On the reliability of Daniels systems

Structural Safety, 7 (1990) 229-243 Elsevier

229

ON THE RELIABILITY OF DANIELS SYSTEMS

*

S. Gollwitzer RCP-Consult GmbH, 8000 Munich, Fed. Rep. Germany

and R. Rackwitz Technical University Munich, 8000 Munich, Fed. Rep. Germany (Accepted May 1985)

Key words: reliability; redundant structures; system reliability; brittle failure; non-linear

member properties; load redistribution.

ABSTRACT The available theoretical results on time-invariant and time-variant Daniels systems with emphasis on asymptotic solutions are first reviewed. It is found that the methodology for static Daniels systems is well developed. Further research is necessary for time-variant Daniels systems where the model for the failure process includes multiple componental failure, strength deterioration caused by load-induced fatigue and the dynamics of load effect redistribution. From numerical studies for small systems some conclusions concerning the extra reliability due to redundancy in general systems are then drawn. Significant extra reliability in small systems is, in fact, only available if the components are weakly dependent, have somewhat ductile stress-strain behavior and if the load variability does not exceed substantially the variability of strength.

1. INTRODUCTION In 1945, H.E. Daniels [1] published a paper with the title " T h e statistical theory of the strength of bundles of threads", which initiated a new branch of research in statistical strength theory paralleling another development started by Weibull in 1939 with his famous paper, "A statistical theory of the strength of materials", based on the weakest-link concept [2]. The mechanical system investigated by Daniels is shown in Fig. 1. * Presented at the Workshop on Research Needs for Applications of System Reliability Concepts and Techniques in Structural Analysis, Design and Optimization, Boulder, CO, September 12-14, 1988. 0167-4730/90/$03.50

© 1990 Elsevier Science Publishers B.V.

230 /22_1

/

/1

/ II1

I I I I

/

i/I/i/iIiIli

l,l i )t/x2 Fig. 1. Daniels system.

The load acts quasi-statically. Daniels assumed independently, identically distributed componental strengths Xi (i = 1 .... , n), a constant modulus of elasticity and perfect so-called equal load-sharing among the unbroken fibers (components, elements, members etc.). Furthermore, the fibers behave ideally brittle. Obviously, the system strength R , under these conditions is n

R,,=max((n-i+ l)Xi}

(1)

where the 2 i's are the ordered elemental strength values of Xi such that X] ~< -~z ~< --. ~< )(,- The system failure probability becomes

Pf=P(R.<~S)=P

{(n-i+l).~i-S~O} i=

tl

~< r n i n P ( ( ( n - i + 1).("i - S ~ 0 } ) i=1

(2)

The second line corresponds to the "strongest" component and represents an upper bound for parallel systems which, unfortunately, is rather conservative in most cases. The contrary case of ideal plasticity is trivial because the system strength R , then is just the sum of componental strengths. For large systems, the central limit theorem holds for the distribution of system strength under suitable conditions (Liapunov conditions). Daniels not only found a recursive scheme for the determination of the probability distribution of system strength which later could be rearranged by several authors to improve its numerical performance. It still is the only compact, exact solution known for a redundant structural system with non-perfectly ductile elements. He also derived the Gaussian distribution as a limiting distribution of system strength for n ---, oo. Although directly a realistic model only in a few cases as, for example, for parallel wire cables, for fiber-reinforced composite materials with a soft matrix or for certain fastening structures, the Daniels system has found repeated interest from engineers, materials scientists and statisticians for various reasons. In what follows the available results are first reviewed with the explicit intention of identifying "flaws" in the theory or where it is still incomplete. In the second part some numerical results for small systems will be presented enabling some general conclusions on the general aspects of structural system reliability.

2. TIME-INVARIANT DANIELS SYSTEMS The statisticians were interested especially in the asymptotics of such systems and in weakening the rather restrictive assumptions originally made by Daniels on the stochastic model.

