A sensitivity analysis for various structural reliability models applied to timber beams in bending

Structural Safety, 5 (1988) 47-65

47

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

A SENSITIVITY ANALYSIS FOR VARIOUS STRUCTURAL RELIABILITY MODELS APPLIED TO TIMBER BEAMS IN BENDING C.B. Pierce Department of the Environment, Building Research Establishment, Princes Risborough Laboratory, Aylesbury, Bucks. HP17 9PX (Great Britain)

(Received November 1, 1986; accepted August 6, 1987)

ABSTRACT

The use of reliability methods is being advocated for determining partial coefficients for limit state codes of practice even though little fundamental work has been carried out on the effects of the many assumptions involved. Many research workers continue to use normal or lognormal distributions to represent both the applied load and material resistance mainly because these cases are the only ones for which simple analytical solutions are available. In order to examine this type of simplification, this study investigates several different distributional assumptions for load and resistance using the classical reliability approach applied to timber beams in bending. Solutions are developed using numerical integration techniques. Another aspect of the study concerns the sensitivity of the failure probability emerging from the model to the estimate of the population location parameter when the material resistance is represented by the 3-parameter Weibull distribution. The behaviour of the models in response to a simple duration of load assumption is also studied.

1. INTRODUCTION The aim of any structural engineer is to design structures that will not become unfit for use during their intended lifetime. The engineer uses the appropriate code of practice for his material or structural type together with a loading code to arrive at a design solution which is in line with good practice and previous experience. He must be aware that absolute safety can never be achieved, and that a satisfactory balance has to be made between adequate safety and the total cost of erection and maintenance of the structure. A structural engineer does not normally make his own assessment of the safety level required for a particular design, but by using existing codes of practice he has a design method which is based on previous experience and consequently has its own in-built safety levels. This design method reflects the successes (and failures) of previous structures of a similar type, and of course © 1988 U.K. Crown.

48 it poses great risks for the pioneers of new structural types and to designers of familiar structures who wish to extend their use beyond known limits. Design codes exist in their present form only because the engineering profession (and the public at large) are not dissatisfied with the failure rates of the structures covered by them, and they are constantly being reviewed and modified in the light of new data. In Limit State Design a more positive effort is made to identify all possible modes of failure and the designer takes all these into account to produce an end product with an acceptably low probability of failure. Except for very specialised structures where separate reliability studies are necessary (e.g. offshore structures), the designer will be concerned only with the deterministic design method itself, and will continue to assume that adequate safety levels are provided as long as he follows the appropriate code of practice. It therefore falls upon the code drafting committee to create codes which provide the necessary safety levels and yet are fairly simple to use. The current solution to this problem is to use a simple partial factor format so that the end products of design are very similar to what would have been produced by the pre-limit state codes. Reliability analysis may have a role to play in assisting code drafting committees to arrive at suitable values for the partial factors. For some years now considerable effort has been channelled into developing mathematical models for estimating the reliability of structures, particularly in North America and Europe. This calculated value of reliability generally relates only to the inherent risk of the design process even though the majority of failures arise due to errors or circumstances outside the provision of design codes [1-3]. Consequently, any theoretical value of reliability calculated from a model must be viewed with extreme caution: it is often termed a notional probability of failure, since it does not represent the "real" situation, only an idealised inodel. At the present time the study of reliability analysis is still in its infancy, partly because the models are still being developed but mainly because of the inadequacy of reliable and accurate d a t a - - o f which more will be said later. The use of reliability analysis should therefore be limited for the present time (apart from some specialised structures where the data are known more accurately) to a background support role where the mathematical models are developed and investigated, particularly in respect of their sensitivity to the various assumptions that inevitably have to be made in the absence of better data. It was with this aim in mind that the following exploratory exercises were undertaken; they are by no means exhaustive, but serve to provide a better understanding of the subject and perhaps to indicate further areas of study. In its full complexity the design problem is one in which all variables in the design equation are subject to random variation and associated with each variable is a statistical distribution defining the frequency of occurrence for each value. Clearly the problem at this level poses considerable difficulties in obtaining a solution. Only in a few very simple cases is it possible to obtain an exact analytic solution. For example in a 2-variable problem where both the load and resistance variables are represented by normal distributions or by lognormal distributions an exact expression can be obtained for the safety index fl in terms of the coefficients of variation of each variable and the central safety factor which is a measure of the separation between distributions. Very often research workers use these particular combinations of load and resistance distributions in order to make the problem soluble--not because their distributions are truly normal or lognormal. When analytic solutions are unobtainable, it may be possible to find solutions by means of multiple integration, Monte Carlo techniques or the so-called "Advanced Level 2" method. In recent years considerable effort has been put into developing the Level 2 methods where the statistical distributions are assumed to be normal and are represented

49 by means and variances. However, doubts have been expressed about certain approximations which are fundamental to the Level 2 methods, particularly when dealing with non-normally distributed variables and in linearising a non-linear failure boundary. The approach taken in this paper has been to develop numerical solutions to the multiple integration models representing the simplest (2-variable) design problem, a timber beam in bending, and to explore the sensitivity of the models to changes in various assumptions. It is important to understand the simple situation before progressing to the more complex multi-variable problems.

