Design of a composite structure to achieve a specified reliability level

Reliability Engineering and System Safety 56 (1997) 273-283 ELSEVIER

PII:

© 1997 Elsevier Science Limited All rights reserved. Printed in Northern Ireland 0951-8320197/$17.00

S0951-8320(96)00095-6

Design of a composite structure to achieve a specified reliability level C. Boyer & A. B6akou & M. Lemaire Laboratoire de Recl~erches et Applications en M~canique Avanc~e, Institut Franfais de M~canique Avanc~e, Universit~ Blaise Pascal-Clermont H, Campus des C~zeaux-B.P. 265, F-63175, AubiOre C~dex, France

Safety factors are widely used in structural design. For composite material structures, however, the lack of experimental feed-back does not allow the use of safety factors optimized from cost and reliability point of view. Reliability methods are one way to achieve the calibration of partial safety factors using a more rational method than judgement alone. First we present the calibration process. The reliability methods FORM, SORM, simulation, are initially applied to a laminate plate under uniform pressure. In this example, we compare three design criteria; the different reliability methods agree with the reference method for all criteria used. We chose the Tsai-Hill criteria and the FORM method to calculate safety factors. Then, a calibration process is undertaken on a composite pipe and this serves to illustrate the different steps in the calculation. Finally, we present a calibration of a general plate structure. The partial safety factors and their sensitivities to the different parameters of the stochastic variables are given according to load type. © 1997 Elsevier Science Limited.

1 INTRODUCTION

choice. Finally, applications are m a d e to pipe and plate composite structures.

Traditional design methods use global safety factors to take into account the uncertainties in manufacturing, loads, materials properties... Their values have been established after m a n y years of experiments and calibration by judgement, but they are not suited to new materials with particular features. Fibre-reinforced composite materials are characterized by their exceptitonally low weight-strength ratio but also by considerable scatters o f mechanical properties and m a n y failure modes. For these materials, the choice of safety factors cannot be only based on p o o r or inexistent experiments. Therefore, their calibration, by means of reliability methods, is one way to determine the new values for reliable optimized structure design. In [22], a m e t h o d to determine safety factors equivalent to those used in metallic structures is presented. The author does not, however, take account of the particular properties of composites such as anisotropy, ultimate limit states. In this paper, second m o m e n t reliability methods which allow us to consider these characteriistic features are used. The first part deals with general calibration methods. The second discusses the mechanical failure criteria and different reliability methods and it allows us to m a k e a

2 GENERAL CALIBRATION PROCESS OF SAFETY FACTORS 2.1 Different design values During the design process, a stochastic variable may be characterized by several values. Each one has its own signification and must be distinguished f r o m the others [10]. - - T h e features of a material, actions .... are represented by characteristic values Yk with a given probability k to be reached in a unlimited set o f tests. They are often particular fractiles of the density function. Each variable may have two characteristic values: upper and lower ones. In m a n y cases, however, only one is used, thus allowing a safer design. Generally, for the material strength, the lower value is used then k is equal to 0.05, for loads the upper one then k is equal to 0.95. A variable with an cumulative function F gives: P r o b ( Y < Yk) = k ¢:~Yk = F - ' ( k ) . 273

(1)

274

C. Boyer, A. Bdakou, M. Lemaire

For the material properties, Yo.05 is generally the minimal value ensured by suppliers. - - W h e n the limit state function is not sensitive to the variable scatters, the nominal value may replace the characteristic one. Its statistical meaning is less explicit and it may be the expected value or a value based on broad experimental utilization. --Characteristic and nominal values do not enable reliable structural design but may be used to determine the design values for which the value of limit state function is zero. There are two ways of obtaining these values, either by using characteristic values multiplied or divided by a partial safety factor, or directly by taking reliability into account by means of a second moment method.

2.2 Partial and global safety factors

Stochastic phenomena are traditionally taken into account by the use of safety factors 3'. The effects E of actions on a structure and the resistance R to these effects must verify a criterion [6, 4, 10] in the form:

2.3 Calibration method Since the calibration tools and the form of the criterion are known, the process is easy to understand. An optimization of the design parameter is carried out to obtain a structure with a specified reliability level. In order to evaluate the reliability of a structure, we need to know which variables must be considered as stochastic: material properties, loads, size... These parameters are called the basic variables Xi. Then, in order to avoid different kinds of failure, the design rules have to be listed from the physical, mechanical or experimental knowledge of the designer. From the basic variables, the design rules allow us to determine the ability of the structure to fulfil its duties. So the basic variables' space is divided into two sets, the safe set Ss and the failure set Sy. It defines a limit state function G [13, 12]. The failure probability Pr is calculated by the integration of the variables' density function fx over the failure domain SI. el = e r o b ( G ( X ) <- O) = I fx dX.