231

For example, Daniels himself obtained an improvement for the mean of the limiting distribution maintaining the original assumptions on the stochastical and mechanical model [3]. Because his result is considered so important, it is given here for easy reference. The component strength is distributed continuously as F x ( x ) with x > 0 and Fx(O) = 0. Then, Daniels' asymptotic result is [1] lim P( R , <~s) = d~[ s -

E. 1

(3)

with E, = nx0(1 - Fx(xo) ) + C,

(4)

D, = xo[nrx(xo)(1 - Fx(xo))] 1/2

(5)

and x 0 the maximum point of x (1 - Fx(x)) or the unique solution of a [ x ( l - Fx(x)) ] = 0 Ox

(6)

If, in particular, the Xi are Weibull-distributed according to F x ( x ) = 1-exp[-?~xt~], one determines x 0 = (?~fl)-a/a. The basic result is best interpreted by observing that (1 - F x ( x ) ) is the proportion of unbroken fibers at level x, and nx(1 - Fx(x)) is their minimum strength in the sense that larger strengths than x are set equal to x. Eqation (6) maximizes this strength. But the actual number of fibers for a given proportion F x ( x ) of failed members is random. It is binomiaUy distributed and, according to the central limit theorem in the Moivre-Laplace version, asymptotically Gaussian. The correction term, C,, derived later by Daniels using quite different statistical arguments is

[31 C, = 0.966n~/3a

(7)

where

a 3 =f2x(Xo)X~/(2fx(Xo) + Xofx(Xo)) Barbour [4] improved the variance of the limiting distribution, but his improvement is numerically less important. Sen and Bhattacharyya [5] found that a limiting Gaussian distribution is also obtained if the strengths of the components are dependent but fulfill certain mixing conditions. Later their result was generalized to so-called continuous systems, i.e. where the strength is a continuous, homogeneous process or field over a certain domain [6]. Hohenbichler and Rackwitz [7] introduced a special type of strong dependence and found that Gaussianity will not be reached in this case. For example, let the fiber strength be given by X i := X 0 Xi, where X 0 is a variable with given distribution which is common to all fibers. Then, the distribution of R n will be dominated asymptotically by the distribution of X 0. An important mechanical generalization was achieved by Phoenix and Taylor [8]. They k e p t the independence assumption but adopted relatively general force-deformation curves for the properties of the components and again were able to prove a limiting Gaussian distribution for the system strength.

232 Harlow and Phoenix [9] developed not only results for so-called local load-shearing rules based on the earlier proposals of Rosen [10] and Zweben [11], but also extended the theory of chains-of-bundle systems (see also Refs. [12,13]). In both cases the limiting distribution now becomes an extreme value distribution. Local load-sharing rules (stress concentrations around broken fibers) imply a tendency to progressive failure once the weakest fiber is broken and, therefore, the Weibull-distribution is a natural candidate of system strength given that componental strength has a power law behavior of Fx(x ) in the lower tail. Then, the asymptotic strength of a chain-of-bundle system should also be Weibull-distributed. This could, in fact, be shown and we refer to Smith [14] for a review and a discussion of the relevant literature. If, on the other hand, there is equal load-sharing in the bundles, the asymptotic strength is Gumbel-distributed [13]. Much of the work just reviewed was motivated by its potential for application to a statistical theory of uniaxial strength of fiber-reinforced composite materials and a number of comparisons between experiments and theoretical predictions support the validity of the models. The authors are inclined to believe that similar arguments, particularly those for the chain-of-bundle model with local load-sharing, are also suitable for other materials such as concrete, with the necessary modifications. But detailed studies are still missing. Whereas asymptotic concepts appear to be appropriate when deriving statistical strength models because n can be assumed to be large, this is rather questionable for small systems. In fact, convergence to the asymptotic distribution is rather slow even in Daniels' original ideal case (see Smith and Phoenix [13] for some interesting theoretical observations concerning the rate of convergence and the tails of the asymptotic distribution, and Hohenbichler and Rackwitz [15] for numerical comparisons). Quite another route of research focused on small to medium systems with the intention of studying structural redundancy for non-ductile materials in general. Daniels' model actually appears to be very attractive for this purpose due to its simple mechanics. From a probabilistic point of view Daniels' model, however, is rather complicated because of its many possible sequences of componental failures all equally important to system collapse. It is also an extreme case for a number of reasons. The most important is that the Daniels system is the system with the largest possible effect of redundancy on reliability due to the equal load-sharing rule. Not surprisingly, the Daniels system has frequently been used as a demonstration example in structural system reliability studies and we will mention only a few references of direct interest in the context of this paper. For example, Shinozuka and Itagaki [16] found the transition probability from one state of the system into the next. Kersken-Bradley [17] applied the model to structural timber, concentrating primarily on a description of system strength by its first and second statistical moments. In Ref. [15] a formulation amenable to the application of modern f i r s t - a n d second-order reliability methods ( F O R M / S O R M , see Refs. [18,19]) was presented. It is based on the so-called order-statistics approach used in eqn. (1), which is implicit already in Daniels' early work and which has repeatedly been used later on (see also Refs. [20,21], where certain special but practically important cases are treated such as random shapes of the componental force-deformation curves, "slip" or "slack" in the anchorage of the components, the existence of defect components, etc.). Gollwitzer [22] carried out numerous numerical studies with particular reference to the circumstances in redundant fastening systems. Those results will be discussed below in more detail.