2. DEFINITION OF SAFETY INDEX Before describing any specific examples it would be as well to define the concept of a safety index. When calculating a probability of failure (PF) from a statistical model the answer will be a very small decimal number, e.g. PF = 0.0000235 or 2.35 × 10 -5. From this starting point, the logical definition for talking about probabilities of failure is in logarithmic terms, which in the above example would have log PF = --4.63. Consequently some people have defined a safety index as - l o g P F since this is a convenient number normally between 1 and 10. However, another definition is now widely accepted. If the load (S) and resistance (R) are normally distributed, uncorrelated variables with failure function M = R - S then the safety index fl was defined by Hasofer and Lind in 1974 to be the minimum distance from the origin to the failure surface in a normalised coordinate system [4]. When the failure boundary is linear in the basic variables there is an exact relationship between fl and PF, namely: pF=~(-/3)

(1)

where • is the cumulative normal distribution function. This relationship becomes approximate for non-linear failure boundaries and also for non-normally distributed variables. As with the - log p v definition,/3 is also a small positive number, usually in the range 1 to 6. The two safety measures/3 and - l o g PF are very similar up to about 3.5 but thereafter - l o g Pv increases more rapidly than/3.

3. SETTING UP A SIMPLE EXAMPLE The example considered throughout the following sections is that of a simply supported timber floor joist under uniformly distributed load w per unit area. When designing timber members it must be remembered that strength is not independent of load, and hence any reliability figure has to be related to a specific reference period over which the load acts. For a safe design using the current design method, the maximum permissible bending stress ( f ) in the joist for a particular grade/species of timber must be greater than the applied stress

f > wsl2 8KZ

(2)

where s is the spacing of joists, 1 is the span, Z is the section modulus and K is the load sharing factor. In Fig. 1 the strength distribution R represents the population of all such beams of that grade and species, adjusted to the particular conditions of service in which the beams are used. Floor

50

Load effect, S / / ~

~

/Resistance R

Area = PF = I ~'FR(x) fs (x) dx

Equivalent stress S.r

Fig. 1. The probability of failure integral Pv = pr(R < S).

joists are designed for a long-term loading situation (say 50 years) and for dry conditions ( < 18% moisture content, mc). From laboratory tests information is available on the short-term breaking strength (modulus of rupture, MOR) of samples of full-sized structural material of certain species/grade combinations at 18% mc. This MOR distribution for structural timber always contains positive values and generally exhibits positive skewness: it can be well represented by the 3-parameter Weibull distribution [5], which can be justified theoretically on the basis that strength is governed by the largest defect (the "weak~st-link" theory). The lognormal has sometimes been used, but it does not have the sound theoretical justification of the Weibull; it also has a fixed shape whereas the Weibull shape is variable and hence more adaptable to fit different data sets. The 2-parameter Weibull is occasionally also used because of its relative simplicity in having zero location parameter. These short-term strength values must then be reduced by a duration of load factor to convert to a long-term loading condition. There are immediately three sources of error involved in converting the laboratory test results to those applicable in the design situation: (i) Duration of load effect. As with some other materials, timber is capable of sustaining higher loads for short periods of time than it is for longer periods. This is currently accounted for by a reduction factor of 9/16th's on the 5th percentile of the MOR distribution to convert from the short-term strength to a long-term reference period (see Madison curve, Fig. 6). This factor is based on research work done on small defect-free specimens about 30 years ago [6] and is assumed to apply throughout the strength range, i.e. it is quality-independent. Because of recent research [7] there is some doubt as to whether this assumption is correct and also about whether the magnitude of the factor is correct, but the model could easily be changed if further evidence becomes available from the extensive research programmes already underway. (ii) Orientation of the test piece. Members are tested in the laboratory with the worst defect in the centre of the beam, although in service the worst defect will be positioned at random. No modification factor has been applied in the following exercises, but there is some evidence [8] to suggest that stresses might be increased by a factor of about 1.15 and further research is planned. Hence the resulting reliability estimates will be conservative. (iii) Sampling variability and sample size. Other samples of the same species and grade will possess different sample statistics, and the magnitude of the differences will depend on the