R E < --. 3'

try " 3"model" 3"reliability"

In order to calculate PI, we could use one of the well-known reliability methods [13]: Monte Carlo, FORM, SORM, Importance Sampling. F O R M is particularly interesting because of its efficiency and its low computational cost. Its aim is the calculation of a reliability index prior to a failure probability evaluation. The reliability index is defined as the minimal distance from the origin O to the limit state function in a normalized space [9]. The point P* which minimizes this distance, is the design point. The approximation of the failure probability is given by:

(3) ef ~-cb(-[3)

There are many processes to design with partial safety factors which corresponds to the evaluation of the design values. If R is a function of two variables A and B, the design values Rd may be calculated by: Rd = R(Ad, Bd)

or

1 Rd = - - R(Ak, Bk) 3"Rd

or Re = - - R YR

(5)

(2)

There are as many criteria as failure modes. Various factors are applied in limit state functions corresponding to each variable. Strength or effect values depend on the variables describing the structure and its environment; so 3" may be considered as a product of partial safety factors linked to materials, geometry, actions, mechanical models and the specified reliability level [18]: ~/ = 3"material" 3"actions " 3"g. . . .

actions), the safety factors modified by the latter alone need to be re-evaluated.

,

(4)

.

The relations between the various safety factors may be easy if R is linear but this is not always the case. For composite materials with differing strengths, these relations may be complex. The last form of eqn (4) sometimes allows each safety factor to be considered separately. In some special cases, for a design with a new design element (material, geometry,

(6)

where • is the standardized normal probability function. The coordinates of P* are important to estimate the safety factors. P* is on the limit state surface and is called the critical design value. Next, the design values obtained by the probabilistic methods are given as equal to those used in a deterministic method. So we can determine the partial safety factors from the characteristic values. The steps in this process are: ---choice of a reliability level, i.e. a target reliability index ~target --probabilistic calculation and reliability optimization. After convergence, the coordinates x* of the design point P* are equal to design values of each variable Xi ---choice of the characteristic values xik ---equalization of design values found by prob-

Design of a structure to specified level abilistic methods and used by deterministic ones to obtain partial safety factors: X i g¢ ~ X d reliability ~ X d deterministic = f ( X i k, ~ti')~:~ ~/i = g ( x ~ ,

Xik )

(7)

where the functions f and g are a multiplication or a division depending on whether the stochastic variable is a load or a strength, to achieve 7i > 1.

3 PROBABILISTIC DESIGN AND COMPOSITE MATERIAL Fiber-reinforced composites are nowadays widely used because of their high resistance to weight ratio and the additional degree of :freedom in the mechanical behaviour they give to the designer through the fiber orientation. Also, t h e ; material properties and especially, ultimate strength reveal more scatter than metallic materials. This is done by applying statistical analysis to the results of testing on samples of the materials. Design data derived in this way are often called 'design allowable properties' or characteristic values. Unfortunately, the properties given in suppliers' data sheets are usually the mean properties with no indication of variability. 3.1 Design allowable properties Statistical analysis performed on the results of mechanical tests produce design allowable properties which take into account the degree of scatter and are related to a specific leve![ of confidence.11 For instance, in the aerospace industv.¢, the usual standards adopted are the A- and B-bases. An 'A' is allowed as defined as the value above which at least 99% of the population is expected to fall with 95% confidence and a 'B' is allowable, as the value above which at least 90% of the population is expected to fall within 95% confidence. The steps of the determination of the design allowable properties are: ----choice of a statistical distribution; generally log-normal or Weibull's distributions are used for composite material --estimation of the parameters of the chosen distribution from test data; maximum likelihood, regression analysis are used --derivation of the design allowable properties; confidence factors, ,~2 distribution, or Student's t-distribution are used. For more general application, design allowable properties are used in a deterministic design format including a global safety factor and no quantification of the safety of the component can be performed.

