Reliability and mechanical design

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Reliability and mechanica1 design Maurice Lemaire Research and Applicarions in Advanced Mechanics Laboralory-LaRAMA. French Itzsiitate of Advanced Blaise Pascal University, campus des Cézeaux. BP 265. F-63175. Alrbiere cedex. Frame

Mechanics and

(Received 1 September 1995: revised 26 October 1995: accepted 30 May 1996)

A lot of results in mechanica1 design are obtained from a modelisation of physical reality and from a numerical solution which would lead to the evaluation of needs and resources. The goal of the reliability analysis is to evaluate the confidence which it is possible to grant to the chosen design through the calculation of a probability of failure linked to the retained scenario. Two types of analysis are proposed: the sensitivity analysis and the reliability analysis. Approximate methods are applicable to problems related to reliabilitv. availabilitv. maintainability and safety (RAMS). 0 1997 Elsevier Science Limited. All rights reserved.

1 INTRODUCTION

When an engineer achieves the mechanica1 behaviour calculation of a structure, he uses numerical methods, which are supposed as being accurate, and he asks the computer to undertake millions of operations with 14 significant digits, whereas the knowledge of data often allows access to only 2 or 3 digits. When an engineer achieves a design in calculation of structures, he evaluates needs and resources and makes a choice that is only a bet, fortunately won most of the time. Rules and norms, as wel1 as the experience of designers, allow one to justify solutions, but practically never to evaluate the leve1 of risk. These two remarks lead to two complementary developments: the sensitivity and reliability analyses. The sensitivity analysis allows one to evaluate the variation of input parameters of a given calculation (displacements, stresses...) with respect to hypotheses on some characteristic parameters, such as averages and variances of data. As part of the finite element method in which we place the present study, because of its important development, the developed methodology is known as the stochastic finite element method, see, for example, Refs 1 and 2. The aim of the reliability analysis is to compare need S (the internal forces in terms of structural mechanics) and resource R (the resistance) al1 along a foreseeable lifetime T. Considering unavoidable imperfections during the fabrication. the failure probability is not null, but very small. The object is then to evaluate the formula: f, = Prob (S 2 R).

(1)

The sensitivity analysis is a natura1 part of the standard reliability analysis with mean value calculation, while the genera1 reliability analysis uses design point calculation. These two analyses depend on the probability calculation (Prob). A wel1 known method to perform this kind of problem is the Monte-Carlo method. The average and the covariance of the response, together with the probability of failure of a scenario, are evaluated by simulation of a great number of realizations. The necessity to use complex non linear models and the very smal1 value of the associated probability, in the order 10~‘-lO~h, render its utilization impossible in most of the practica1 situations. However, simulation techniques may be applicable for low probabilities if variante reduction techniques are applied. This is why methods of approximations have been developed and have now demonstrated their efficiency in many problems in the field of structural mechanics. First of all, we present principles of the sensitivity analysis and then the principles of reliability analysis. Finally, we give an example of an offshore platform design.

2 SENSITIVITY

ANALYSIS

2.1 Presentation

of the problem

Let US work in the context of the finite element method (FEM) with the hypothesis of static linear behaviour. Non linear static or dynamic behaviours can equally be taken into account by combining the

M. Lemaire

164

probabilistic algorithm and a solution method such as Newton-Raphson. According to classica1 FEM notation, the equilibrium equation is: Kq=F

(2)

in which: K is the stiffness matrix, ?? F is the vector of equivalent nodal forces, ?? q is the vector of nodal displacements. ??

These vector or matrix quantities are represented by simple letters, without specific indication of their components. Indices are reserved for random variables. The stiffness matrix K and the known components, from a mechanica1 point of view, F and q, are functions of random variables Xi, i = 1, II, whose realizations are Xi* These random variables represent: random data on the geometry: dimension, location of loads, random data on materials: elasticity modulus, plastification thresholds..., random data on imposed external actions (forces and prescribed displacements). is also convenient to introduce centred random variables, CZ;,deduced from Xi by the translation: (Y~= X; - m(X,) in which m(.) is the average question. Equation (2) becomes: K(a;)q(ai)

of the variable

aai

= F(ai).

q;l = (K’)-‘(F;;

and

-

(4)

E=Bq

e’) = BqO E; = Bq;

2 k=l

&k

+

;

a=()

2 K$Y~(Y/ + ... II

I

~F&Ct,+**.

kg,

E;; = Bq$.