51 material variability and the sample size. The only way of reducing between-sample variability is to increase sample sizes, but this would be at considerable expense. A further factor is extremely important with regard to the reliability analysis: (iv) S~Ustical distribution for R. Many researchers currently use normal or lognormal distributions to represent the strength of timber beams in reliability analyses, but it is essential to find out how sensitive the model is to other choices of distribution, particularly the Weibull because it is so widely used in determining characteristic stresses. Items (i) and (ii) are assumed inherent to the problem and they are dealt with in the same way throughout this paper. More research is required into these two areas to minimise the errors introduced. However, items (iii) and (iv) will be the subject of sensitivity analyses which follow. The applied stress or load effect variable in Fig. 1, S = wslZ//8KZ is assumed to take all its random variation from the load variable w, with the values of all other variables being fixed. For the purposes of the examples to follow, assume s = 0.6 m

l = 2.5 m

K = 1.1

so that S = 0.426 w/Z (Z will be fixed for each sample, but will vary from sample to sample in later examples). Certainly the likely errors in s, l and Z will be very small, though the load sharing factor K may well have a somewhat larger variability [9] in reality. Consequently, the problem is reduced to a 2-variable case with a linear failure boundary, the failure function being: f-

0.426 w ----~-- - 0

(3)

Similar to item (iv), another factor must be investigated with regard to the reliability analysis: (v) Statistical distribution for w (and $). Generally speaking the current knowledge of load distributions is rather poor. CP 3, Chapter V, Part 1, 1967 [10] (recently superseded by BS 6399, Part 1, 1984 [11]) specifies dead and imposed load levels (other than wind loads) to be used when designing buildings in the U.K. and Part 2 of the same document specifies wind loads for design. Dead loads are calculated from the unit weight of materials given in BS 648 [12] or from the actual known weights of the material used. Most weights are quoted as fixed values in BS 648 with no variability (and are presumably approximate mean values), but some materials are given tolerances or a range of weights. Likewise BS 6399, Part 1 specifies fixed values for the imposed loads to be used on a wide range of structural types; there is no indication of the random variability to be expected, but presumably they were chosen as being unlikely to be exceeded and therefore correspond to high percentiles of the underlying distributions. CP 3, Part 2 on wind loads is the only part of the loading code where the mass of information on the natural variability of the load has been reduced by statistical analysis to a single discrete characteristic value. Large databases do now exist for some categories of imposed loads (e.g. domestic premises [13]) but they still need to be analysed in depth. Most researchers feel that dead loads are approximately normally distributed, although for small coefficients of variation the differences between normal and lognormal distributions are small. Imposed loads exhibit a positive skewness, and a variety of distribution types have been suggested including perhaps most frequently the lognormal and extreme type 1 for maximum values. In the simple example considered here the load will be represented by a single distribution, even though in reality the situation may be more complex with a combination of two or more loads (e.g. dead and imposed). The load may be continuously varying in time (as in Fig. 2), or it

52

Xmax

/ E×

x~ f ~ , ~

t

T

f×

fx(x),fy(V)

Fig. 2. Illustration of continuous time-varying load.

TABLE 1 Distribution of kX (k constant)

Distribution of X Normal Gamma Extreme type 1 max values Weibull

- NOt, o) - G ( a , fl) - El(U, l / a ) -W(a,

b, c)

Normal Gamma Extreme type 1 max values Weibull

- N(k/~, ko) - G ( a / k , fl) - Ea(ku, k / a )

- W ( k a , kb, c)

may consist of discrete steps at random intervals. However, when dealing with single time-varying loads (when failure occurs if and only if the load exceeds some threshold value) the form of the arbitrary point in time distribution fx (x) is not of immediate relevance. The random variable which is of importance is the magnitude of the largest extreme load fr (Y) that occurs during the reference period T. The distribution of the largest extreme load can be thought of as being generated by sampling values of Xm~, from successive reference periods T. In the example being considered, T would be taken as 50 years, and provided the correct distribution is used, the time element may then be neglected in the reliability calculations. Results from statistical theory show that if the variable for load w (or strength R) follows a distribution listed in Table 1, then so does the load effect S = kw (or factored strength kR).

4. EFFECT STRENGTH

OF

DIFFERENT

DISTRIBUTIONAL

ASSUMPTIONS

FOR

LOAD

AND

Figure 1 illustrates the formation of the probability of failure integral pF =

FR(x)fs(x)dx

(4)

the derivation of which may be found in any standard text [4]. Because this integral is formed in the overlap region of the R and S distributions, P F is often strongly dependent on the shape of the upper tail of the load effect distribution S and the lower tail of the strength distribution R. The type of distributions used in the model may therefore be of crucial importance. The following six models for S/R have so far been considered. The first two are shown with their analytic expressions for the safety index fl, and the remainder with their appropriate probability of failure integral. In the usual notation ~, o, V represent mean, standard deviation and coefficient of variation; ~ is the central safety factor t~R/~s; a, b, c are location, scale and

53

shape parameters for the Weibull distribution; the Extreme type 1 for maximum values. Normal/Normal

and u, l/a

are location and scale parameters for

(N/N)

Z-l

(5)

P=

(Z2V,2 + V;)1’2

Lognormal / Lognormal (LN / LN)

l+[~)“‘]

p=

(ln[(l+

Vi)(l+

Normal/3-parameter pF

=

/ml

(6)

Vz)])1’2

Weibull (N/W3)

exp( $)[l

- exp( -t’)]

(7)

dy

Afi

where t = ( yu, + ps - a)/b

and A = (a - ps)/us.