275

However, probabilistic design enables the quantification of the reliability of a structure in terms of failure probability or partial safety factors by using the entire probability distribution, or at least the tail, of the material strength [15]. But the accuracy of the analysis depends also on a realistic mechanical model of the structure. On this point, the following detail should be noted: a reliability method cannot give greater accuracy on mechanical models. Models used in a probabilistic design are the same in a deterministic approach. In the case of composite material, the accuracy of the failure criteria is very important. 3.2 Choice of the failure criteria As far as the scale level and mechanical modelling are concerned, failure criteria used in structural design are macroscopic. Certainly, probabilistic formulation can express the strength of a ply in term of constituents and interface failure. At the sample scale level, the results of this formulation may be validated by mechanical tests and then used in a macroscopic criterion for a design process. There are many failure criteria of composite materials related to different modes: matrix cracking, fiber failure and delamination. Some of these criteria require a better understanding of the physical phenomena and experimental validation. In the general case, a n-ply laminate will imply n possible ply failures which can act as a series system (weakest link) or a parallel system [7]. Here, we consider the first case as it represents the lower bound for the reliability of the laminate and is confirmed by experimental observations. So the laminate is considered to fail when one ply fails. For the reliability of a single lamina, Tsai-Hill's interactive criteria will be chosen below in order to take into account longitudinal, transverse and shear modes. In a recent paper [5], it is shown that many limit state functions can be considered in a probabilistic design. So, all the failure modes known nowadays in the mechanical modelling of composite structure could be taken into account to achieve a more accurate result.

4 RELIABILITY METHODS AND FAILURE CRITERIA ASSESSMENT In this part, we present an application of the reliability methods. We also present a comparative study of the various design criteria usually used in composite materials. We consider a symmetric orthotropic plate (1 × i m 2) simply supported, u n d e r a uniform pressure Po, with [0°/90°/90°/0 °] stacking sequence. Each lamina has a thickness a.

C. Boyer, A. B~akou, M. Lemaire

276

4.1 Mechanical modellisation The classical laminate theory allows us to calculate the strains and stresses in each lamina of the plate [1]. The global behaviour law of the laminate is obtained from the geometrical features and the stacking sequence. For the chosen structure, this law is:

(N)=(aA'

0

" "e°"

where e ° and k are the strains and the curvatures of the middle surface of the plate; N and M are the generalized forces and moments. A' and D ' depend on the elastic features of the material. The fundamental mechanical equations in classical laminate theory are solved by using Fourier's expansion. The expression of the displacements of the middle line is:

Wo(X,y) _16Po ~

~ sin((2m - 1)zcx)sin((2n - 1)ny)

g6a3m=, n=,

(2--~= 1)(2~- 7

1)S(2,,----~-,,~2~_,---~)

(9)

with: S,,m = (D 'll m4 + D'22 n4 -t- 2(D '~2 + 2D'66)n2m2), and Wo(X,y) is the displacement along z, the displacements in the plane are insignificant. The derivation of the displacements gives the expression of the strains, and from Hooke's law, the stress expression becomes in each lamina k:

{ O'L(X'Y)~ ~ r k = ~ O'r(X,y) /

where the R parameters are the strength of the materials, the indices C and T differentiate between the compressive and the tensile strength. Here we use these three criteria but others can be chosen.

4.1.1 Stochastic model We use the first ply failure theory, i.e., the weakest link theory. Each lamina may crack independently and when one of them has failed, the plate is considered to have collapsed. It is a series system. In this example, we give results only for the weakest-ply because of its predominance in the failure process. Each lamina is made of 67% glass S in a XP-251 epoxy resin. The elastic moduli [3] are usually considered as constants: Ec = 57 GPa, ET = 20 GPa, G C T = 5 . 9 G P a and VLT=0.26. The mechanical properties' expected values are taken from test data. On the other hand, the strength, the lamina thickness a, the pressure Po are considered as stochastic variables. Their expected values, their coefficients of variation (CV) and their probabilistic laws are summarized in Table 1. The coefficients of variation are within the scale of values found through testing [8]. The stochastic strength laws are taken as Weibull's two parameter law [20] which is often used to characterize composite material strength because of its simplicity and its link to brittle material theory:

F(x) =

z.po{ A(x'Y) ~ = - - a3 / / x ( x ' Y ) ]

\OrLT(X'Y)/k

(10)

