Inverse problems in structural safety analysis with combined probabilistic and non-probabilistic uncertainty models

Engineering Structures 150 (2017) 166–175

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Inverse problems in structural safety analysis with combined probabilistic and non-probabilistic uncertainty models K. Karuna, C.S. Manohar ⇑ Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India

a r t i c l e

i n f o

Article history: Received 29 November 2016 Revised 1 June 2017 Accepted 14 July 2017

Keywords: Structural safety Inverse reliability Convex functions Reliability based structural optimization

a b s t r a c t In problems of structural safety analysis, depending upon the nature and extent of availability of empirical data, uncertainties could be quantified by using probabilistic and/or non-probabilistic models. The options for non-probabilistic representations include intervals, convex functions, and (or) fuzzy variable models. We consider a few inverse problems of structural safety analysis aimed at the determination of system parameters to ensure a target level of safety and/or to minimize a cost function for problems involving combined probabilistic and non-probabilistic uncertainty modeling. The treatment of this problem calls for combining methods of uncertainty analysis with finite element structural modeling and numerical optimization tools. Development of load and resistance factor design format, in problems with combined uncertainty models, is also presented. We employ super-ellipsoid based convex function models for representing non-probabilistic uncertainties. The target safety levels are taken to be specified in terms of indices defined in standard space of uncertain variables involving standard normal random variables and/or unit hyper-spheres. A class of problems amenable for exact solutions is identified and a general procedure for dealing with more general problems involving nonlinear performance functions is developed. Illustrations include studies on inelastic frame with uncertain properties. Accompanying supplementary material contains additional illustrations. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The forward problem of structural safety assessment consists of quantifying a measure of safety given a performance function and a mathematical model for the uncertainties relevant to the problem. Extensive studies on this class of problems have been reported in the existing literature for the case when uncertainties are modelled using probabilistic framework [1–3]. These models work well when the empirical data needed to model the uncertainties are available adequately, and satisfy the requirements of being random samples. When these conditions are not met, alternative modeling frameworks (for example, those based on theories of intervals, convex functions, and fuzzy variables) become appropriate and these have been explored in the engineering mechanics literature [4–14]. The monograph by Moller and Beer [15] extensively covers

⇑ Corresponding author. E-mail addresses: [email protected] (K. Karuna), [email protected]. ernet.in (C.S. Manohar). http://dx.doi.org/10.1016/j.engstruct.2017.07.044 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

methods for uncertainty modeling using fuzzy random variables in the context of civil structures and computational mechanics. In the broader context of science and engineering, the fact that, both aleatoric and epistemic uncertainties typically co-exist while modeling a given physical variable, has been recognized, and, this has lead to several modeling frameworks, such as, probability box, random sets, evidence theory, fuzzy random variables, and the more general polymorphic models for uncertainty [see, for example, 16–24]. The importance of these developments has been recognized for problems of engineering mechanics and this is evidenced by thematic issues of research journals focusing on these topics [25–28]. The inverse safety analysis problems, on the other hand, consists of determining one or more of the system parameters (which could be deterministic system parameters and (or) parameters of an uncertainty model) when a target level of safety is specified, or, more generally, to minimize a cost function with constraints on safety measures. Such studies, for the case when the uncertainties are characterized probabilistically, include

K. Karuna, C.S. Manohar / Engineering Structures 150 (2017) 166–175

the development of load and resistance factors which aid reliability-based design [1,3,29–31], procedures based on first order reliability method (FORM) [32–34], and reliability-based design optimization method which minimize specific cost functions under the constraints of target reliability with respect to specified limit surfaces [35]. The papers [36–37] present extensive reviews of literature pertaining to problems of reliability based structural optimization. In the context of non-probabilistic, or, combined probabilistic and non-probabilistic description of uncertainties, studies in the existing literature addressing problems of inverse safety analysis are very limited. The study [8] has shown the common threads that exist in the problems of safety assessment when uncertainties are specified using alternative frameworks (viz., random variables, intervals, convex functions and fuzzy variables). In all these contexts, the study has shown that a measure of safety can be obtained by solving a constrained nonlinear optimization problem as in structural reliability studies. The recently published series of papers [38–43] considers options for non-probabilistic modeling of limited empirical data based on theories of intervals and convex functions (ellipsoids and super ellipsoids). These studies contain illustrations on skeletal structures as well as finite element based structural models. Questions on how to decide on choice of non-probabilistic model, and, how newly acquired data can be incorporated into an existing convex model are also addressed. Problems of design optimization, in which, uncertainties are modelled by using combined probabilistic and convex function models, have been discussed in [44]. Computational issues arising in problems of structural optimization involving non-probabilistic and (or) probabilistic uncertainty models have been explained using perturbational strategies [45], surrogate modeling methods [46], and sequential optimization strategies [47]. The study [48] has considered problems of reliability based design optimization in which the uncertain variables are characterized in terms of random variables with interval valued distributional parameters. The present authors have recently discussed the use of ellipsoids/super-ellipsoids with minimum volumes, and, convex functions modelled as contours of Nataf’s probability density function (pdf), in modeling uncertainties using convex function and fuzzy variable approach [14]. In this context, these authors have introduced the idea of a unit hyper-sphere as the standard uncertainty space and have shown that the problem of quantifying safety measure in a wide range of contexts (involving alternative uncertainty modeling frameworks) reduces to an identical form, namely, that of finding the shortest distance from the origin of an unit hyper-sphere to the limit surface in the standard transformed space. The present study aims to extend the idea of standard space introduced in this earlier work to address problems of inverse safety analysis when uncertainties are modelled, in general, using combined non-probabilistic and probabilistic frameworks. The study specifically considers the following three problems: (a) development of load and resistance factor design (LRFD) design format, (b) determination of a set of design variables so as to achieve a target level of safety, and (c) determination of a set of design variables which minimize a cost function under the constraints on safety measures. Illustrations include studies on linear/nonlinear beam and simple frame structures. It is pointed out that, while the study allows for both probabilistic and non-probabilistic modeling frameworks to be employed within the ambit of a single problem, the question of treating simultaneous presence of aleatoric and epistemic uncertainties with respect to a single uncertain variable (via the use of polymorphic models of uncertainties), on the other hand, has not been addressed.