Normal /P-parameter Weibull (N / W2)

As for N/W3, but with a = 0. Extreme type 1 /Normal

(E,/N)

dy dz

(8)

where z = - a( x - u) and B = (x - 1_1~)/u~. Extreme type 1 /Sparameter

exp[ z PF=JB -CO

Weibull (E, / W3)

exp z] [l - exp( - t’)]

dz

where z = -a(x - u), t = (x - a)/b and B = -c~(a - u). Several computer programs have been written to calculate pF for the two-variable case using the analytic solutions for /3 where possible and either a Chebyshev or Simpson’s rule integration procedure in the other cases. In the following study, one particular data set has been utilised for representing the MOR of the timber joist; this sample of 50 X 150 mm Swedish redwood and whitewood has a mean strength of 39.3 N/mm2 and a standard deviation 9.46 N/mm2 and is the same sample reported on by Pierce [5]. Normal, lognormal, 2-parameter and 3-parameter Weibull distributions have all been fitted to this data set. The following procedure was adopted for calculating pF. The load distribution (which in reality would vary depending on the reference period) was assumed arbitrarily to have a

54 TABLE 2

Comparison of long-term failure probabilities under different distributional assumptions for the example in Section 4 (n = 164 sample) Nominal applied load "

Mean applied load b

Nominal applied stress ~

( k N / m z)

( k N / m 2)

( N / m m z)

0.5 1.0 1.5 2.0 2.5 3.0 4.31

0.377 0.753 1.130 1.506 1.883 2.259 3.245

1.136 2.273 3.409 4.546 5.682 6.818 9.793

Probability of failure, p v N/N

25.85 12.93 8.62 6.46 5.17 4.31 3.00

LN/LN

3 . 1 x 10 -5 6.3 x 10 - s 1 . 2 x 10 - a 2 . 4 x 1 0 - 4 9.0 4 . 5 x 1 0 - 4 5.9 8 . 4 x 1 0 a 1.3 3 . 7 x 1 0 -3 2.1

- 1 0 -2~ - 1 0 -16 - 1 0 -12 x 1 0 l0 ×10 8 xl0 6 xl0 4

N/W2 d

N/W3 ~

E~/N

El/W3 ~

7 . 1 x 10 -7 1.4x10 -s 7.9x10 -s 2.8x10 4 7.2x10 -4 1 . 6 x 1 0 -3 7 . 5 x 1 0 -3

3 . 7 x 1 0 7a 1 . 3 x 1 0 -29 8.9x10-16 4 . 8 x 1 0 -1° 3 . 4 x 1 0 -7 4.2x10 -4

3 . 2 x 10 - s 6.2x10-s 1.2x10 -4 2.3x10-4 4.3x10 4 5.4x10 4 4.1x10 3

2.7 x 10 -29 1.3x10-15 7.8x10 -u 2.5x10-8 9.3x10-7 1.1xl0-S 6.5x10-4

Assumed to be 95th percentile of distribution. b Calculated from the nominal applied load (column 1) on the basis of a normal distribution: column (1) divided by 1.328. For an extreme type 1 (max value) 1.328 should be replaced by 1.373. c C o l u m n (1) multiplied by = 2272.7 for joist size 50 x 150 ram. d Scale and shape parameters for W2 fitted to this sample are 43.037, 4.317. e Location, scale and shape parameters for W3 fitted to this sample are 13.466, 28.945, 3.048.