\p(x,y) /k

where the indices L and T correspond to the longitudinal and transversal directions in the orthotropic axis of the k lamina. A(x, y), /~(x, y), /z(x, y) functions depend on material elastic properties. So the expressions of the stress are known in each lamina and at each point of the plate as a function of the pressure Po and the thickness a. Several criteria are used in material structural design. The most commonly used are maximal stress, maximal strain, Tsai-Hill's and Tsai-Wu's criteria. The two former are equivalent. The maximal stress 11, Tsai-Hill's 12 and Tsai-Wu's 13 criteria are expressed in each lamina:

2

2

(11)

( LT)2

1--\~--~L/ - \ ~ - ~ 1 +(-~L)2L)2--\~77cr/ > 0 1_(1,,,

1

1

(14)

Table 1. Featuresof random variables

Rr

]O-LTmaxl < Rt.T

~o

where xo is the scale parameter and 77 is the shape parameter. The criteria are applied at the most critical point of the plate. According to eqn (10), stresses are maximal when IZl is maximal. The criteria are therefore expressed in each lamina at the surfaces furthest from the middle surface of the plate, i.e., at z = la where l E { - 2, - 1 , 1, 2}. For the maximal stress criterion, the points where Io'i.... I, IO'TmaxI, I~LTmax[ are highest, are located. The longitudinal and transverse stresses are at their highest level in the centre of the plate, the shear stress in the corner. For Tsai Hill's and Tsai Wu's criteria, by using the expected value of stochastic variables, we search for the coordinates x and y where

IO'Lm,xl< RL IO'Tmaxl<

1 - exp -

(12)

1

v c -- + RIr RI.c RTRT R2cr RLRc >

0

(13)

Variable

Expected value

CV in %

Type of law

R~ Rcc Rr RCr RLT P, a

1.99 GPa 1.17 GPa 76 MPa 200 MPa 62 MPa 5000 Pa 1 mm

5 6 5 6

Weibull Weibull Weibull Weibull

5 2.5

log-normal normal

5

Weibull

277

Design o f a structure to specified level

the criteria are minimal. If the lamina is loaded with tensile stress, the criteria are minimal in the centre of the plate; if it is loaded with compressive stress, in the corner. Finally for each criterion, stresses are expressed by:

P0

(15)

o'i/ = --~ parij

where parij depend on the stress type (transverse,...) and on the lamina. Substituting expression 15 in eqns (11)-(13) leads to an explicit form of the limit state functions. 4.1.2 Results The reliability calculations have been made with C O M R E I A . 0 [19] software. Table 2 shows the different failure reliability estimations. The maximal call numbers to limit state functions (MCN) are also given for all methods. The probability estimation with the second m o m e n t method agrees with the reference method. The n u m b e r of calls is much lower than 100 000, i.e., the number of calls which has to be done in the Monte Carlo method. In fact, if the failure probability value ( ~ 1 0 -2) is not acceptable for a structure, often more than a million simulations have to be done. The number of calls for the F O R M method depends on the number of stochastic variables and on the shape of the limit state function, not on the failure probability value. The corrective methods such as S O R M or importance sampling method do not modify significantly the estimation given by F O R M . The scale of magnitude is the same ;and the different equivalent reliability indexes are almost equal. So the linear approximation of the limitt state function at the design point is acceptable. The importance sampling simulations are a good way to validate this approximation. The call rmmber of limit state function shows the calculation cost of each method. The reliability estimation with Tsai-Wu's criterion is smaller than the others, because of the difference between the compressive and tensile strength in the unidirectional material. It demonstrates the importance of the choice of the criterion. The more accurate the modellisation, the better the reliability estimation. Now, we apply the calibration process to a simple structure such as a pipe with features which allow an analytical solution. Then we apply it to a more general

structure using a numerical treatment. We are interested in the search of the partial safety factors that a designer must apply to material strength to obtain a reliable structure with a new material. Following Eurocode 1 [4], one may put some partial factors equal to 1.0 and include the required safety margin in another factor. In our case, this leads the loads and geometric parameters as deterministic. Tsai-Hill's failure criterion is retained for its simplicity, its interactivity and its coherence compared with Tsai-Wu's.