167

2. Problem statement Consider the problem of structural safety analysis in which the uncertainties in specifying load and structural parameters are collectively represented through a n
1 vectorQ. We consider the situation in which the empirical data available to model Q could be inadequate, to varying degrees, and a probabilistic model encompassing all components of Q may not be possible. Consequently, a modeling framework involving a combination of random variables, intervals, convex functions and fuzzy variables is proposed to be employed. To allow for this, we write Q ¼ ð H X W Þt such thatH = nH
1 vector of random variables, X = nX
1 vector of variables modelled using a convex function X (X) such that X e X(X), and W = nW
1 vector of fuzzy variables characterized in terms of the membership function Xf ðW; aÞ; 0 6 a 6 1; it is noted that nH + nX + nW = n. Let g(Q) denote a scalar valued performance function such that the region in the space spanned by Q1, Q2,   , Qn in which g(Q) < 0 and g(Q) > 0, respectively, denote the unsafe and safe regions with the surface g(Q) = 0 representing the limit surface. The specification of elements of Q ¼ ð H X W Þt is taken to be as follows: (a) The vector H is specified through its nH-the order joint pdf pH(h). In case the complete specification of pH(h) is unavailable, it is assumed that the first order pdf-s, pHi ðhi Þ; i ¼ 1; 2; . . . ; nH and the nH
nH matrix of correlation coefficients are available so that a Nataf’s model for the joint pdf can be constructed. Using standard methods of transformations, the vector H can be transformed to the vector of standard normal random variables denoted byU. One could, in principle, consider more general models for representing probability distribution of H, such as, those based on Pboxes, or, the more detailed polymorphic uncertainty models. As has been already noted, the present study does not consider these options. (b) The specification of the convex model for X follows the procedure described by [14]. These authors have considered two alternatives for representingX(X): the one based on super-ellipsoids and the other based on convex regions fashioned after contours of a Nataf’s pdf. In the present study, we limit our attention to the first of these alternatives. Box 1 summarizes the relevant steps involved. As may be observed, the vector X here is transformed to Z so that, in the transformed space, the convex region X(X) gets mapped to a unit hyper-sphere. It is also noted that with ni  4, i = 1, 2,   , nX, the convex region reasonably well approximates an n-dimensional hypercube, thus providing a means to deal with interval models. In view of this, in the present study, the interval modeling strategies are not separately considered. (c) The specification of fuzzy variables wi, i = 1, 2,   , nW is through a membership function 0 6 lðw1 ; w2 ; . . . ; wnW Þ 6 1 such that for any a e [0, 1], the function lðw1 ; w2 ; .. . ; wnW Þ ¼ a is a convex function in the space ofwi, i = 1, 2,   , nW. Thus, the fuzzy variable model can be interpreted as a parametered set of convex functions. Consequently, the procedure for transforming the fuzzy variable to standard space of nested hyper-spheres, with radii varying between 0 and 1, follows the procedure as has been used in the case of convex functions. Here the convex function corresponding to the a-cut with a = 0 is mapped to a hyper-sphere of unit radius and the remaining functions corresponding to a-cut with 0 < a 6 1 are mapped to inner hyper-spheres with radii lesser than unity.

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Box 1 Convex function model for X using super-ellipsoid and transformation to standard space.

(A) Forward transformation: from data to the standard space. r X i ¼ ½maxðX i Þ  minðX i Þ; mX i ¼ 0:5½maxðX i Þ þ minðX i Þ; i ¼ 1; 2; . . . ; nX (b1.1) C ¼ nX
nX sample covariance matrix of X.
 C/ ¼ k/ ) U ¼ /1 /2 . . . /nX 3 Ut U ¼ I &Ut CU ¼ diag½ k1 k1 . . . knX  (b1.2) Yi ¼

X i mX rX

i

i

; i ¼ 1; 2; . . . ; nX (b1.3)

v i ¼ pUffiffiffik ; i ¼ 1; 2; . . . ; nX (b1.4) ¼ ½maxðv i Þ  minðv i Þ; mv ¼ 0:5½maxðv i Þ þ minðv i Þ;

U ¼ U Y; t

i

i

rv i i ¼ 1; 2; . . . ; nX (b1.5) wi ¼

v i mv i rv i

Remarks.

i

; i ¼ 1; 2; . . . ; nX (b1.6)

Super-ellipsoid with minimum volume model: PnX wi ni i¼1  ai  ¼ 1  ni 1   2 Z i ¼ waii  ) wi ¼ ai ðZ 2i Þni sgnðZ i Þ; i ¼ 1; 2; . . . ; nX (b1.7) PX 2 Model in standard space: ni¼1 Zi ¼ 1 (B) Inverse transformation: from standard space to the X space.   1 pffiffiffiffi PX X i ¼ mX i þ r X i nj¼1 Uij kj rv j aj ðZ 2j Þnj sgnðZ j Þ þ mv j (b1.8) @X i @Z k

(E) In case combined probabilistic and non-probabilistic models are used, the interpretation of safety margin varies with nature of models used. Thus, if random variable models are combined with convex function models, an interval model for Hasofer-Lind reliability index emerges and safety margin can be specified in terms of the lowest Hasofer-Lind reliability index. Alternatively, safety margin as defined for a convex function model here can be interpreted as a random variable and safety measure can be specified in terms of a suitably chosen characteristic value of this measure. These interpretations can also be extended, as shown in [14], to the case when random variable and fuzzy model are combined to describe input uncertainties.

1 1 pffiffiffiffiffi ¼ r X i Uik kk ð2Z k r v k nak ðZ 2k Þnk sgnðZ k Þ k

1

þ2rv k ak ðZ 2k Þnk dðZ k ÞÞ

(b1.9) Here dðÞ is the Dirac delta function and in the

 2 1 exp  2Zr2 . numerical work we take dðZÞ ¼ limr!0 pffiffiffiffi 2pr

3. Measures of safety As a prelude to the discussion on inverse problem, we briefly summarize measures of safety which we propose to use while tackling the inverse problems. (A) When all the uncertainties relevant to the problem can be modelled probabilistically, we transform the specified random variable model to equivalent space of standard normal random variables and subsequently employ the HasoferLind reliability index, bHL, which is defined as the shortest distance from the origin to the limit surface in the space of standard normal random variables. (B) We limit our attention to convex function models in which the convex region is specified as a super-ellipsoid. As indicated in Box 1, we transform X(X) to a unit hyper-sphere centered at the origin. This transformation also entails corresponding transformation of the limit surface. The safety measure,b, is again defined in terms of the shortest distance from the origin to the limit surface in this standard space. (C) As has been noted already, interval models for input variables can be viewed as special case of super-ellipsoid models and the approach mentioned above can be employed for characterizing the safety margin and no separate consideration is given to treat interval models. (D) Given that fuzzy variables can be viewed as a parametered family of convex functions, the safety measures for system with fuzzy uncertainty models can be obtained as in the case of convex function model. The safety margin here would be a function of the membership values.