sl2/8KZ

coefficient of variation of 20% throughout. A nominal load (such as might be specified in a code for imposed loads) was assumed to represent a high percentile of the true underlying distribution and was taken arbitrarily as the 95th percentile. Values for the mean and standard deviation were then calculated based on the underlying distribution type (normal, lognormal or extreme type 1). These values were then input to the appropriate computer model which calculated the required parameters for the applied stress distribution, and finally the probability of failure was calculated using eqns. (5) to (9). Table 2 shows the results for PF in the long-term load case (i.e. assuming the strength distribution R represents the long-term breaking stress, reference time approximately 50 years). Also shown in the table are the nominal applied stress (i.e. 95th percentile value) and the central safety factor h. These values of PF are plotted in Fig. 3 together with the values of PF obtained under a short-term load assumption (reference time approximately 5 minutes). Several parallel scales have been marked along the horizontal axis because different measures of load or stress are useful in different circumstances. The scale marked "equivalent stress at 95th percentile" is based on a normal distribution and is therefore only approximate for an extreme type 1. The vertical scale is based on - l o g PF, but for comparison purposes a parallel scale based on eqn. (1) gives an approximate fl value. Other combinations of load and strength distributions are clearly possible but have not been analysed at this stage. The combinations described are thought to be the ones most likely to be considered, and from Fig. 3 several important features emerge. (i) With all models P v increases (B decreases) as the load and strength distributions become "closer", i.e. as the ratio of the means h = ffR/~S becomes smaller, but both the level of PF and its rate of increase vary enormously depending on the model. For example under long-term load, Table 3 shows that as n is reduced from 10 (a low load) to 4 (design load region) for the N / N model Pv increases only slightly from 10 - 4 tO 10 - 3 whilst for the N / W 3 model it increases dramatically from about 10 -6o to 10 - 6 . (It is worth noting that the definition of safety index as

55

Safet,

measure

- Lo

~

PF

\ \

20

\ \ \

\

LN/LN

15 -

\

\ \\

8

\

N/W3

\ I ', \\ \ \\ \,,\ \\

\, 'x

x\

10

~

~

-

.

.

.

-

-

N¢~#2&EI/N

a ~ S h o r t t e r m load -Long term load

o

i'.o

21o

31o

Mean applied load- kN/m 2 l

I

I

1

I

A

I

2 3 5 Equivalent mean stress-N/mm 2 Equivalent

L

25

9 5 th p e r c e n t i l e

1~0

stress ]

5

I

6

- N/mm 2 4

5 Fig. 3. Comparison of failure probabilities for different models based on a single sample of 50 x 150 m m r e d w o o d /

whitewood (n = 164).

- l o g PF is far more sensitive a measure than fl: this is particularly noticeable at low probabilities of failure.) On the basis of their behaviour with respect to changes in ~, it is possible to separate the models into two distinct groups. The values of PF for Group 1 models (N/N, El/N, N / W 2 ) do not vary much with the level of applied load, whereas the Group 2 models ( L N / L N , N / W 3 * and E l / W 3 *) are very sensitive to changes in ~. For the Group 1 models therefore, because the * Section 5 shows that the response of these models is strongly dependent on sample size, and for larger n they can behave as Group 1 models.

56 TABLE 3 Effect of the central safety factor ~ of PF for different models under long-term load (n = 164 sample) Model type

Probability of failure at =10 (low load)

N/N E1/N ) N/W2 LN/LN N/W3 El/W3

10 -4 5 X 10 - s 10 13 - 10 -60 10-11

h =4 (design load region) 10 3 2 X 10- 3 4x10 6 3 X 10 -6 4 X 10 -5

slopes of the lines are shallow, if a design were being done to a specified reliability, the/3 value would have to be defined within close limits otherwise a wide range of loads would be permissible. (ii) The N / N and E1/N models produce almost identical probability of failure values in this example and they cannot be distinguished on Fig. 3. This is not because the tails of the normal and extreme type 1 distributions are similar (they are not), but because the major contribution to the failure integral comes from within the main body of the load distribution and this is not too dissimilar in the two cases. Hence the form of the tail region is relatively unimportant in this example. This is perhaps a surprising result in that one thinks intuitively that the shape of the upper tail of the load distribution is always important. The P F distribution may lie almost entirely within the main body of either the S or R distribution or anywhere between them - - depending on the relative magnitudes of V R, V s and ~. Figure 4 illustrates a selection of these cases. At high VR the P F distribution is usually governed by the body of the S distribution: hence the tail of S is unimportant and as Fig. 5 shows the two models E 1 / N and N / N are virtually indistinguishable. However, for low VR the P F distribution can be almost entirely within the body of the R distribution when V s is high and h low (and would then be relatively insensitive to the lower tail of R but very sensitive to the upper tail of S), but as V s is reduced or n increased the PF distribution will "move" back towards the S distribution. At this intermediate position the PF value will be strongly influenced by both the upper tail of S and the lower tail of R. Therefore for the design of individual timber members where the coefficient of strength variation is normally high, it would appear that the E 1 / N model may be disregarded since the simpler N / N model provides a very good approximation to the probability of failure. (iii) The very low values of PF obtained with the N / W 3 model are due to the 3-parameter Weibull having a cut-off value at its lower end, rather than extending to zero as in the 2-parameter Weibull or to minus infinity with a normal distribution. The value of this cut-off (the location parameter) has a very large effect on PF and this feature will be explored in greater depth in Section 5. When the 3-parameter Weibull distribution is fitted to this same sample of strength data a fairly high location parameter (a) is estimated, and since the PF distribution must lie above a, it lies entirely within the R distribution. Consequently for a high value of a the value of P v from the N / W 3 and E l / W 3 models will be very much smaller than that from the corresponding