5 ILLUSTRATION OF CALIBRATION PROCESS 5.1 Calibration of partial safety factors for composite material pipe 5.1.1 Mechanical m o d e l The structure is a thin pipe made of glass/epoxy. Its middle radius R is much greater than its thickness e. The material is a woven fibreglass [+45°]. An axial compressive load F and an internal pressure p are applied to the pipe (Fig. 1). The thin cylinder hypothesis leads to the stresses in the axis (er,eo,z): pR O'oo = - -

(16)

e

F trzz - 2 z R e "

(17)

The stress transformation from (er,e0,z)into the material orthotropic axis (e,X,Y)gives: N ~rL=~rr=-e

and

T ~Lr=--

(18)

e

where N = (1/2)(pR + (F/2zR)) and T = ( 1 / 2 ) ( ( F / 2 r c R ) - p R ) . H e r e the feature of the woven

A-

e0/ \

Table 2. Failure probability of structure

maximal stress Tsai-Hill Tsai-Wu MCN

FORM

SORM

IS (reference)

2.95 10 2 2.95 10-2 1.43 10-2 46

3.35 10-2 3.96 10 2 1.62 10-2 491

3.30 10-2 3.77 10 2 1.62 10 -2 591

S4o° Fig. 1. Structure description.

C. Boyer, A. B~akou, M. Lemaire

278

Table 3. Features of random variables

Variable

Expected value

CV in %

Law type

RL

127 MPa 27 MPa

6 6

normal normal

Rcr

G(x)

,

e2-(RY---x)2

X

fibreglass and the load gives RL = R r and o'r (see Table 3). The failure criterion becomes: Fig. 2. Limit state function vs x (N > 0, T = 0).

G(RL, RLr) = e 2 -

-

.

(19) (N = 0, T > 0). We fnd:

5.1.2 Stochastic model The stochastic variables are normally distributed to allow an analytical solution. The target reliability index/3,arge, is equal to 3.8. So the failure probability is about 10 .4 as recommended by Eurocode 1. The optimization process adjusts the thickness e to obtain

/3 =/3,o,ge,.

T

/3 = ~3target• t - eRL----r- 1 -/3CVLr.

General loads N > 0, T > 0: the previous results introduce two non-dimensional parameters n and t. The limit state function becomes: n 2

5.1.3 Solution Gaussian law allows an analytical transformation of the basic variables.

RL - Rc x--R----[L. CVL

and

y

RLr -- RLr RLr. CVLr"

(20)

The limit state function in the standardized space becomes:

(1 + x. CVL) 2 (1 + y . CVLT) 2

.

G(x, y) = e 2

(1 + x. CVL)

.

.~/

(21)

(22)

x < - - 1 / C V L has no physical meaning because RL > 0. Moreover the design value must be smaller than its expected value, otherwise the mean point is in the failure set. So x must be negative. The limit state function looks like Fig. 2. G(x) is zero at the design point. It leads to/3 value:

(l+y[-~VLr)2 ].

vl 1 -

t -

n

2

1

j2.

(23)

A numerical search on x is carried out to obtain /3 = min(d). Then e is adjusted to obtain/3 ~3target" It is straight forward due to the regularity of function d. Calculation of partial safety factors. For practical reasons, the parameter r = n/t is introduced. Results are presented for a set of r values with n constant. To simplify partial safety factors calculation__z, the characteristic values are the expected values R/rand Rcr. So the expression of the safety factors becomes: =

1

1

TL - I + x*.CVL

t-vx

"

N

/3 =/3targe, • n = - - = 1 -/3CVL. eRL

(24)

For a fixed value of N, the thickness e must be adjusted to satisfy the equation n = 1-/3CVL. Then the thin cylinder hypothesis must be controlled. The same calculation could be made for the load case

(27)

( I + xCVL )

and

TLr--I+y*.C"r'VL

(28)

I

/3 = ~-V-]L 1 - e-~L

(26)

In the standardized space, the design point is on the limit state surface G(x, y) = 0 with x E ] - 1/CVL, ~[ and with y E ] - 1/CVLr, ~[ (Fig. 3). The design point P* minimizes the origin-to curve distance d. Using the eqn (26), it can be deduced that:

d2 = x 2 + C V L T

The analysis of particular cases allows us to specify the method and to fix the upper bound of the generalized forces N and T Particular load N > 0 , T = 0: this load corresponds to that found in a thin tank.

t2

G(x,y)=e2[1-(l+x.CVL)2

1

G(x, y) = e 2

(25)

~i

i Safety

--~ '

Failur~

!~ (l-t>~,,r >

~/'

~Y*

i

I

x:

~'- ':

Fig. 3. Limit state function G(x, y) =0 (N > 0, T > 0).

Design of a structure to specified level ..........