233

3. TIME-VARIANT DANIELS SYSTEMS The late 1950s saw a number of papers by Coleman (see, for example, Ref. [23]) where the fibers were allowed to suffer strength deterioration with time especially due to load-induced fatigue. The simplest model assumed an exponential distribution of the time to failure of a component

P(T<~t)=

1- exp[-f0t~(s('r))de]

(8)

where x(s(¢)) is the hazard rate which here does not depend on the load history s(¢) prior to time t. In particular, Coleman used the so-called power law breakdown rule

tc(s('r))=(s/lo)°;

10>0, 0>~1

(9)

which appears to be realistic for static fatigue under sustained loading. Assume now that initially the load per fiber is s, where s = S / n . After the failure of k - 1 fibers the load in the remaining fibers has increased to n s / ( n - k + 1). Therefore, the hazard rate for the time T~, k simply is the number of surviving elements multiplied by the hazard rate for one fiber X..k= (n-- k + 1 ) ~ ( n s / ( n - k + 1))

(10)

The corresponding time to failure has an exponential distribution due to the lack of memory

P ( T . , k ~< t) = 1 - exp[--X.,kt ]

(11)

Coleman then assumed that the time to bundle failure is n

T~=

E

T~,k

(12)

k=l

where the T.,k'S are independent. It is important to remark that this implies'that the components fail one after the other and one at a time. The times Tn, k do not depend on the age of the structure. According to Liapunov's form of the central limit theorem one then has

p(T
as n ~ oo

(13)

with Ell

~

( lo/s ) ° P

-

(t0/s) p

Dn= ( n ( 2 0 - 1 ) ) 1/5 after some algebra and reductions valid for n--* oo. If the load is a stationary and ergodic process, (¢), so that fatigue under spectrum loading must be considered, the breakdown rule may be replaced by

= (e and a crude approximation for the system probability is obtained by making also the appropriate changes in eqn. (13).