(14)

Now, calculating stresses, taking into account possible random variability in Hooke’s law H: ~=HE which is similarly developed

a

(15)

into a series expansion:

H” is the elasticity matrix evaluated last, developing the stress vector: lTL(Yk+ ; g h

at & = 0. At

2 @;$(Ykff,+ ‘**

(17)

11-1

it follows then: # =

H”@

(18)

(5) aaj

F;CQ+;$ k

(12)

in which B is a matrix dependent on only fixed coordinates. The deformation vector is developed into a series expansion:

1/

The first two statistical moments of the response, the average E[.] and the covariance Cov[.], are now obtained. In case of displacements q:

(6) (7)

1

cov~q~ql=E[(q - 9’%4- 4’))l

+bZ~$&7%7~ h

+

(11)

CV!/

around the average point,

k=l

q”

- (K;q: + K;q; + Kf,q”)).

2.3 Average and covariance

d2V

aai

K = K” + i K;CQ +; 2 k=I k

=

(10)

k=l

it follows, by developing denoted by”:

q

- Kkq’)

The main interest of these equations is that the same matrix K” is used to determine q”, qi, q;!,... This results in a gain of calculation time. Deformations and stresses are calculated from q for each element. Using standard techniques of finite element method, the vector of deformation is then:

fJ.= ,‘) + c

a=O

F=F”+i

q; = (K”)-‘(F;

(9)

in

Because of the smal1 probability involved, the random variables, ai, are considered as infinitely smal1 in Taylor’s expansion. This is calculated for the three quantities considered and developed to the second order. In the first approximation, the first order can be retained alone. Noting that, for a variable V: = V!

q” = (K”)-‘Fn

(3)

2.2 Taylor’s expansion

dV

Application of eqn (4) makes it possible to determine the components of q at order “, ’ and “:

,$,

qf,aka,

+

.“.

(8)

Il-11-1,

I

X (E[akaj]E[a/a;]

+

E[akai]E[a,aj]).

(22)

Rrliahilit_v md

This procedure variables.

is only exact for Gaussian random

mechmicrrl

165

design

%(...) is the representative function of the mechanica1 model, there is a relationship such as: Pfi(i,F(f),K(i),S(I))

2.4 Conclusion This analysis makes it possible to determine the first two statistical moments of the response of a mechanica1 system whose model is formulated according to the FEM. It needs the calculation of the first derivatives (and possibly the second ones for the order “) of the rigidity matrix and force and displacement vectors. But this analysis does not allow one to approach the problem of reliability correctly. because Taylor’s expansion is calculated around the average point. but not around a specific point. the most probable failure point that wil1 be defined in the following paragraph. Except in very particular situations, an approximative failure probability cannot be derived.

3 RELIABILITY 3.1 Description

ANALYSIS of the prohlem

From a structural mechanics point of view, need S is a function of variables ~1, E and (r. The resource R is itself a function of random variables characterizing the section resistance capacity, an equivalent stress in a continuous middle... The aim of design of a mechanica1 structure is then to ensure that the resource is superior to the need. However. the set of realizations of possible designs is continuous. Contrary to well-known situations in safety, it is not possible to distinguish two binary states: the safe state and the failure state, but only an infinity of states, some of which belong to the safety domain and others to the failure domain. Each mechanica1 structure runs in a damaged mode. An important question is then to know whether damage leve1 is acceptable or not. The answer depends on the functioning conditions when the damage concerns serviceability or ultimate states: it also depends on politica1 decisions. In simple words, it is sufficient to verify that the resource is superior to the need and the probability of failure P, is given by eqn (1). Usually, the custom was to choose a safety coefficient which expressed the incapacity to estimate the resource as wel1 as the need. Today. it is unthinkable to justify al1 designs by the evaluation of P, which remains reserved to exceptional situations. Rules and standard codes are there to propose formulae insuring sufficient reliability, demonstrated by a calibration undertaken by evaluating an approximation of P, . More generally, a design is insured from a model of calculation of S and a model of calculation of R. If

= 0.