57 Frequency

a) VR = 0.25 VS = 0.25

i

,#o

S

-

fi=2

~

o c @ 20

~~_ "r. .o 10 t O"

-3

-2

-1

0 ~s

1

2

3

4 2~s

No. of std. devs above us

Frequency

b) V R = 0.05 V s = 0.25 fi = 2

i

~ o i0

N×c ~

o

~

5 0

-3

-2

-1

0 #s

1

2

4 2~s

No. of std. devs. above ~s

Frequency

? o ®

c) VR = 0 . 0 5 Vs = 0.05 5=2

x o

.-~ .~ 10

0 0

10

20 2~s

No. of std devs. above ~s

Fig. 4. Relative position of PF distribution for N / N model ( # s is the mean value of the S distribution).

N / N and E 1 / N models. Hence when R is represented as a 3-parameter Weibull, the shape of the upper tail of the S distribution becomes critical and consequently there is a much bigger difference between E l / W 3 and N / W 3 than between E 1 / N and N / N . (iv) Duration of load (DOL) affects different models to different degrees. For example, if the material were stressed to the design value for SS redwood/whitewood (7.3 N / r a m 2 in CP 112, Part 2, 1971) by applying the long-term design load at the 95th percentile of the load distribution, it is possible to plot a curve of - l o g PF against time showing how the reliability of a timber floor joist varies with time under various model assumptions (see Fig. 6). This is based on the Madison DOL curve where the effect is assumed to be quality independent, but could equally well be obtained for other load duration functions. For Group 1 models the member has a somewhat greater safety in its early life than in its later life (up to 20% in fl terms) and for the Group 2 models reliability in the short term is increased by about 40% for L N / L N (fl = 4.5 to 6.3) and has almost doubled for N / W 3 (fl = 4.6 to 9.0) * These are huge differences in reliability: a fl value of 9 corresponds to a failure rate of 1 in 1019!

* This result is applicable to the sample with n = 164. The analysis in Section 5 shows that this effect is strongly dependent on sample size.

58

10

~=25 ~ VR = 0 15 -

o=

n 15 """ "-~.

~ = 10

6i

=5

i 5

fi=25 4

VR=O24 f I ~------""--

3 -----

2

n

5

N/N model E1/N model

2'0

1'o

3'0

VS " ~o

Fig. 5. Sensitivityof N/N and E]/N models to changes in in VR, Vs, ~ (under short-term load). Other researchers appear to be suggesting acceptable levels of safety in the/3 = 3 to 4 region, and the Group 1 models would appear to fall roughly into line on this point. Once again it appears that/3 is a fairly insensitive safety measure, this time in terms of the duration of load. However, if the Group 2 models are used it is evident that the joist spends most of its life with a reliability far in excess of what is required--between 2 and 5 times the desired reliability measured in terms of - l o g PF, or up to 3 times the desired reliability measured in terms of/3.

5. EFFECT OF SAMPLING ON WEIBULL PARAMETERS AND RELIABILITY ESTIMATES It is intuitively obvious that as the sample size increases the strength of the weakest piece in the sample will decrease towards some lower limit greater than zero. Since the location parameter is always slightly less than the lowest value in the sample, that too tends to decrease with increasing sample size n. In order to investigate this aspect the 3-parameter Weibull distribution has been fitted to the bending strength of 21 samples of timber of varying section size: there were three grades of each of eight sizes of Finnish and Swedish redwood/whitewood, but only samples with n > 20 were included in this study. Table 4 shows the parameter estimates for each sample after having adjusted to a common depth of 200 mm using the factor (200/h) °4 as in Fewell [14]. The modified graphical method of parameter estimation was used, as described by Pierce [5]. A plot of the location parameter against sample size in Fig. 7 confirms the expected trend. Unfortunately a plot for each grade separately does not contain enough points to show any significant trend, but when all grades are combined a curvilinear relationship is evident. A

59 Safety

measure

- Log PF /3 20 9 18 16

8

14 12-

7

10 6 8 6

5

4

4 3 2

~20

(n =164)

/ N/W3 & El/W3 (n = 220) 01.1 I I 1

N/N,E1N

i

N/W2

iJO

I

102

i 103

L 104

5 rains

day

1 year

5 mins

1 day

1 ear

t 105 Hc Jrs under load 50 ,ears

1.0 0.9 08

=.j :~0

0.7 0.6 0.5 50 ,rears

Fig. 6. Madison DOL curve and its effects on reliability for different models at design stress.