1. It is well known in probabilistic analysis that, for systems with Gaussian variables and linear performance functions, the problem of reliability estimation is exactly solvable and the HasoferLind reliability index is related to the exact measure of reliability. It turns out that when convex function models are used, and the convex functions are taken to be ellipsoids, and for linear performance functions, the measure of safety is again exactly determinable. 2. When random variables are correlated and Gaussian, the transformation to standard normal space is through a linear transformation, and, this preserves the Gaussian nature of the pdf. Similarly, in the case of convex models, ellipsoidal nature of the convex regions is preserved when transformation of variables to standard space of unit hyper-sphere is carried out through a linear transformation. 3. The transformation of non-Gaussian variables to standard normal space involves nonlinear transformations. Similarly, the transformation of super-ellipsoids to standard space of unit hyper-sphere involves nonlinear transformations. This would mean that the non-Gaussian nature of uncertainties in probabilistic modeling and non-ellipsoidal nature of convex functions can be considered to be analogous as they present similar difficulties in the safety analysis.

4. Load and resistance factor design format for mixed uncertainty models Given a performance function g(H), the pdf pH(h) of random vector H, and a specified measure of safety, LRFD format aims to establish an equation of the form g(H⁄) = g(b1l1, b2l2,   , bnln) = 0 with bi li ¼ Hi ; i ¼ 1; 2; . . . ; nH , where H⁄ denotes the design point, and bi, i = 1, 2,   , nH are deterministic constants to be determined so that a target reliability level is achieved [1,31]. The use of the equation g(b1l1, b2l2,   , bnln) = 0 by a designer ensures that the resulting design meets the specified safety level and the design calculations proceeds apparently in a deterministic manner. The components of vector H are thought of as made up of load and resistance variables and the associated constants bi, i = 1, 2,   , nH are accordingly called, respectively, as load and resistance factors. In this section, we aim to extend the problem of developing LRFD format when uncertainties are modelled by using combined probabilistic and non-probabilistic frameworks. Towards this end, we consider the uncertain vector to be of the form Z ¼ ð H X Þt where H is specified through the nH dimensional joint pdf pH(h) and the vector X is specified through a convex functionX(X). As has been already noted, the safety measure can be specified either in terms of a lower bound on the Hasofer-Lind reliability index, or, in terms of a characteristic value of the safety measure

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associated with convex function model. The associated LRFD formats can be developed accordingly. In each case we first present results for exactly solvable cases and this is followed by more general formulations.

with the associated design point given by

U k

1 ¼ K2K3

4.1. Exact solutions




Here we consider H to be a set of Gaussian random variables with mean vectorl, covariance matrixC, and matrix of correlation coefficients q, and X(X) to be an ellipsoid given by  i2 PnX hPnX X i mi ¼ 1. The matrix of eigenvectors of the correlaj¼1 i¼1 T ij ai tion coefficient matrix of H is denoted by U and the corresponding eigenvalues by the vector k. The performance function is taken to be PH PX of the form gðH; XÞ ¼ a0 þ nj¼1 bj Hj þ ni¼1 ci X i . The cases of safety  margins specified in terms of bHL or in terms of a characteristic value of the possibilistic safety measure b are considered separately.

ð1Þ

nX nH X X Hi þ Xi ¼ 0withc Hi ¼ 1  bHL eH di bi ri ; b i li c ci mi c i¼1

eH ¼

i¼1

j¼1

j¼1

j¼1

nX nX X X ci ai T ij Z j þ mi i¼1

!

; di ¼

nH X 2 bi r2i i¼1

which, respectively, transform the vector H to the standard normal space and X(X) to a unit hyper-sphere. The performance function is now rewritten as

þ

i¼1

bHL ðZÞ ¼ min

GðU;ZÞ¼0

pffiffiffiffiffiffiffiffiffi UtU ¼

a0 þ

nX nX nH X X X bi li þ ci ai T ij Z j þ mi i¼1

i¼1

j¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u nH nH uX pffiffiffiffiX t ki bj rj Uji i¼1

"

nX nX nH X X X a0 þ bj lj þ ci ai T ij Z j þ mi




j¼1

i¼1

j¼1

nH nH X pffiffiffiffiX ki bj rj Uji i¼1

nH pffiffiffiffiffiX

k¼1

!2

j¼1

ð4Þ

bHL ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u nH nH uX pffiffiffiffiX t ki bj rj Uji

j¼1

bi Hi þ

nX X

ci mi

i¼1 i¼1 bðHÞ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi un nX X uX X t ai ci T ik k¼1

k¼1

from which, it follows that b is a normal random variable with mean and standard deviation given respectively, by

k¼1

i¼1

k¼1

ð5Þ

ð8Þ

i¼1

nX nX uX X t ai ci T ik

" !# nX nX nX X X X ci ai T ij ck ak T kj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 v !2ffi un u nH nX nH X uX uX pffiffiffiffiX X t ki bj rj Uji t ai ci T ik i¼1

nH X

i¼1 i¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lb ¼ v !2ffi &rb u

j¼1

j¼1

Case-2: Safety margin specified in terms of characteristic value of b

nX nH X X a0 þ bi li þ ci mi

i¼1

i¼1

i¼1

These factors clearly depend upon parameters in the uncertainty model and the specified safety measure.

a0 þ

nX nH X X bi li þ ci mi

i¼1

ð7Þ

that

The lower bound, bHL ¼ minZt Z61 bHL ðZÞ, can now be shown to be given by

i¼1

i¼1

ai mi

GðH;ZÞ¼0

!#

j¼1

a0 þ

X c2i a2i

X X t ai ci T ik

; xi ¼

define the safety margin as bðHÞ ¼ max

bj rj Ujk ; k ¼ 1; 2; . . . ; nH

kk

k¼1

Here we consider the performance function as in Eq. (2) and pffiffiffiffiffiffiffiffi Z t Z . It can be shown

j¼1

with the design point obtained as

~  ðZÞ ¼  U k

ð3Þ

j¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cXi ¼ 1  eX xi ci ai ; eX ¼ v !2ffi un nX u X nX

