 (%)
(m)
9.194 102
9.263 102
9.260 102
-0.748 -0.714
u
(m)
1.190 102
1.194 102
1.205 102
-0.339 -1.208
u (m)
1.689 102
1.700 102
1.699 102
-0.628 -0.616
u (m)
1.931 103
1.953 103
1.946 103
-1.111 -0.793
M (kNm)
92.652
92.757
92.790
-0.113 -0.148
M (kNm)
9.655
9.613
9.685
0.440
-0.311
t com (s)
71.765
9.470 102
1.767 104
N/A
N/A
167, x
167, x
70, y
70, y
1
1
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u
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For Case 1 of Example 3, both UPMP and Monte-Carlo Simulation (MCS) method are implemented to investigate the random behaviour of the structural responses of this complex frame. For the MCS approach, two simulations with 10,000 and 100,000 samples have been adopted to verify the results obtained by the proposed method. The means and standard deviations of the horizontal displacement at node 167 ( u167, x ), vertical displacement at node
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70 ( u70, y ), and the bending moment at node 1 ( M 1 ) calculated by both approaches are
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outlined in Table 7, where R1 and R2 denote the relative differences between the results obtained by UPMP approach and MCS method with 10,000 and 100,000 samples,
R
UPMP MCS % MCS
(74)
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respectively. The relative difference in this case is defined as:
It is indicated in Table 7 that the results calculated by the UPMP approach agree well with the statistical information obtained from the MCS approach with various sample sizes,
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but of course in a much more computational friendly manner. Therefore, the applicability, accuracy and computational efficiency of the proposed UPMP approach for pure stochastic analysis have been rigorously justified through this investigation.
34
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(a)
(b)
Fig. 9 Bounds of (a) u70, x and (b) M 1 of frame structure of Case 2.
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For Case 2, interval analyses of the displacement and force responses of the frame structure have been conducted by both UPMP and Quasi-Monte-Carlo Simulation (QMCS) approaches with various sample sizes. The reason for adopting QMCS in this investigation is due to the advantage of QMCS approach in generating more uniform samples than the MCS method, which could potentially enhance the effectiveness of the QMCS method. For this
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particular case, 100, 1000, 10,000 and 100,000 simulations have been implemented through the QMCS scheme. For the demonstration purpose, the upper and lower bounds of u70, y and
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M 1 obtained by both approaches are shown in Fig. 9, where the colour scale in the plots indicates the computational time consumed by the QMCS approach. It can be easily observed
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that the UPMP approach surpasses the capability of the QMCS method by providing a larger upper bound and smaller lower bound within 20 seconds. Indeed, limited improvements of
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results obtained from the QMCS approach can be observed by exponentially increasing the sample size, but the possibility of engineering application for such global search based
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computational approach is quite challengeable due to the inevitably substantial consumption of computational effort.
Table 8 The bounds of statistical characteristics of the frame structure in Case 3. MCS-QMCS UPMP
u
167, x
(m )
1.016 101
Scenario 1
Scenario 2
9.351102
9.351102 35
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(m )
1.127 102
1.041 102
1.048 102
u
(m )
8.350 102
9.105 102
9.105 102
(m )
9.640 103
1.019 102
1.027 102
(m )
1.815 10 2
1.734 102
1.734 102
(m )
1.764 10 3
1.691103
1.702 103
(m )
1.578 102
1.654 102
1.654 102
(m )
1.550 103
1.616 103
1.627 103
M (kNm)
110.832
101.412
M (kNm)
10.568
9.581
M (kNm)
76.804
85.679
M (kNm)
7.140
7.919
t com (s)
2.031103
167, x
u
167, x
u
70, y
u
70, y
u
70, y
u
70, y
1
1
1
1
101.416 9.657
85.682 7.983
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167, x
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u
6.399 104
6.366 104
For Case 3, the uncertain-but-bounded statistical characteristic of selected displacement and force responses obtained by both approaches are listed in Table 8. For the MCS-QMCS
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method, 100 simulations have been implemented for generating realizations for interval variables through the QMCS scheme, and subsequently, 10,000 simulations have been
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conducted at each interval realization for the stochastic analysis. By inspecting the results reported in Table 8, the upper and lower bounds of statistical information regarding concerned
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structural responses obtained by MCS-QMCS method are enclosed by the results determined from the UPMP approach for both Scenarios 1 and 2. Since that the proposed approach
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simply requires the mean and standard deviations of the random variables, it leads to the same UPMP formulations and subsequently the results for the structural responses for both