hyperbolic curve provides a reasonable fit to the data apart from one outlier. As one would expect, any horizontal asymptote representing the population value for the location parameter is likely to be very low, perhaps in the region of 2 N / r a m 2, and probably represents the stress above which pieces will not be broken by routine handling operations. Whilst there are also significant relationships (p < 0.01) between the scale parameter and sample size (r = 0.69) and between shape parameter and sample size (r = 0.68), there are no apparent trends to show that these parameters are approaching any limiting population value. From a reliability viewpoint it is unfortunate that these population values are elusive because it would be so much simpler in any reliability study to use a single set of fixed population parameters rather than variable sample parameters. It must be accepted therefore that any reliability calculations using a 3-parameter Weibull distribution to represent material strength will be subject to considerable variation as a result of sampling the material. Using the same example of a timber joist described in Section 3, a study was made of the reliability of these joists based on the description of the material given by the 21 samples. For this study a long-term normally distributed load with 20% coefficient of variation was applied so that its 95th percentile produced the design stress in each joist. (The joists were of different sizes

60 TABLE 4

Parameter estimates for 3-parameter Weibull distribution fitted to 21 samples of redwood/whitewood bending strength (N/mm 2) adjusted to 200 mm depth Section size

Origin

Grade

(mm)

a

Sample size n

50 x 200

Finnish

50 x 200

Swedish

70 x 195

Swedish

58 x 170

Swedish

38 x 150

Finnish

50 x 150

Swedish

38 x 150

Swedish

38 x 100

Finnish

S10 $8 $6 S10 $8 $6 S10 $8 S10 $8 S10 $8 S10 $8 $6 S10 $8 $6 S10 $8 $6

53 66 83 206 161 91 38 41 38 35 49 69 168 138 106 220 119 100 26 73 102

Estimated parameters b location

scale

shape

14.80 17.54 10.81 3.18 8.69 2.85 11.12 11.12 18.14 12.02 25.60 4.13 8.06 16.77 11.23 1.70 2.10 4.99 26.53 11.13 8.52

39.03 25.28 24.96 47.28 34.30 35.22 32.88 27.91 22.80 23.67 18.29 35.86 38.57 23.97 25.42 46.81 39.83 31.73 15.63 24.18 22.70

3.92 2.47 2.76 5.70 3.63 4.07 3.27 3.35 3.88 3.05 2.18 3.29 4.38 2.78 3.39 4.70 3.86 3.17 2.02 3.60 3.36

a European grades. b Modified graphical method as described in BRE Current Paper CP 2 6 / 7 6 [5].

and grades and therefore different load levels had to be applied for each joist type to reach its design stress.) The resulting probability of failure values for each of the 21 samples are shown in Fig. 8, and it may be seen that there is a huge variation in PF at the 50 year level--between about 10 -3 and 10 -19. The reference data set used for the study in Section 4 fits well into the general trend of an increasing probability of failure with increasing sample size. The samples with a very small P F are exactly those samples which had high location parameters. Indeed, a further plot (Fig. 9) of k times the location parameter against - l o g PF (where k = 9 / 1 6 as in the long-term case) shows a strong positive correlation (r = 0.89), and the best fit by a power curve using maximum likelihood is: Y = 3.066 + 0.096 X 1684 where Y = - l o g PF and X = k X location parameter (k = 9/16). So as the location parameter becomes smaller and smaller and approaches its lower bound, so the value of - l o g PF approaches a value of about 3 (a probability of failure of 1 0 - 3 ) . This value is still subject to considerable error: on the basis of this data the mean + 2 standard deviation has a range 1.456 to 4.676, i.e. a probability of failure between 3.5 x 10 -2 and 2.1 x 10 -5. This section began by showing that there appeared to be no easily obtainable description of population strength in terms of a 3-parameter Weibull distribution, with only the location

61 3O ~•

to

•

$10

•

S 8

•

$6

• 20 E m

•

n 0"826 a = 318.9 (r = -0.65)

Ill

• •

•

?, I

0

I

L

n

i

•

I

50

40

•

•

•

~

•

n

20.20 (r = 0.69)

÷

E 30

2O a

I

I

i

I

I

.....-,-

5

"~ E m

4

I

~

2

0.0102 n ÷ 2.51 ( r = 0.68)

•

4/0i

I

I

80

I

120

Sample

size

]

160 -n

I

200

240

Fig. 7. Effect of sample size on Weibull parameters.

parameter a showing any trend towards a limiting value as the sample size increased. However, Fig. 9 implies that the values of the other parameters b and c are unimportant as far as the reliability is concerned (except in contributing to the sampling variation). As the sample size

- Log PF

/~

20-

\

16-

•

$10

•

S8

* S6 \ \

12-

•

Reference in section

~.

data set used 4 and Fig 6.

e~

8ee

4-

O'

•

• •

em

1~o

o Sample

•

2~0

size- n

Fig. 8. Effect of sample size on 50-year reliability using N / W 3 model.