The success of a design is translated veritïcation of an equality such as: G(S(t),R(t))

(23) into

> 0,t’t E [O,T]

the

(24)

in which T is the required lifetime. Only considering the time independent case or choosing a given prescribed date t,,, the probability of failure is translated into the relationship: Pi = Prob(G(a,)

50) = 1 .tV,(cu,)do, do’...da,, fi

(25)

in which it must be remembered that G is a random function of variables (Y, and 9, is the part of the random variable space corresponding to failure realizations. The expression of J’, is simple to state but its evaluation involves many operative difficulties taking into account the complexity of the expression of il, and the determination of the density of probability ,fl, (ff). 3.2 Evaluation

of the prohability

of failure

Presently available methods aim to avoid the Monte-Carlo method (crude or improved) by considering the mechanica1 model complexity and the heaviness of its numerical solution. The retained approach involves two steps (Fig. 1): ??

first of all, calculating a reliability index B (First mul Second

??

Order Relinhility

Methods),

and then, evaluating an approximation of the probability of failure (see, for example, Ref. 3).

3.2.1 Reliahility index The first definition of a reliability index can be attributed to RzhanitzynJ who defined the mfety chrrrmteristic. It has been resumed and popularized by Cornell. Suppose that G(Lu,) is written under the form of a variable of margin Z between the resource and the solicitation: Z(a,) = R(o,) - S(o,)

(26)

then. in the case of random Gaussian variables and with a linear function G(cw,). we obtain the exact following result: f, = Prob(Z 9 0) = ‘P( - p) with /? = F

(27) /

in which III, and W: are. respectiv,el!z. the average and

166

M. Lemaire

domain

ÖKb? physical

space

*tangent

normalized

hyperplane

space

pf

by SORM or simul .ation V evaluation Fig. 1.

of Pf Approximation of the probability of failure.

the standard deviation of Z and @. (.) is the distribution function of the Gauss’ law of a random variable of nul1 average and unit standard deviation. This approach was first of al1 erroneously generalized, then correctly defined by Hasofer & Lim? who proposed choosing a space of Gaussian normal functions (nul1 average, unit standard deviation N (0, 1)) to render the definition of p independent of the representation in the physical space. Rosenblatt’s transformation T makes it possible to transform random variables CX;to variables Ui (N (0, 1)) and the reliability index is correctly defined as the smallest distance from the origin to G(ui (ai)) = 0. The calculation sequence (Fig. 1: physical space+ normalited space) is then: lli =

T(~j);

j3 = min

G(c~~) = G(T(aj)) = 0;

2 uj? ; JL )

??

??

by approximation of Pr from a hypersurface defined by its curvatures to the most probable failure point P*, which is called the SORM approximation Second Order Reliability Method’ by application of methods of simulation around P*: conditional simulation, directional simulation, importante sampling. Some simulations confirm - or not - the good approximation of c,..’

Figure 1 (appruximation of Pr) illustrates this method and Fig. 2 gives the correspondence between P( and p. In some particular situations, a direct calculation of the probability of failure is possible and, then, the reliability index is obtained by: p = - @-‘(Po.

(28) with G(u;) = 0

The solution point P* is the most probable failure point. This procedure is now very wel1 defined and software performs this calculation. 3.2.2 Prohahility of ,failure lf the function G(ui) is a hyperplane, the result of eqn (27) is exact and the calculation of cf is immediate. Most commonly, the approximation P[ = @( - p) is known by the name of First Order Reliahility Method (FORM).

The difficulty is then to appreciate the quality of the approximation, and methods are now proposed:

3.3 Results of a reliability analysis

As compared to a sensitivity analysis, the reliability analysis introduces the idea of failure scenario and of most probable failure point (design point). The sensitivity analysis keeps al1 its interest when Taylor’s expansion is calculated around P* and not around the average point. The reliability analysis gives more genera1 results such as the elasticities of p in relation with the following variables:x

167

1

.I

2

L

(1

i

.‘i illclt,’

Fig. 2. Approximation

??