' '--~"l J t

1.25 t 1.2 1.15

"~LT

-

-

7L ---

1.1 1.05

,"/ . ~ ,

1

ill

0.01

0.1

1

|

i

10

y

.....

100

Fig. 4. Safety factors YL and YLr VS r.

Figure 4 shows the existence of the point of inversion of the safety factor values. The value of r is 1 at this point. This means that the two loads have the same importance in Tsai--Hill's criterion. There are three sets of r values. In the first, YL and YLr are almost constant and equal to 1 and 1.3 respectively. In the third, the values are swapped over. An approximately linear relation between YL, YLr and r can be found in the second set around the r inversion vahle. Now we shall try to evaluate the different parameters influences on the safety factor value on a more general structure. 5.2 Calibration o f partial safety factors for thin c o m p o s i t e material structures

Many composite structure',; such as plates or pipes are thin structures. In thin structure theory, normal stress is insignificant. So, we consider a thin composite material structure with a thickness e submitted to longitudinal L, transverse T and shear S generalized stress in its orthotropic axis. Tsai-Hilrs criterion is kept as the limit state function. Nevertheless, any other criterion could be used. Apart from multi equation criteria such as I-[ashin's or maximal criteria, the various failure modes must be taken into account as parallel or series system elements [2, 3, 21, 17]. The material is the same as in the first application. Calculations have been made with tensile strength but the features of loads do not matter for this application. Obviously a buckling failure needs to be considered in some cases when there are compressive loads. The previous paragraphs lead to the introduction of two non-dimensional parameters r~ and rE: N rl

eRr T eRr

These parameters are a good way to measure the relative importance of each term of Tsai-Hilrs criterion, N2/(eRD, T2/(eRr) 2 and S2/(eRLr) 2. For unidirectional composites the coupling term (NT)/(eRL)Zis negligible in all load cases because of the great difference between RL and Rr. For a balanced composite, this term could be important so it must be introduced in r~ and r2 with RL terms. The parameters are studied in [0.05,20] interval, so all load cases are listed if we consider that terms lower than 5% of the maximum value are considered insignificant. For extreme values of rl and rE, Tsai-Hill's criterion is close to the stress maximal criterion: --(rl; r2)= (20; 20) correspond to the criterion for longitudinal stress. --(r~; r2) = (0.05; 20) correspond to the criterion for transverse stress. --(r~; r2)= (20; 0.05) correspond to the criterion for shear stress. So the calibrated safety factors may be used for some other simpler design criteria in these extreme cases. If not, the adaptation of the safety factors to design criteria may be undertaken. So the safety factors for two different criteria are not interchangeable in the main set of rl and r2 values. For each criterion, a calibration ought to be done and their own safety factors should be calculated to take the different parameters and the considered failure scenario into account. 5.2.1 Calibrated factors and their variations with stochastic parameters The safety factors YL, YT and Yrr vary with the non-dimensional parameters r~ and rE (Figs 5-7). As in the previous results, the set of r~ and r2 is divided into three parts. Each factor is equal to 1 in two parts and to a constant in the third. A sheer drop transition exists between them when rl or r2 are close to 1. These parts correspond to the predominance of a term in Tsai-Hilrs criterion: --r~ and r2 greater than 1, predominance of R L term --rl smaller and r2 greater than 1, predominance of Rz term --r~ greater and rz smaller than 1, predominance of RLr term

N NRr TRL

and

eRr NTLT r2 = S SRL eRrr

(29)

279

10 10 Fig. 5. Safety factor YL VS rl and r2.

280

C. Boyer, A. BOakou, M. L e m a i r e

4i

111

3.5 0.1

0.1

0.1r

0.1 l

[

~

~

~N 10

10 10 Fig. 6. Safety factor Yr vs rj and rz.

Fig. 8. Reliability index fld~,VS r~ and r2.

In these three sets, the safety factor of the predominant strength i is equal to a value ~i_max o Ti_max=max(Ti)

with

iE{L,T, LT}.