ð2Þ !

ri li

" !# nX nX nX X X X ci ai T ij ck ak T kj

j¼1

It follows that the Hasofer Lind reliability index, as a function of Z, is given by

;

j¼1

ð6Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u nH nH uX pffiffiffiffiX t ki bj rj Uji

j¼1

!

i¼1

If it is specified that bHL ¼ bHL , a design format, akin to the traditional LRFD format, can be obtained as

i¼1

nX X X i ¼ ai T ij Z j þ mi ; i ¼ 1; 2; . . . ; nX

i¼1

i¼1

!2

i¼1

k¼1

pffiffiffiffi Uij kj U j þ li ; i ¼ 1; 2; . . . ; nH and

pffiffiffiffi Uij kj U j þ li

k¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi un nX uX X X K3 ¼ t ai ci T ik

nH X

nH X

j¼1

nH pffiffiffiffiffiX kk bj rj Ujk ; k ¼ 1; 2; . . . ; nH

j¼1

We first use the relations

ri

i¼1

nX nH nH nH X X X pffiffiffiffiX K 1 ¼ a0 þ bj lj þ ci mi ; K 2 ¼ ki bj rj Uji

a0 þ

nH X GðU; ZÞ ¼ a0 þ bi

" !#) nX nX nX X X X K1K3  ci ai T ij ck ak T kj

j¼1

Case-1: Safety margin specified in terms of bHL

Hi ¼ ri

(

i¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u nH nH u XX u bi bj qij ri rj u u i¼1 j¼1 ¼u !2 u nX nX uX X t ai ci T ik k¼1

ð9Þ

i¼1

Assuming that b is specified in terms of a fractile P½b 6 b  ¼ p, the design format, similar to the one in Eq. (7), can be deduced with

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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX nH u nH X t bi bj qij ri rj

pffiffiffi

cHi ¼ 1 þ 2 erf 1 ð2p  1ÞeH di bi ri ; eH ¼

i¼1 j¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi un nX X uX X t ai ci T ik

cXi ¼ 1  b eX xi ci ai ; eX ¼

i¼1

k¼1

nX X c2i a2i

nH X 2 bi r2i

nH X

; di ¼

ri li

ai mi

ð10Þ

Eqs. (7) and (10) represent alternative version of the design formats and these are expected to lead to similar designs if bHL and b are calibrated to achieve the same level of safety.

 U k

@G @U k

 ; s ¼ 1; 2; . . . ; nX Z s ¼  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nX  2 X @b ðZÞ @Z i

i¼1

with " 

@bHL @Z s





nH X

# ðak Þ2

k¼1

¼

! ! ! nH n nH nH o X X X @ a @ a @U  U k @Zks þ @Zsk ak U k ak ak @Zks þ k¼1

"



nH X 2 ðak Þ

#32

k¼1

k¼1

;

k¼1

s ¼ 1; 2; . . . ; nX nX nH X @ ak X @ 2 g @ Hj ¼ @X i @ Hj @U k @Z s i¼1 j¼1

2 @ a

! 

@X i ; k ¼ 1; 2; . . . ; nH ; s ¼ 1; 2; . . . ; nX @Z s

" # !3 nH nH X X @ a ðaj Þ2  ak ak @Zks 7 7 j¼1 j¼1 7 7; " #3=2 7 n H X 5  2 ðak Þ

ð12Þ

; j ¼ 1; 2; . . . ; nH ; ð13Þ



"

1

1

lk

PHj 

@G @U j

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi U t U ¼ min bHL ðZÞ with bHL ðZÞ ¼ min U t U over all U such that GðU;ZÞ¼0

ð14Þ The details of the solution to this optimization problem are provided in Appendix A (supplementary material) and here we present the key steps. The optimal bHL(Z) and the associated design point are first obtained by using Lagrangian multiplier method as

nH X pffiffiffiffi U  Ukj kj aHj bHL

!#

; k ¼ 1; 2; . . . ; nH ;

j¼1



 ffi aHj ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  H X

and write the limit surface as G(U, Z) = 0. Here U() denotes the cumulative probability distribution function of a normal random variable with zero mean and unit standard deviation. The lower bound bHL is given by

ð17Þ

In the above equations a subscript/superscript ⁄ implies that the concerned quantity is evaluated at the optimal value ðU  ; Z  Þ. The design factors for a target bHL are now obtained as

cHk ¼

j¼1

over all Z such that Z t Z61



k ¼ 1; 2; . . . ; nH ; s ¼ 1; 2; . . . ; nX

!#

nX 1 X n T ij sgnðZ j Þ½ðZ j Þ2  j ; i ¼ 1; 2; . . . ; nX

over all Z over all U such that such that Z t Z61 GðU;ZÞ¼0

ð16Þ

HL

ð11Þ

p¼1

bHL ¼ min min





6 @Zks 6 @U k 6 ¼ bHL 6 6 @Z s 4

To proceed further, we first transform (H, X) to the standard space of ðU; ZÞ using the relations

X i ¼ mi þ ai



j¼1

x X i ¼ i ; i ¼ 1; 2; . . . ; nX ¼ ; j ¼ 1; 2; . . . ; nH ; c lj mi hj

Hj ¼ PHj U

~ U¼U

@G @U k

k¼1

and the required design factors are denoted by

pffiffiffiffiffi Ujp kp U p

ð15Þ

 ffi ; j ¼ 1; 2; . . . ; nH ; ¼ bHL sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nH  2 X

@bHL ðZÞ @Z s

H1 l1 ; c  H 2 l2 ; . . . ; c Hn ln ; c X1 m1 ; c X2 m2 ; . . . ; c X n mnX Þ ¼ 0 gðc H H X

nH X

~ U¼U

~  ðZÞ; k ¼ 1; 2; . . . ; nH produce the optimal b ðZÞ. Followwhere U HL k ing this, the determination of bHL ¼ minZt Z61 bHL ðZÞ, again using the Lagrangian multiplier method, is considered. It can be shown that the optimal value of ðU; ZÞ, denoted by ðU  ; Z  Þ, is given by