??

of the prohability

averages and standard deviations of random variables, deterministic parameters of the function G(cy,).

4 APPLICATION

of failure.

normal force N represented by a square in the N, M space. Geometrical effects of the second geometrical order are represented by a negative stress hardening. Under these hypotheses, corresponding to each step of the damage of the structure, there is a relationship of the form: (29)

4.1 Presentation in which: An example of application is proposed to illustrate this comment. It concerns the comparison between two technical solutions for the design of frames. This example has been processed by Lemaire et al.,” more complete data and results are in Refs 10 and ll. Two basic solutions are proposed, one with the frame braced by K and the other braced by X. They are represented in Fig. .3. 4.2 Mechanica1 behaviour

[K] is a tangent stiffness matrix taking into account the elastoplastic damage, {q} is the vector of displacement to the nodes of the structure, {F} is the vector of equivalent actions, at a given leve1 of damage. The description of the mechanica1 behaviour frame is then the following: ??

To represent mechanica1 behaviour precisely, a piece-wise linear model is proposed. The elastoplastic behaviour is taken into account by an interaction diagram between the bending moment M and the

the initial state corresponds to an elastic behaviour. It is limited by the achievement of plastification in any section among al1 potential sections. The safety of a structure is measured by the probability to reach a failure state in any

K K-frame

13.4 111

X-frame Fig. 3.

K and

of a

X frames.

168

M. Lemaire

section. Resistance of a section is given by a limit state relation: f(N,M)

??

- 1= 0

(30)

in which f(N, M) is a function of (I, itself a random function of (Y;which represent hazards of the rigidity K and of external actions F. Therefore m functions Gj;)((~~),k = 1.m exist so that G$$)(ai) = 0 represent the first achievement of plastification in a specified section k. Reliability indexes pi” and associated probabilities measure the safety in elastic behaviour. Suppose now that a section i is plastified. The structure can again insure an acceptable serviceability in damaged mode. Equation (29) is updated taking into account the condition ‘the section Si is plastified’, and functions @,“‘(a,) = 0 represent the behaviour until the plastification of a new section Si, j # i, knowing that S, is failed.

This calculation sequence can be repeated as often as desired. The number of events of failure to consider grows very rapidly and only the most significant events must be retained. This diagram is called branch and bound.‘”

At a given leve1 (I), the probability obtained by the union of probabilities: Prob~(~~‘~OnG)‘-“~O...nGji’~O).

of failure is

(31)

4.3 Data 4.3.1 K-frame

The K-frame is constituted of 7 members. The cross sections are circular tubes of three different sections (Table 1). 4.3.2 X-frame The X-frame is constituted of 8 members. The cross sections are circular tubes of three different sections (Table 2). 4.4 Sensitivity

analysis

The sensitivity analysis is applied to compare the influence of the thickness variability when the first plastic limit state is reached. The thicknesses ti, i = 1, 3 modify the cross section areas and the inertia

moments but also the plastic moments Mp, and the plastic axial strengths N,,. The load P is such that the limit state (eqn (30)) is equal to 0 when the variables are calculated with the average thicknesses. By using the finite differente calculation, the gradient and the Hessien of the limit state are approximated around the average point. G(P,t;) =f(P,N(ti),M(ti))

G(P,ti) = G(&JI,,(,,,

+5

dti 1 I nz(t,)

+& a*G

2 afiafj

- 1 =O

dti dti +

Let US consider the thicknesses ti as independant random variables with coefficients of variation equal to lO%, then average and variante are obtained, for the K-frame: E(G) = 0.0104

Cov(G,G)

= V’(G) = 0.0102

Cov(G,G)

= a’(G) = 0.0092.