234

It is seen that one again obtains a Gaussian distribution, but on quite different lines of thought. Surprisingly, E n does not depend on n, which must be attributed to the progression of the failure rate in time. Coleman already recognized that the exponential distribution is not very realistic in many cases and proposed the following as an alternative to eqn. (8)

P(T t)=FT

K(s(

(14)

Phoenix [24] was able to derive the asymptotic distribution for the time to failure of a bundle of threads with equal load-sharing as a normal distribution also in this case. Because the failure times now are no longer independent, this has been a much more difficult task than for the original model. The same reference gives formulae for the mean and variance of the system failure time and discusses in some detail the cases of static fatigue, of so-called stress-rupture and of high-cycle fatigue. The formulations are in the so-called damage-indicator space. Multiple failures at a time are not taken into account. Also, the damage history can be considered only for the time T,, k. To some extent the same way of reasoning as before was also successful for chain-of-bundle models and local load-sharing for this more general model of the failure process [13,251. In all the studies reviewed so far the loading is quasi-static. If it causes dynamic effects in the structure the reliability analysis still could essentially proceed as described before. The componental failure event can be the crossing of the resistance level by the load-effect process. For example, Grigoriu [26] still used the original "break-down law" proposed by Coleman but assumed dynamic system behavior. But dynamic effects are also present during load-effect redistribution even if the time-variant load does cause only negligible dynamic load effects. The magnitude of the dynamic effects during load redistribution depends on the system state, on system damping and on the energy absorption during rupture of a component. For ideal brittle materials there is no energy absorption and system damping usually is small. Then, the ultimate dynamic effect in static systems is determined by reducing the strength of the system by the strength of the component which just failed. Eqn. (2) is modified into

pf=e(R° s)--e Nl{(n-k÷l) k- k 1-S 0}

(15t

Inspection of this equation reveals that for n ~ oc the quantity Xk-~ vanishes since for the dominating failure mode, i.e. with k 0 = nFx(xo) failed components, the first term, (n - k 0 + 1) J(ko, is much larger than Xk0-1- Therefore, dynamic effects during load-effect redistribution need not be considered asymptotically. The contrary is valid for small systems as shown in Refs. [27,28]. See also below for some numerical results. Recently, Fujita et al. investigated the dynamics of the Daniels system especially during load redistribution in some more detail [29,30]. They found that the concept underlying eqn. (15) is realistic for small system damping (~< 3%), conservative for medium damping, and far too conservative for high system damping (>t 8%, say). In this case any dynamic overshooting can be neglected. Guers et al. considered the details of the time-dependent failure process of a Daniels system which is necessary if one wishes to include inspection and repair into the analysis (see Refs. [27,28,31]). They further considered the possibility of more than one component failing "in a large load-wave". As an alternative to an investigation of the entire set of sequences of componental failure to system collapse, they proposed investigating only the most important failure sequences in line of the so-called branch-and-bound methods used elsewhere in structural

235 reliability. They showed that fairly narrow reliability bounds can be achieved by this method. Furthermore, in [27] a fracture mechanics approach for the degradation of strength in the components was considered. This makes the individual failure times load-history dependent. All these studies essentially rest on a formulation similar to eqn. (12). The individual times to failure are determined by using the theory for rare crossings of resistance thresholds by ergodic load-effect processes. The approaches mentioned in Refs. [27,28,31] are highly numerical due to the considerable complexity in the model for the failure process in the system and the investigations are not yet completed. They are designed for more general systems. Asymptotic results such as for the simpler models for the failure process of the system are not yet available and appear difficult to obtain. Numerical results have been produced only for relatively small systems (see, however, Ref. [32]). The results available so far indicate that there is a relatively small likelihood of having multiple failures in "large load-waves" if the load-effect redistribution in the system takes much longer time than a typical period of the "load-wave" (delayed redistribution). If this is not the case (immediate load redistribution) and, especially, if the dynamics of load-effect redistribution must be taken into account because componental failure is brittle and the system damping is small, multiple failures and progressive collapse are fairly likely. This implies that the effect of redundancy is almost vanishing and system failure must be considered as being nearly identical to the failure of the weakest component in those cases. It remains to be shown that for componental failure processes which are not perfectly brittle there is still some extra gain in reliability by redundancy. Although this review of the main steps in the development of a probabilistic theory for the Daniels system is not complete and is biased by the authors' own evaluations and more recent work, it might demonstrate that this system is one of the most studied systems in structural reliability theory. The results obtained are relatively rich, and may be richer than for any other system.