The exact solutions obtained in the preceding section are for linear performance functions with Gaussian models for H and ellipsoidal convex function model for X. Typically, the performance functions would be nonlinear, random vector H would be non-Gaussian, and X would be described by more general convex function models. In this section, we take that the pdf of H is described in terms of Nataf’s model (with specified first order non-Gaussian pdf-s and matrix of correlation coefficients; [49]), X is modelled using super-ellipsoidal convex functions, and the performance function is nonlinear in H and X. In present study attention is restricted to the case when the safety margin is specified in terms of bHL and it is aimed to determine the design factors associated with each of the elements of (H, X) to achieve the specified bound bHL . Hj , applied to the mean By introducing the design factors c X i to the centroid values value lj of Hj (j = 1, 2,   , nH), and, c mi associated with Xi (i = 1, 2,   , nX), we rewrite the performance function as

"



@G @U k

4.2. Approximate solutions

1

@G @U k

k¼1

i¼1

cHj



k¼1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bHL ðZÞ ¼  s nH  2 X

i¼1

; xi ¼

~ U k

k¼1

@G @U k

cXs ¼ 1  xs

2 

! nX 1 X n T si sgnðaX i Þ½ðaXi Þ2  j ; s ¼ 1; 2; . . . ; nX ; j¼1

HL ðZÞ ð@b@Z Þ i  aXi ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n  H X @b ðZÞ 2

ð18Þ

HL

@Z s

s¼1



The details of the derivation of Eqs. (15)–(17), along with details Hj ; j ¼ 1; 2; . . . ; nH and of iterative steps for determining c cX i ; i ¼ 1; 2; . . . ; nX , are provided in Appendix A (supplementary material).

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K. Karuna, C.S. Manohar / Engineering Structures 150 (2017) 166–175

The details of the relevant steps closely parallel those provided in the context of Eqs. (15)–(17) and a summary of these steps are available in Appendix B (supplementary material).

5. Problem with multiple design variables and multiple performance functions Here we consider a design problem with a r number of design variables denoted by h and a set of r performance functions. The uncertain variables H and X are described by pH(h) (jointly nonGaussian in general) and X(X)(a super-ellipsoidal function in general) respectively. The design variables could include parameters associated pH(h), X(X), and (or) deterministic parameters associated with the definition of gi(H, X, h), i = 1, 2,   , r. It may be noted that earlier studies [33,49] consider this type of design problems when uncertainties are modelled within entirely probabilistic framework. Using the transformations given by Eq. (13), the performance functions gi(H, X, h), i = 1, 2,   , r in the standard space are denoted by Gi(U, Z, h), i = 1, 2,   , r and the associated safety margins bHLi ; i ¼ 1; 2; . . . ; r are taken to be specified. The problem on hand can be stated as finding the r
1 vector h using the conditions

bHLi ¼ bHLi where bHLi ¼ min

over all Z such that Z t Z61

min

over all U such that

pffiffiffiffiffiffiffiffiffi U t U ; i ¼ 1; 2; . . . ; r

ð19Þ

6. Design based on minimization of a cost function Here we consider the problem of determining the optimal values of r
1 design vector h which minimize a specified cost function C(h) under the constraints

bHLi P bHLi ; where bHLi ¼ min min

pffiffiffiffiffiffiffiffiffi U t U ; i ¼ 1; 2; . . . ; p

over all Z over all U such that such that Z t Z61 Gi ðU;Z;hÞ¼0

ð21Þ

This type of problems, within the framework of probabilistic modeling, has been tackled [35]. As in the preceding sections, the uncertain vector (H, X) is transformed first to the standard space (U, Z) using Eq. (13). In the solution scheme, the problem of minimizing C(h) is tackled using the genetic algorithm (for which purpose the ga toolbox available on the Matlab platform is employed) and the optimization problem embedded in each of the constraints is solved using an iterative procedure which is similar to the one used in the preceding section.

Gi ðU;Z;hÞ¼0

These equations are equivalent to the r conditions Gi ðU i ; Z i ; hÞ ¼ 0; i ¼ 1; 2; . . . ; r where U i and Z i are the design points (to be determined) corresponding to the i-th performance function. An iterative strategy is developed to tackle this problem. Here we consider the functions F i ðhÞ ¼ Gi ðU i ; Z i ; hÞ ¼ 0; i ¼ 1; 2; . . . ; r and set up Newton-Raphson’s iteration at the k-th step as kþ1 hi

¼

k hi

þ

Dki ; fDj gkr
1

"  # k @F i ¼ @hj

k

k

k

fF i ðh1 ; h2 ; . . . ; hr Þgr
1 ;

r
r

i; j ¼ 1; 2; . . . ; r

7. Numerical illustrations We present a set of nine examples to illustrate the ideas presented in the preceding sections: three of these are included in the body of the paper and the remaining are included in Appendix C (supplementary material). Four of these examples are fashioned after illustrations contained in [50] which involve forward problems of reliability modeling given probability models for uncertain variables. In our study, these have been modified to allow for alternative uncertainty modeling frameworks and to treat inverse safety modeling.

ð20Þ 7.1. Example 1

Fig. 1. Example 1; statically loaded frame; AD (ISHB 250); DG (ISHB 250); BE (ISHB 350); EH (ISHB 300); CF (ISHB 250); DE (ISLB 550); EF (ISHB 250 A); GH(ISHB 300); Q1, Q2, & Q3 act at the mid-span of DE, EF, and GH respectively.