(30)

rl,r2

In the set where R+ is not predominant, each y+ is equal to 1. The different Yi-max are equal in Figs 5-7, because of the equality of the different coefficients of variation (5%). As in the previous application, the safety factors vary sharply at the bounds between the sets. A little variation of the non-dimensional parameters of about 5% leads to a large variation in the value of the factors. It explains the chaotic aspect of this area. Shao et al. [14, 16] have proposed methods to solve these kinds of problems. We have also represented the values of fide, which is the reliability index calculated using constant safety factors (ys, = Yr = Ycr = 1.4) (Fig. 8). Further, the difference d expressed as a percentage of the thicknesses (design variable), e, calculated before and after calibration, is plotted in Fig. 9. For these two values, the existence of the three subsets is also clear. These figures show the circumstances for which a probabilistic design must be performed and the interest of such a design. Hence, no reliability calculations are needed in any of the three subsets. The convergence rate is good and the second moment methods give good approximation of the failure

1.4 1.3 "~LT 1.2 1.1

probability. The use of the calibrated safety factors makes the design and the reliability index firm and optimizes the thickness. It may be also cost effective. But at the bounds, the safety factors must be used carefully because of the poor convergence rate. Hence the limit state function is strongly non-linear. It may be safer to use a corrective model factor. A finer probabilistic calculation with simulations may be interesting because the variation of the optimized thickness and reliability index are higher at the bounds. Here, with a coefficient of variation of 5%, fidet is equal to 5.9 and d is greater than 21%. Such a saving in material or weight may be highly cost effective. The difference between the deterministic and probabilistic design methods are all the more important as the coefficients of variation of the stochastic variable are great. The more scattered the strength, the more reasonable and the safer the use of probabilistic methods or calibrated safety factors is. It is also important to know how the stochastic law parameters influence the safety factors. Figure 10 shows the difference expressed as a percentage between YL calculated with two different expected values of R+. (1.99 GPa and 1 GPa). So, a variation of more than 50% leads to a maximal variation close to 6% at the bound between RL and RT subsets because of the coupling term in Tsai-Hill's criterion. In the rest of the set, the difference is not significant, y.+, presents exactly the

-15 0.1 0.1

0.1

Fig. 7. Safety factor YLr VS r~ and r2.

0.1

-'~'~:~_

/

1

rl

10 Fig. 9. Difference expressed as a percentage between the thickness e,l,calculated with constant safety factors and calibrated safety factors.

281

Design o f a structure to specified level 4.5 IZ % 3.5 k 0

1

5

~<~::z'-.~

~"~

. e ~ g ~ . ~ ; ~

',

0

.

1

10

~

0.1

10

Fig. 10. Variation expressed as a percentage of the safety factor YL for a variation of the expected value RL (1 and 1.99 GPa). same variation with a negative sign. YLT does not vary throughout the set. So, if the material is unidirectional, i.e., the coupling term is low, and the observations about the bounds are carefully examined, it is possible to use calibrated safety factors whatever the expected value of each variable. On the other hand, the safety factors for quite balanced composite materials for which the coupling term may be high and RL is not equal to RT (i.e., different from the first application case), may vary qualitatively at the bounds with expected value of strengths RL and RT. Nevertheless, the m e t h o d has a wide application for unidirectional composites. The safety factor variation with the second parameter of Weibull's law is also interesting for the designer. Manufacturing improvements or new raw materials may strongly modify the coefficient of variation of the material strengths. So the designer must be able to adapt the safety factors without carrying out a new process of calibration. We obtain the same shape as for 3', vs r t and r2, if we plot the difference between the R~ safety factors calculated with two coefficients of variation vs r~ and r2. In the predominant R i subset, the difference is constant and equal to the difference between the y~-m~, anywhere else the difference is zero. The shape of 3' does not vary with the coefficient of variation except for the maximal value 3'. . . . The influence of change in Ri second parameter on other strengths' safety factors is also interesting. Figure 11 plots the ,difference expressed as a

percentage between two YL calculated with (RL, R r , R L r ) coefficients of variation equal to (5,5,5) and (5,6,5). This difference for a little variation is important at the bound of the subsets (between 10 and 20%) and insignificant anywhere else. For the same variation of the R r and R L r coefficients of variation, the difference is not symmetrical because of the coupling term. The level of difference for small variations shows the great dependence of safety factors with Weibull's second parameter. From the previous results we should be able to predict the values of Y~-,,ax vs the R i coefficient of variation. Figure 12 allows us to adapt the value of Yi-max for three current distribution laws over a wide interval of the coefficient of variation. Yi-,,ax calculation is easy since only one stochastic variable influences the structural reliability. So the reliability index could be written as a function of standard coordinate ui of Ri [13]: fl = < u*,u* > 1/2 ~. +u*.