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Scenarios 1 and 2. Comparing with the adopted sampling method, the proposed UPMP approach is capable of offering sharper bounds on the means and standard deviations of all the considered structural responses with dramatically less computational cost. Therefore, the accuracy and computational efficiency of the proposed approach in hybrid uncertainty analysis have been evidently demonstrated. In addition to the illustration on the applicability and effectiveness of the proposed UPMP approach, both parametric and non-parametric statistical inference analyse methods which are widely adopted [55 - 59] in structural engineering for identifying the probability distributions of random variables are selected to construct the probability density functions 36
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(PDFs) and cumulative distribution functions (CDFs) of the bounds of the concerned structural responses demonstrated in Table 8. The selected parametric methods include loglikelihood test, Akaike information criterion (AIC) test and Bayesian information criterion (BIC) test. The non-parametric analyses adopted involve Kolmogorov–Smirnov (KS) test and Anderson–Darling (AD) test. These methods are implemented on the simulated results of
u167, x , u10, y and M 1 at each realisation of interval parameters from MCS-QMCS for both
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Scenarios 1 and 2. According to the test statistics, the types of probability distribution which are closest fit for describing the selected structural responses are summarised in Table 9. For demonstration purpose, the test statistics for the samples corresponding to the 50th realisation of interval parameters for Scenarios 1 and 2 are respectively summarised as in Table B1 and B2 in Appendix B.
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Table 9 Probability distribution types identified by statistical inference analyses in Case 3. Probability Distribution Scenario 1
Scenario 2
Normal
Gumbel
u167, x
Normal
Gumbel
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M1
Lognormal Lognormal
M
u70, y
Subsequently, the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the demonstrated displacement and force responses of Scenarios 1 and 2
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are shown in Figs. 10 and 11, respectively. It can be observed from Figs. 10 and 11 that the PDFs of the MCS-QMCS results are bounded by the upper bound of PDF (PDF-UB) and
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lower bound of PDF (PDF-LB) constructed based on the results of UPMP approach. Moreover, the upper bound of CDF (CDF-UB) and lower bound of CDF (CDF-LB) of the
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selected nodal displacement and elemental force are rigorously enclosing all CDFs constructed basing on the results of the MCS-QMCS approach.
37
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CE
PT
ED
M
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Fig. 10 PDFs and CDFs of selected structural responses of Scenario 1 of Case 3 of the frame structure.
38
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CE
PT
ED
M
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Fig. 11 PDFs and CDFs of selected structural responses of Scenario 2 of Case 3 of the frame structure.
39
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5. Conclusions This paper presents a computational scheme named as unified perturbation mathematical programming (UPMP) approach for structural analysis involving random and interval parameters. The proposed approach offers a potent non-sampling strategy for investigating the static response for various engineering structures. The integration of matrix perturbation with mathematical programming within the proposed computational approach transforms the
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intricate hybrid random interval analysis into a series of NLP problems. By treating interval parameters as inequality constraints in the mathematical programming, the interval dependency is eliminated from the algorithm.
The applicability of the proposed UPMP approach is critically validated by investigating various types of engineering structures through numbers of uncertain circumstances. By
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comparing with analytical solution and results obtained from sampling method, the accuracy and computational efficiency of the proposed uncertainty analysis scheme are evidently illuminated. Additionally, hypothesis test is implemented in this study for identifying the probability distribution of the structural responses.
M
From the computational benefits associated with the proposed UPMP approach, it is possible to further extend the proposed computational scheme for other types of engineering analyses, such as hybrid uncertain linear and nonlinear buckling analysis, dynamic analysis of
ED
structures involving random and interval uncertainties, etc. Furthermore, the concept of the proposed approach can be also extended to other areas of study where the effects of
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uncertainties must be evaluated.
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6. Acknowledgement
The work presented in this paper has been supported by Australian Research Council
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through projects DP140101887 and DP160103919. The authors would like to sincerely express our gratitude to the anonymous reviewers for
their valuable comments and constructive suggestions.