Here we present a set of five studies with reference to the building frame model shown in Fig. 1; the first four studies consider the frame to be linear while the last study accounts for material and geometric nonlinearity. The nomenclature of the member cross sections used in Fig. 1 is as per the Bureau of Indian Standards classification. The structure is modelled on the Abaqus finite element analysis software and this model is interfaced with Matlab based programs to model uncertainties and compute quantities related to safety. To start with a model with 171 degrees of freedom by using Timoshenko beam elements of length 0.6 m is obtained and with Young’s modulus = 200 GPa and Poisson’s ratio = 0.3. The loads Q1, Q2,   , Q5 are taken to be uncertain in nature. A performance function in terms of allowable stress at the right outerP most fiber of the column EB, given by gðQ Þ ¼ Q u  5i¼1 C i Q i , is considered. Here Qu is the allowable stress and Ci, i = 1, 2,   , 5 are the induced stresses due to the application of unit loads along the directions of Q1, Q2,   , Q5. The performance function accordingly is given by g1(Q) = Qu  140.6489Q1  32.1436Q2 + 133.6245Q3  1010.3814Q4  1098.6429Q5. Table 1 shows the LRFD formats

Table 1 Example 1; design format for alternative uncertainty models and target safety measures. Problem

Safety margin

LRFD format

Study-1 Study-2 Study-3

bHL ¼ 2:5 b ¼ 2:5 bHL ¼ 2:5 P½b 6 2:50 ¼ 0:05 bHL ¼ 2:5

0.8373lu  1.0544l1  1.0182l2 + 0.9015l3  1.3981l4  1.2164l5 = 0 0.8535mu  1.0490m1  1.0164m2 + 0.9113m3  1.3584m4  1.1948m5 = 0 0.9064mu  1.2041l1  1.0682l2 + 0.6303l3  1.2290m1  1.1245m2 = 0 0.8507mu  1.0126l1  1.0042l2 + 0.9771l3  1.3653m1  1.1986m2 = 0 0.9160mu  1.2038l1  1.0681l2 + 0.6309l3  1.2402m1  1.1134m2 = 0

Study-4

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Table 2 Example 1; Description of the convex set model. Variables

Description

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 X24 X25

Load (kN) Load (kN) Load (kN) Load (kN) Load (kN) Yield stress (MPa); members AD, DG Yield stress (MPa); members BE, EH Yield stress (MPa); member CF Yield stress (MPa); members DE, EF Yield stress (MPa); member GH Ultimate stress (MPa); members AD, DG Ultimate stress (MPa); members BE, EH Ultimate stress (MPa); member CF Ultimate stress (MPa); members DE, EF Ultimate stress (MPa); member GH Young’s modulus (GPa); members AD, DG Young’s modulus (GPa); members BE, EH Young’s modulus (GPa); member CF Young’s modulus (GPa); members DE, EF Young’s modulus (GPa); member GH Ultimate strain; members AD, DG Ultimate strain; members BE, EH Ultimate strain; member CF Ultimate strain; members DE, EF Ultimate strain; member GH

obtained for a few alternative models for uncertain loads. The details of studies 1–4 are as follows. Study-1 Here, we adopt a probabilistic model for the loads and the target safety measure is specified as bHL ¼ 2:5. The components of Q are modelled as Q 1 Nðl1 ¼ 169 kN; r1 ¼ 25:35 kNÞ; Q 2 Nðl2 ¼ 89 kN;r2 ¼ 22:25 kNÞ; Q 3 Nðl3 ¼ 116; r3 ¼ 29ÞÞ; Q 4 Nðl4 ¼ 62 kN;r4 ¼ 15:50 kNÞ; Q 5 Nðl5 ¼ 31 kN;r5 ¼ 7:750 kNÞ; Q u Nðlu ¼ 250 MPa;ru ¼ 69 MPaÞ with qQ 1 Q 2 ¼ 0:6; qQ 1 Q 3 ¼ 0:4; qQ 1 Q 4 ¼ 0:3; qQ 1 Q 5 ¼ 0:2; qQ 2 Q 3 ¼ 0:3; qQ 2 Q 4 ¼ 0:2; qQ 2 Q 5 ¼ 0:5, qQ 3 Q 1 ¼ 0:1; qQ 3 Q 2 ¼ 0:3; qQ 4 Q 5 ¼ 0:6. Here ‘ ’ is taken to mean ‘‘distributed as” and N denotes normal distribution. Study-2 Here, the safety measure is specified as b ¼ 2:5.The components of Q are modelled using an ellipsoidal convex function with m1 ¼ 169 kN, m2 ¼ 89 kN, m3 ¼ 116 kN, m4 ¼ 62 kN;m5 ¼ 31 kN, mu ¼ 250 MPa;a1 ¼ 25:35 kN, a2 ¼ 22:25 kN, a3 ¼ 29 kN, a4 ¼ 15:50 kN, a5 ¼ 7:750 kN;au ¼ 69 MPa. The transformation matrix T used in the calculations is made up of the eigenvectors of the correlation matrix given in study-1. Study-3 Here a hybrid uncertainty model is employed with the performance function assuming the form g 1 ðH; XÞ ¼ X u  140:6489H1  32:1436H2 þ 133:6245H3  1010:3814X 1  1098:6429X 2 . Here, the random variables H1 ; H2 and H3 are modelled as H1 Nðl1 ¼ 169 kN;r1 ¼ 25:35 kNÞ; H2 Nðl2 ¼ 89 kN;r2 ¼ 22:25 kNÞ; H3 Nðl3 ¼ 116 kN;r3 ¼ 29 kNÞ; with qH1 H2 ¼ 0:6; qH1 H3 ¼ 0:4; qH2 H3 ¼ 0:3; X 1 ; X 2 and X u are characterized by an ellipsoidal convex function model with m1 ¼ 169 kN;m2 ¼ 89 kN;mu ¼ 250 MPa;a1 ¼ 15:50 kN;a2 ¼ 7:750 kN;au ¼ 69 MPa and the elements of the transformation matrix are given by T 11 ¼ 0:7071; T 13 ¼ T 21 ¼ T 23 ¼ T 11 ; T 32 ¼ 1; T 12 ¼ T 22 ¼ T 31 ¼ T 33 ¼ 0. Two alternative safety margins, given by bHL ¼ 2:5 and P½b 6 2:50 ¼ 0:05 are considered. Study-4 Here a hybrid uncertainty model, as in study 3, is used with H1 ; H2 and H3 being modelled as H1 ln Nðl1 ¼ 169 kN; r1 ¼ 25:35 kNÞ; H2 Gumbelðl2 ¼ 89 kN;r2 ¼ 22:25 kNÞ; H3 Nðl3 ¼ 116 kN;r3 ¼ 29 kNÞ; and X 1 ; X 2 and X u are charac-

Model parameters mi

ai

ni

169 kN 89 kN 116 kN 62 kN 31 kN 300 MPa 300 MPa 300 MPa 300 MPa 300 MPa 400 MPa 400 MPa 400 MPa 400 MPa 400 MPa 200 GPa 200 GPa 200 GPa 200 GPa 200 GPa 0.35 0.35 0.35 0.35 0.35