(31)

As the reliability index is linked to one standard variable, the inverse transformation T -1 from standard variables to physical ones leads to the expression of yi_,,~x expressed as follows: Rik Rik ~/i- max = ---- -. R~' T - ' ( - ~target)

(32)

If R~is the expected value, 7i-max becomes for the normal, log-normal and Weibull laws:

1 3'i

max

for normal law

~

1 - CVi. fitarget

(33)

7i-max = V I + C V 2 . exp(/3,~rge, • Vln(1 + CV2)) for log - normal law

(34)

~'~-max -- ( - - I n ( 1 -- 6 ( -- ~,or~e,))) t ~ " ~ )

for Weibull's law

(35)

where rl(CV~) is the Weibull's second parameter in terms of coefficient of variation and F is the G a m m a function. The y~_,~,, expressions prove its independence from the expected value of stochastic variable. -20

0.1

O. 7"2

1

5.3 Summary of design process

-"~_....~ lO

i0

Fig. 11. Variation expressed as a percentage of the safety factor YL for modification of the Rr coefficient of variation (5 and 6%).

From the previous results, the steps of a structure design process can be stated as follows: - - T h e specifications give the structural loads and

282

C. B o y e r , A . B ~ a k o u , 0

I

4.5 -

I

I

I

M. Lemaire I

. . . . . . inor-m~t taw::. . . . ..... l o g } n o r m a l law! ........

I

: i

I

i :

: Ji/"~

4 3.,5 "~i--max

3

2.5 2 1.5 1 2

4

6 8 10 12 14 16 coefficient of v a r i a t i o n in %

18

20

Fig. 12. Safety factor Y~-,,,,x vs coefficient of variation of the strength R~ for three kinds of distribution law.

Table 4. Summary table of safety factor values

rl

0.01

1

100

r2 ~ll = l

0.01

bounds Yr = 1 ~ILT

:

~LT

max

1

7L = 1 100

")sT= Yr-max

TL = YL-m,× TT = l

TLT" = 1

Yt.r = 1

the failure scenario. Material strengths, geometrical values and mechanical models are chosen by the designer. So the generalized stresses N, T and S are calculated. - - T h e non-dimensional variables r~ and r2 are determined from N, T and S and material strengths. - - A c c o r d i n g to r~ and r2 (Table 4) and the coefficient of variation (Fig. 12), the choice of safety factors YL, TT and Yt_r and other factors usually applied in modelization, loads and geometry. If the design takes place at the bound of a subset, the use of charts YL, YT, Y t . r in terms of rt and r2 will be of great engineering significance. - - S a f e t y factors, characteristic values and the limit state function lead to final determination of design variable value.

6 CONCLUSION The first two parts deal with calibration and stochastic methods. Next, three design criteria are compared using various methods: F O R M , SORM and Importance Sampling. First order approximation leads to acceptable results and corrective methods allow us to verify this first approximation. Several calculations have shown

that the scale of magnitude is respected for all approximations. Nevertheless the number of limit state function calls is quite limited for the first order method in relation to the Monte Carlo Method. Moreover, all design criteria give the same order reliability evaluation. So, Tsai-Hill criteria and the F O R M method are retained to calculate safety factors but calculation may be extended to any explicit design criterion. Then we presented two applications of safety factor calibration method for the design of composite materials structures from the probabilistic method. The first application was an example for a particular structure with an analytical treatment of the problem. It is an introduction of the following more general application. The process is then easier to understand and non-dimensional variables are introduced. The second one shows the existence of the three subsets of r~ and r2 for the safety factor value. Generally the reliability of structures depends on one strength even if all strengths are stochastic variables. So this proves the interest of the safety factor method on the main set of rl and r2 values where safety factors are fairly constant and equal to 1 or a value yi ........ No expansive probabilistic methods need be applied and y i - , .... is easily determined as a function of the strength Ri coefficient of variation. The study also shows yi_, ...... independence of expected values and the coefficients of variation of other stochastic variables. Nevertheless, the probabilistic methods should be applied at the bounds of the subset because of the great instability of safety factors and their dependence on the coefficient of variation and the expected value of the other stochastic variables. Use of these methods may ensure the design and make a significant saving on its structural weight and cost. Obviously the validity of results depends on the accuracy of mechanical and stochastic models and data. Depending on his line of manufacturing, each supplier may justify his own safety factors and establish the

Design o f a structure to specified level p a r a m e t e r which may be modified in order to improve his products.

ACKNOWLEDGEMENT O u r thanks go to M m e S. D a v e y for her linguistic advice.

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