4. NUMERICAL RESULTS FOR SMALL DANIELS SYSTEMS In the remainder of the paper we will concentrate on formulations and results for not too large Daniels systems with an equal load-sharing regime and whose components do not deteriorate with time. Furthermore, dynamics can be neglected except, possibly, for the phase of load redistribution after componental failure. Numerical results for this system with fairly general mechanical and stochastical characteristics for the components will be used to draw some general conclusions on the effect of redundancy on structural reliability. Extensive numerical studies will be reported partially based on the material in Refs. [15] and [22]. A numerically feasible formulation is particularly simple if the stochastic dependence between the different components can be described by a common set of variables. This type of dependence may be called "equidependence". Then, the order-statistics approach as in Ref. [15] is still applicable after some modification (see Ref. [27]) in conjunction with m o d e m F O R M / S O R M techniques for the determination of the resulting probability integrals. For an arbitrary force-deformation curve, as in Fig. 2, the componental failure event for a given imposed deformation 8 is

R,(8)-s

o

(16)

236

ti, Rsys

.R 1 I

I

) Y-6

Fig. 2. Typical componental force-deformation curves.

where S denotes the possibly uncertain load and Ri(8 ) denotes the uncertain componental force at deformation 8. System failure occurs if the maximum system resistance is exceeded by the load which can be described as follows F~ys=

max E R , ( 8 ) - S ~ < 0 (8) \i=1

=

ER~(8)-S<~O

(17)

i=1

Figure 2 gives typical force-deformation curves of components together with the curve of system resistance Rsy s = ~R i. For a finite number of components it is always possible to define the order statistics (Y1,..., Y,) for the vector (II1. . . . , II,) of deformations where the components reach their maximum bearing capacity, X~. In this case formula (17) can be rewritten as

Fsys~{minalx(Ri(Yi'Qi) + ~ Rk(Yi' Qk)) -

(18)

where the Qi's denote random vectors describing further properties of the force-deformation curve. The inequality sign now reflects the possibility of a larger system resistance for deformation states in between [ ~ , ~ ÷ 1 ] for i = 1 . . . . . n - 1. For the following parameter studies the components of the system are designed for a reliability index of flk = 2.0 as if no system effect exists.

4.1. Ideal brittleness and Ideal plasticity in independent components as limiting cases The foregoing formulation can easily be specialized to the case of ideal brittleness and ideal plasticity. Clearly, ideal brittleness together with a linear elastic behavior in non-failed componental states corresponds to the least extra reliability provided by redundancy. On the contrary, ideal plasticity provides the largest extra reliability achievable by redundancy. These statements hold for all possible dependencies between the variables characterizing the components. Figure 3 first demonstrates the influence of the mechanical behavior of the components on system reliability for independent properties of the components. In this figure the system reliability index, flsys = - ~ - l ( p ( F s y s ) ) versus the number of components is given. Two limiting

237

parallel system ideal ductile

J

fJ sys

I

i m e d i u m ductiLe

6£,

J

~ m e d i u m bri

e

50 /

,'.l,

brittle

3.(.

/

/

J

20

---.... 1.o

/

j

/

ideat elasti brittle

---...._

ideal series, system 3

5

10

/5

)n

Fig. 3. System reliability index versus numberof components for different force-deformation curves. curves are also included, corresponding to an ideal series system and an ideal parallel system and whose failure events are given below for easy reference n

//

= U F,.,

= f"1 F,

i=1

i=1

Note that the ideal parallel system with so-called hot redundancy has no plausible mechanical interpretation. The three other curves correspond to different degrees of ductility (see below). As expected, the reliability of the series system decreases with n. The reliability of the parallel system increases significantly with n. For the brittle system one observes first a decrease of reliability below the reliability level of a single component. Only for a larger number of components is this level exceeded by the system reliability and a significantly larger degree of redundancy is necessary to produce higher reliabilities, but at a much smaller rate than for the more ductile systems. Obviously, for a small number of components the brittle Daniels system behaves like a series system. This is interpreted as follows: if the weakest component breaks, it is unlikely in a small system that there are resistances of the remaining components which can carry the extra redistributed load. This means that a small degree of redundancy does not necessarily produce extra reliability in systems with brittle components. It is better to design a single but strong component to carry the load rather than to distribute its material among several components. For larger componental variabilities this unfavorable "series system" effect vanishes, however.