25.35 kN 22.25 kN 29 kN 15.50 kN 7.75 kN 21 MPa 21 MPa 21 MPa 21 MPa 21 MPa 16 MPa 16 MPa 16 MPa 16 MPa 16 MPa 6 GPa 6 GPa 6 GPa 6 GPa 6 GPa 0.021 0.021 0.021 0.021 0.021

3.4807 1.5129 2.7268 2.3931 3.9081 2.8322 1.6753 1.4787 2.5052 1.3829 3.7815 1.6797 1.8380 2.5793 1.4540 1.5352 2.5228 2.9559 3.8059 2.8000 3.4878 3.7272 2.4631 1.7404 3.8925

terized by a super-ellipsoidal convex model with n1 ¼ 2:1; n2 ¼ 2:5 and n3 ¼ 2:3 and the remaining model parameters being the same as in the preceding study. The target safety margin is taken to be bHL ¼ 2:5. It may be observed from Table 1 that, in all the studies the resistance factor (multiplier of mu ) is less than unity, while the factors multiplying the loads are greater than unity except the factor associated with the load X 3 . This is to be expected since the load X 3 tends to oppose the bending stress produced by other loads. Study 5 The structure under study and details of FE discretization are is as in the preceding example except now we take the beam material to be elasto-plastic and include nonlinear straindisplacement relations. Element type B21 (two-dimensional Timoshenko beam element) were used. Table 2 lists the uncertain variables considered and a super-ellipsoidal convex function model with parameters as in Table 2 is assumed. Also, we take the transformation matrix, T ¼ I. The problem on hand consists of estimating the highest axial load that can be transferred to the foundation (of size 3.5 m
3.5 m
0.5 m and self-weight 147 kN) so as to achieve a safety margin of b ¼ 2:225. The implementation of the proposed procedure leads to the conclusion that a bearing capacity of 26.68 kN=m2 is required to achieve the target safety measure. The values of the loads X i ; i ¼ 1; 2; 3; 4; 5 at which this solution was realized were 146.14, 48.35, 82.16, 57.55, 25.93 kN respectively. Other variables pertaining to system parameters were observed to lie close to their respective centroidal values. 7.2. Example 2 (fashioned after [50]) Here we consider a simply supported beam having a span L = 5 m carrying random static loads comprising of dead load (g) and live load (q). The permanent load g determined by the weight of concrete floor with a uniform floor thickness h equivalent to 0.25 m (including both the slab depth and floor layers) and the self-weight of the beam. The imposed load q is made of long -term and short-term loads. Additional uncertain parameters considered include yield strength and factors which account for

173

K. Karuna, C.S. Manohar / Engineering Structures 150 (2017) 166–175 Table 3 Example 2; Description of convex set model. Variables

Description

Model parameters

X1 X2 X3 X4 X5

Slab depth Imposed long-term load Imposed short-term load Uncertainty of resistance Uncertainty of load effect

modeling uncertainties in computing dead and live loads. In the present study, for the purpose of illustration, we take the yield strength and density of beam material to be random variables modelled as H1 ln Nðl1 ¼ 280 MPa;r1 ¼ 19:6 MPaÞ; H2 Gumbelðl2 ¼ 0:024 MN=m3 ; r2 ¼ 0:00096 MN=m3 Þ with qH1 H2 ¼ 0 and the remaining variables are characterized using

2

0:37720698 0:32207 0:21218 0:10565

0:59828

mi

ai

ni

0.25 m 0.9 kN/m2 0.4 kN/m2 1 1

0.01 m 2.15 kN/m2 1.42 kN/m2 0.05 0.2

1.0 2.4 2.3 2.6 2.5

7.3. Example 3 (fashioned after[50]) Here we consider a simple two storey steel frame with concrete floors (Fig. 2). Tables 4 and 5 provide the details of system parameters, notations used and models for uncertainty adopted. The matrix T for the variables X i ; i ¼ 1; 2; . . . ; 9 is taken to be given by

0:00375

0:323496

0:38181 0:29904

0:547216

0:04476

0:316262

3

0:25966 7 7 7 0:237209 0:326616 0:27445 0:076486 0:45711 0:52591 0:027657 0:3191 7 7 7 0:265048 0:089753 0:460842 0:29665 0:06879 0:477627 0:200901 0:53749 7 7 0:17677 0:614809 0:02397 0:201738 0:0396 0:346657 0:354174 0:463202 7 7 7 0:29347 0:130574 0:032828 0:60193 0:17723 0:057642 0:61312 0:20937 7 7 0:50919 0:37789 0:450008 0:01782 0:25546 0:3191 0:395372 0:092189 7 7 7 0:471818 0:54066 0:26635 0:08532 0:15249 0:215931 0:115032 0:399077 5 0:40021983 0:284595 0:085762 0:424301 0:144675 0:604457 0:3416 0:20536 0:162108

6 0:27603645 6 6 6 0:40958596 6 6 6 0:24331086 6 T¼6 6 0:2962395 6 6 0:28097038 6 6 0:2513966 6 6 4 0:40556083

0:29917 0:05236 0:49599 0:34156

super-ellipsoid convex functions. Table 3 provide details of the super-ellipsoidal convex model adopted. In this model, the transformation matrix T is taken to be

2

0:47252 0:11464

0:1901

0:82688 0:209053

3

7 6 6 0:43116 0:5457 0:301261 0:385777 0:526049 7 7 6 7 6 T ¼ 6 0:58338 0:13715 0:44748 0:30639 0:58885 7 7 6 7 6 4 0:2016 0:02665 0:810598 0:19705 0:51259 5 0:4581

0:818257

0:125674

0:186367

It is assumed that the plastic section modulus is the design variable to be determined. Specifically, the problem of determining the model parameter m1, the centroidal value of the plastic section modulus X1, so as to achieve a specified target safety margin is considered. Following [50] the performance function is taken to be specified in terms of allowable bending moment and is expressed as G = R  0.16X9h(W1 + W2 + W3) where the resisting bending

0:264721

The problem on hand consists of determining the section modulus W, so as to achieve a target safety measure with the performance function given in terms of bending moment as

G ¼ X 4 W H1  0:125X 5 dðg þ qÞL2

ð22Þ

Here g = hH2 and q = X2 + X3. The associated specified target safety measure is set as bHL ¼ 1:5. Based on the iterative optimization procedure outlined in the preceding sections, we obtain W ¼ 1389:4 cm3 as the desired design. In the numerical work, an initial value of W ¼ 1543:2 cm3 (corresponding to the section ISLB 500) was made and the calculations were observed to converge within 5 iterations. The calculations were repeated for target safety measures of bHL = 0.5, 1.0, 1.5, 2.0, 2.5, and 3. The corresponding section moduli were found to be respectively, 1284:7 cm3 , 1291:3 cm3 , 1383:4 cm3 , 1424:8 cm3 , 1482:6 cm3 and 1514:1 cm3 . The observed trend here is seen to be in qualitative agreement with the results in the study in [50](see Fig. 2.2).