and for the X-frame: E(G) = 0.0097

For this example, the application shows that the X-frame is slightly less sensitive to the thickness uncertainty than the K-frame. 4.5 Reliability

analysis

Reliability analysis is then applied to the study of the frames with two different levels of damage. Random variables are the axial force P, plastic moments and axial strengths. As an illustration, they are represented by Gaussian variables of respective coefficients of variation of 0.10 for the action and 0.11 for resistances. The results are given in Fig. 4. Without exploring al1 possible situations, the tree of failure probabilities is presented for the first two levels. Situations of failure are constituted by the achievement of plastification in some sections Si or in the event of an unacceptable displacement. Conclusions of such an approach can help in the choice between the two structural designs, according to a criterion of reliability and not only weight or tost criteria: ??

the frame braced by X is far more rigid than the

Table 1. K-frame data

Type : 3

Member 2-4,6-8 l-2,5-6 2-6,3-4,7-8

.s.

fn(l,)nz(tl)

Area (cm’)

Inertia (cm’)

N,, (kN)

M,, (kN.m)

988 76.8 94.1

843 6 000 690 11620

241920 700 2 350

6 500 161 235

Rrliahility

rrnd nwchanicrrl

design

169

Table 2. X-frame data

Type 1 2 3

Member

Area (cm’)

1-3.6-8 1-2.3-4.5-6.7-X 3-8.1-6

988 76.X 94.1

Inertia

(cm’)

,V,, (kN)

833 000 6 690

24 700 1 920

11 h20

7 350

M,, (kN.m) 6 500 161 235

reliability

indexes

displacement

leve1 failure

second leve1 failure

first

first

X-frame

K-frame Fig. 4. Comparison

??

second leve1 failure

leve1

failure

of frames

frame braced by K, as shown by the results of the displacement the first failure leve1 of the frame braced by K is superior to the one of the frame braced by X, but at the second leve], a slight advantage is given to the K frame.

From the point of view of tost, the K design is the most profitable. However, this study shows that the same reliability leve1 is not obtained. The choice depends on considerations of expertise and varies according to whether damaged behaviour is accepted leve1 failure, too large or not: e.g.. a second displacement...

5 CONCLUSION When the safe behaviour or the failure of a mechanica1 system can be represented by a set of performance functions of a certain number of random variables of the project, approximate methods make it possible to evaluate the success or failure probability. They result in the exact calculation of a reliability index and in thc approximation of a probability. shows how such a Thc presented example methodology is applied in the two types of frames

braced

by K and by X.

studied. These methods are currently increasingly used in the case of exceptional projects, but equally for the design rules calibration of current works. Principles of calculation are not linked to problems related to mechanics. They are applicable each time a function of performance can be defìned, explicitly or implicitly. however, the control of the approximation of f, must be insured with care. Under this reserve, the reliability indexes method is certainly a methodological tool which can be put forward in evaluations of safety and serviceability in many problems.

ACKNOWLEDGEMENT The author wishes to acknowledge Elf Aquitaine Production for the financial support of this research and his colleagues Dr A. Mohamed and Dr 0. Florès Macias for their participation.

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M. Lemaire M. & Hien, T. D., The Stochastic Finite Element Method, Basic Perturbation Technique and Computer Implementation, John Wiley & som, NY,

2. Kleiber,

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Monte 119-126.

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procedure.

sampling: Struct.

an iterative Safety,

5

fast

(1988)

8. Hohenbichler,

importante

M. & Rackwitz, R., Sensitivity measures in structural reliability.

and Ciuil

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J. P., Model of negative strain-hardening to evaluate the reliability of beam-columns. In Sirth Int. Conf Applications of Statistics and Probability, Mexico, 17-21 June 1991, pp. 201-208. 10. Mohamed, A., Lemaire, M., Muzeau, J. P. & Birades, M., Negative strain hardening to evaluate reliability in offshore engineering. In Proc.OMAE’93, Glasgow, UK, ASCE vol. 2,20-24 June 1993. ll. Mohamed, A. & Lemaire, M., Linearized mechanica1 model to evaluate reliability of offshore structures. Struct. Safety, 17 (1995) 167-193. 12. Thoft-Christensen, P. & Murotsu, Y., Application of structural systems reliability theory, Springer Verlag,

Berlin, 1986.