4.2. Influence of the degree of componental ductility Next the effect of the degree of ductility is studied. We introduce the following measure of ductility m=

(So

28°R ((~) d8 --

2

/ ( R m a x 80)

(19)

238

~=Y= 1

19xy =1,9 k: o ]

5.o~ ideal parallel system !

t

ideal ductile

n/

i

3O

ideal elastic-brittte zOl ideal series system O01

/ O0

t

025

~

/

I i

050

I

075

ductility I

50

:

&

~25

Rrnox . . . . . . . ~6

6o 260 Fig. 4. System reliability index versus ductility.

It is further assumed that the maxima in the componental force-deformation curves are fully correlated with the deformation at that point. Figure 4 then shows, for a five-component system, the increase of reliability with ductility. It is recognized that the increase in the reliability index is roughly linear with ductility implying an exponential decrease of the corresponding failure probability up to relatively high ductilities around A = 1 beyond which the conditions for the fully plastic case prevail. One can conclude that only relatively little ductility will provide

13sys

l n=5,1] =2.0, g,y=l ]

z,.O

3.0

2.0

1.0

c

0

.....

0.25

0.50

0.75

1.0 K

Fig. 5. System reliability index versus correlation Pk of strength between components.

239

considerable extra reliability. It is easily visualized and can be demonstrated numerically that for smaller ductilities this positive effect, however, is true only if the variability of the deformation at the m a x i m u m force is small (see below).

4.3. Stochastic dependencies We now investigate the influence of stochastic dependencies within the variables characterizing the force-deformation curve of a c o m p o n e n t and between components. For example, if full correlation is still maintained between m a x i m u m force and deformation but a non-zero correlation coefficient is assumed for the m a x i m u m forces in the components, one can easily produce Fig. 5. It shows that the redundancy effect is largest for zero correlation and ideal ductility and vanishes for p = 1.0. For the elastic brittle case one observes that m e d i u m positive correlations can make the situation even worse for small systems. Again, any effect of redundancy vanishes for full correlation, which is as it should be. The reliability of the series system increases with correlation. In Fig. 6 one can recognize that correlation of the properties within a c o m p o n e n t appear to have relatively little importance, at least for brittle systems.

4.4. Relative influence of load and resistance variability It is also interesting to investigate the effect of different ratios of the coefficients of variation for the loads and the componental resistance. In view of Fig. 5 one expects a substantial decrease of reliability with increasing dominance of the load variability. This is, in fact, true as can be

,sys 8.0

~idea!.

80

porotlel system

7 0 . ~ 7 0 \

60' ,50.

50

4.0

40

30 20

10

_20

F idea[

_

_

_~<=_2.0

~0- series system

9 " Y % o - o ~ - a s - 0 2 5 o .~es.os .o75 .~o .9,~y Fig. 6. System reliability index versus correlation Pxy between maximum strength and corresponding deformation between components.

240 p

[

sys

I n=5,

Pk =2.0

V j : 9k 4.0

3.0

20

-Pk

~'~'---~--..4-~deal elastic - brittle series system

ZO

. 0

.

.

05 zo

.

.

20

.