Fig. 2. Example 3; two storey steel frame; frame to frame distance, b, is taken to be 5 m.

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Table 4 Example 3; Description of probabilistic model (all variables are taken to be independent). Variables

Description

Distribution type

li

ri/li

H1 H2 H3 H4 H5

Thickness of concrete floor slab Steel yield stress Mass density concrete Long term live load Short term live load

Normal Lognormal Normal Gumbel Gumbel

0.20 m 300 MPa 2400 kg/m3 0.50 kN/m2 0.20 kN/m2

0.03 0.07 0.04 1.15 1.60

Table 5 Example 3; Description of convex set model. Variables

X1 X2 X3 X4 X5 X6 X7 X8 X9

Description

Model parameters

Plastic section modulus Aerodynamic shape factor Gust factor Roughness factor Reference wind speed 1 Reference wind speed 2 Model factor wind pressure Model factor resistance Model factor load effect

moment R, self-weight W1, live load W2, and wind load W3 are given by:

R ¼ X 8 X 1 H2 ; W 1 ¼ abg H1 H3 ; W 2 ¼ abðH4 þ H5 Þ; W 3 ¼ 2hbX 2 X 3 X 4 ð0:5qa X 7 X 26 Þ

ð23Þ

Here g is acceleration due to gravity and qa is the air density (1.25 kg/m3). The target safety measure is taken to be specified as bHL ¼ 1:5. The parameter m1 associated with plastic section modulus is taken as design variable and the proposed procedure leads to the solution m1 = 2.7857
104 m3. 8. Limitations of using the distance metric as a measure of safety Within the framework of probabilistic modeling, the merits/ demerits of using the Hasofer-Lind reliability index as a proxy for the true (but unknown) reliability are well established [1–3,51]. Some of the positives here are (a) the approach replaces the computationally difficult problem of finding probability of failure via a multi-fold integration over an irregular and implicitly defined region by a relatively more tractable problem in constrained nonlinear optimization, and (b) the approach yields, as a by-product, the point of highest probability of failure (i.e., the design point, which plays crucial role in reliability based design), and sensitivity measures associated with notional probability failure with respect to the basic random variables. Some of the drawbacks are: (a) the method is equivalent to linearization of the limit surface at the design point, and, hence, does not fare well when the failure surface has large curvatures at the design point (this difficulty can be overcome, to some extent, by using quadratic approximations to the limit surface at the design point; with this modification in place, for large bHL, the estimated probability of failure becomes asymptotically exact), (b) the reliability index lacks orderability characteristics, and is not simple to use if one has to deal with multiple design points, and estimation and model errors (suggestions for overcoming some of these limitations are contained in the works of Ditlevsen [52], and Kiureghian [53]), and (c) the method remains inapplicable if failure regions are multiply connected. Notwithstanding these limitations, presently, bHL based reliability models are widely used in reliability based design and design code

mi

ai/mi

ni

To be determined 1.10 3.05 0.58 5 m/s 30 m/s 0.80 1.00 1.00

0.02 0.12 0.12 0.15 0.60 0.10 0.20 0.05 0.10

1.5 2.4 2.3 2.6 2.5 2.4 2.6 2.4 2.8

calibration. The study reported in this paper essentially widens the scope of bHL in tackling inverse problems to cover uncertainty modeling using not only random variables, but also, intervals, convex functions, and fuzzy variables. This would mean that the work automatically inherits the above listed weaknesses of the bHL based reliability modeling. An alternative to this distance-based geometric approach, in problems of combined probabilistic and non-probabilistic modeling, would be to deal directly with probability of failure (estimated, for example, by using Monte Carlo simulations) and establish upper and lower bounds on this probability by using numerical optimization. Such an approach would be consistent with procedures followed in enhanced probabilistic modeling (such as, those based on probability box and other polymorphic uncertainty models). One could expect that sampling variance reduction schemes (such as, the subset simulation and particle splitting methods) would play an influential role in developing viable schemes especially while dealing with large-scale structural models residing in professional finite element packages. Work on achieving these developments merits further research.

9. Conclusions The study investigates a class of inverse problems which arise in modeling safety of structures when the uncertainties present in the problem are treated using combined probabilistic and nonprobabilistic tools. The uncertainty models considered include super-ellipsoid based convex function models and non-Gaussian dependent random variables and the analysis permits both geometric and material nonlinear structural behavior. Issues related to the definition of target safety margins, when combined probabilistic and non-probabilistic modeling frameworks are used, are discussed. Three types of problems, including development of load and resistance design format, determination of design parameters to meet specified safety measures, and design parameter determination based on minimization of a cost function under the constraints of target safety measures, are considered. In the context of uncertainty models being treated using probabilistic models, these types of problems have been already tackled in the existing literature and the present study essentially extends these earlier investigations to situations involving the use of alternative uncer-

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tainty models. For linear performance functions and uncertainty models involving ellipsoidal convex function models and Gaussian random variables, exact solution to some of these problems are shown to be possible. For more general class of problems, numerical optimization schemes are needed to obtain approximate solutions. Illustrations include studies on a nonlinear steel frame, modelled using Abaqus finite element analysis software, wherein the codes for solving inverse problems, developed on the Matlab platform, are interfaced with Abaqus software based models using DOS commands. Acknowledgement The work reported in this study has been supported by funding from the Board of Research for Nuclear Sciences, Department of Atomic Energy, Government of India [grant number 2012/36/35BRNS/1674]. We also thank the anonymous reviewers for their insightful comments.

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