40
ACCEPTED MANUSCRIPT Appendix A. Details of reformulation of Eqs. (35) – (38) The details of the reformulation of Eqs. (35) – (38) are illustrated by the following proof, which provides a reference for the construction of the two NLP problems indicated by Eqs. (39) and (40). Proof.
reformulation of Eq. (35) is demonstrated as:
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By implementing the alternative FE formulation presented in Section 3.1, the
(A1)
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CT q F Ah h 1 T I A h K 0 Fh C S(μ ψ R , ξ )CA h Fh CA h e A h I S(μ ψ R , ξ )e A h q A h
where q A h w and e A h w denote the two pseudo-vectors corresponding to A h . The two pseudo-vectors are introduced here for numerical decomposition purpose. By adopting the same concept, Eq. (36) can be decomposed as:
(A2)
ED
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CT q n Bh h 1 T I B h K 0 n h C S(μ ψ R , ξ )CB h n h CB h e B h S(μ ψ R , ξ I )e B h q B h
where q B h w and e B h w denotes the two pseudo-vectors corresponding to B h .
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Similarly, Eq. (37) is reformulated as Eq. (A3) by adding two pseudo-vectors q m w
AC
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and e m w . That is,
CT q F m 0 1 T I m K 0 F0 C S(μ ψ R , ξ )Cm F0 Cm e m I S(μ ψ R , ξ )e m q m
(A3)
Then, Eq. (38) could be decomposed into Eq. (A4) by introducing a pseudo-vector
q n h w , such that: CT q n h n h n h K h m CT S h Cm n h Cm e m S e q nh h m
(A4a)
41
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where
S(ψ R , ξ I ) Sh hR ψ R μ
(A4b) ψR
The combination of Eqs. (A1) – (A4) constructs the equality constraints listed as Eqs. (39c) – □
(39m). This completes the proof.
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Appendix B. Results of parametric and non-parametric analysis for u167, x of Case 3 in Example 3
The test statistics of the implemented parametric and non-parametric statistical inference methods for u167, x Case 3 in Example 3 are summarised in Tables B1 and B2. The maximum
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test statistics obtained by the log likelihood approach indicates the best fitted probability distribution for the given samples, while the distribution with minimum test statistics obtained by other methods is inferred as the most suitable.
Table B1 Parametric and non-parametric test statistics for u167, x for Scenario 1 of Case 3.
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Scenario 1- u167, x Parametric Method
AC
CE
PT
ED
Distribution Log Likelihood 10 4 Normal 3.0879 Gamma 3.0862 Lognormal 3.0821 Gumbel 3.0195 Logistic 3.0769 Log logistic 3.0742 Weibull 3.0461 Rayleigh 1.9963
AIC 10 4 -6.1755 -6.1720 -6.1638 -6.0386 -6.1533 -6.1480 -6.0918 -3.9925
Non-parametric Method BIC 10 4 -6.1740 -6.1706 -6.1624 -6.0372 -6.1519 -6.1465 -6.0903 -3.9918
AD Test
KS Test
0.9730 3.1464 8.3085 97.6199 10.4052 13.6244 59.0456
0.0114 0.0135 0.0191 0.0581 0.0225 0.0229 0.0428 0.4340
2.8050 103
Table B2 Parametric and non-parametric test statistics for u167, x for Scenario 2 of Case 3. Case 2- u167, x Parametric Method Distribution Log Likelihood 10 Normal 3.1477 Gamma 3.1754 Lognormal 3.1859
4
AIC 10 -6.2949 -6.3503 -6.3714
4
Non-parametric Method BIC 10 -6.2935 -6.3489 -6.3700
4
AD Test
KS Test
83.7019 46.1837 31.6121
0.0623 0.0475 0.0399 42
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Gumbel Logistic Log logistic Weibull Rayleigh
3.2039 3.1645 3.1866 3.0184 2.0533
-6.4074 -6.3286 -6.3727 -6.0363 -4.1064
-6.4060 -6.3272 -6.3713 -6.0349 -4.1057
4.8649 48.6919 22.4190 296.2260
2.8695 103
0.0157 0.0381 0.0243 0.1078 0.4648
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