>Vs -

3.0

4.0

50

-

vx

=

-

vs -

Vy

Fig. 7. Systemreliabilityindex versus ratio of load variability to strength variability.

seen from Fig. 7, where the extra reliability due to redundancy reduces to an insignificant amount for ratios of the coefficients of variation V s / V x larger than two, say. The same type of investigation also allows to quantify the relative effect of the coefficient of variation of the componental resistances. In Fig. 8 the coefficient of variation, Vx, is varied ( Vs = 0). One clearly sees that an increasing coefficient of variation has a positive effect on the extra reliability for both the ductile and the brittle systems. For the brittle system one can even see that the unfavorable effect of redundancy for small systems is less pronounced, or even non-existent. Under these circumstances, the remaining components in a Daniels system are likely to have sufficient extra strength to avoid progressive collapse once the weakest component fails. Of course, this does not mean that a larger strength variability is generally better than a small variability because the larger variability has to be compensated for by a larger mean in structural design.

n=5, 13K:2.0, g~=O,~y=l 1 Xi,Yi. : tognormat dist,

~sys

I/ideal .ductile I 4.0 ~

!/

u

c

2°I,' [/ 0.0

!J+

0.2

t

ile

~

I I ,deose,es s te 0.4

0.6

0.8

10 -'vx

Fig. 8. System reliability index versus componental strength variability.

241 /gsys

50

/f-ideal

ductile

4.0 3.0

~ m e d i u m

brittle

20

n=5

, pk=ZO

qk =0, 9xy = O

50

~=sv

Yi = lognormol

)

q

o oh5 d~o oi~5l~I' o'.3o o'.~o O~

O~

Q~

Fig.9. System reliability index versus coefficient of variation of deformation at maximum componental strength. Another important factor for brittle systems is the coefficient of variation of the deformation at maximum strength. From Fig. 9 one can see a dramatic decrease in reliability with V r . Fortunately, this parameter does not vary too much for most materials.

4.5. Dynamics of load-effect redistribution Finally, the dynamic effect is quantified in Fig. 10 according to eqn. (15) which is seen to be significant even for relatively large systems.

s~ffsys l,.O g.,=l, 9,=0 I~, =2.0

I / 2.0

~

1.0

Fig. 10. System

1

"

J

5 10

,,Z~deol

eUostic-

brittle(dynomi

t 20

50

100

reliability index versus numbers of components with and without dynamic effects redistribution.

during

242

5. DISCUSSION AND CONCLUSION The above results and further parametric studies not presented herein allow us to draw some general conclusions. It is important to bear in mind that the Daniels system, as defined above, must be considered as the system where structural redundancy produces most extra reliability due to its equal load-sharing regime. However, if the correlation between components is high a n d / o r the coefficient of variation of the load significantly exceeds the coefficient of strength, the gain in reliability by redundancy soon becomes insignificant. If, on the other hand, there is insignificant correlation between components and the coefficient of variation of loads and resistances are comparable in magnitude, the amount of extra reliability depends on the mechanical behavior of the components. For elastic brittle behavior there is relatively little effect for small systems and the largest relative gain in reliability can be achieved for large coefficients of variation of strength. For small brittle systems there is even a negative effect of redundancy for small coefficients of variation. In small to medium size brittle systems the dynamic effects during load-effect redistribution must be expected to be non-negligible and can reduce reliability quite significantly unless the damping in the system reduces the dynamic effects to a sufficient degree. Only if the components behave in a ductile manner a considerable increase in reliability with the number of components can be expected. In summary, redundant structural systems do provide significant extra reliability only if the components are not highly correlated, if the load variability does not substantially exceed the resistance variability, and if at least a moderate degree of ductility is present. These observations limit the practical significance of structural redundancy with respect to extra reliability. If one now remembers that realistic structural systems tend to fall into the category of systems with distinct local load-sharing and which, therefore, behave like weakest-link structures, one is led to the conclusion that use of the extra reliability of redundant structures must be made with the utmost care. The factors in favor of extra reliability by redundancy, such as independence of componental resistances and ductility, need to be verified. In particular, the assumption of full plastic componental behavior made in many structural system reliability studies may lead to gross overestimations of system reliability.

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