62 2O

16 o.. _J o

J O 6 6 + O O 9 6 X

1684

12

8

•

4~-~,,,,,~

•

0

•

eference

data

set

used

un section 4 and Fig 6.

i i i i 4 8 12 16 k x location parameter ( k • 9/16)

i 20

Fig. 9. Relationship between location parameter and 50-year reliability using N/W3 model.

increases, the location parameter decreases towards a small positive value and the reliability based on a N / W 3 model approaches a value of about 10 -3. (This is a composite value for all three grades considered and of course it may vary depending on the grade of timber, but there are insufficient data here to find out.) The analysis therefore suggests that perhaps the 2-parameter Weibull distribution (which has zero location parameter) or the 3-parameter Weibull with arbitrarily small location parameter would provide an estimate of reliability that is more representative of the material population and that it would be unwise to use the 3-parameter Weibull distribution in reliability analyses unless n is greater than about 250. Therefore the analysis in Section 4 for N / W 3 must be reassessed because that was based on a data set with n = 164. The largest of the 21 samples has been selected for the reassessment--a sample of Swedish 38 × 150 mm S10 grade with n = 220. Once again the design stress is assumed to be applied at the 95th percentile of the load distribution, which is described by N(2.518, 0.504) k N / m 2. When this reliability information is added (by a dashed line) to Fig. 6 and to the long-term reliability curves from Fig. 3 which are replotted in Fig. 10, it may be seen that the N / W 3 and E l / W 3 models for large sample sizes really belong with the Group 1 models, whilst for small sample sizes they belong with the Group 2 models. The location parameter for this n = 220 sample is very small (1.069 for long-term load) and therefore the P F distribution, although entirely within the R distribution is also well within the S distribution. In this situation the relative behaviour of E l / W 3 and N / W 3 is very similar to the behaviour of E 1 / N and N / N described in Section 4(ii): the shape of the tail of S becomes unimportant and hence it matters little whether the load distribution is assumed to be normal or extreme type 1. Because of this behaviour with large samples (and small location parameters), it would appear that in analysing the reliability of timber beams the complication of using the extreme type 1 distribution for loading may be disregarded and a normal distribution may be a good enough representation. This would also simplify the analysis of designs where (very commonly) the load comprises two parts, an imposed load and a dead load. The dead load is probably nearly normal though the imposed load is often postulated as extreme type 1. From the foregoing discussion it would seem to make little difference in the reliability analysis which of the two distributional types is used, and therefore one might just as well use normal distributions for simplicity. This would have the further

63

E,/w.\ \iN/, / ,n=l.,\ \

\./.3

/ /

\ \

\

,oL \ N/NE.

\ \

\

J

tt N/W3 & E~/W3 fsample o r sma//ers size

N

~ Approx design

o

,'o

2'o

3'.o

Mean applied load- kN/m 2

Fig. 10. Comparison of long-term failure probabilities showinghow El/W3 and N/W3 are affected by sample size. advantage that since the sum of two normal distributions is also normal the appropriate curves in Figs. 3, 6 and 10 are all applicable to this more general case. The analysis reported in this section poses questions reaching beyond the realms of reliability analysis. It questions the accuracy of current practice in using the 3-parameter Weibull distribution to assess 5th percentile characteristic stresses of populations from small samples, and suggests that either the 2-parameter Weibull (which has zero location parameter) or a 3-parameter Weibull with an arbitrarily small location parameter might better represent the whole population of material. More investigation is required on this subject.

6. CONCLUSIONS The use of structural reliability models to represent the design process completely disregards about 90% of all failures [5-7]: these occur as a result of poor workmanship, abuse or gross human error. Yet their use has been advocated for determining partial coefficients for limit state codes with very little fundamental investigation work having been carried out on the effects of the many assumptions involved. Some of the main assumptions are concerned with the type of statistical distributions with which to represent loads and resistances. The classical approach to reliability has been used here to develop a model with six different combinations of distributional type in a simple two-variable problem of a timber beam in bending, in order to find out how sensitive the model is to the various assumptions. Several other simple assumptions have been made about duration of load, the variability of the load distribution and the relative positioning of the applied stress (S) and resistance (R) distributions in order to limit the number of variables in this investigation. The choice of distributional types for a reliability model should be made primarily on an understanding of the material behaviour and the nature of the loading. Even so, many researchers persist in using normal or lognormal distributions because they are the only ones for

64 which simple solutions are available. It is widely accepted that the bending strength of timber may be well represented by a 2-parameter or 3-parameter Weibull distribution (W2 or W3), that dead loads are approximately normally distributed (N) and that imposed load distributions are positively skewed and may be represented either by lognormal (LN) or extreme type 1 for maximum values (El). Figure 3 shows that some models for load/resistance (e.g. N / N , E l / N , N / W 2 ) produce similar probability of failure (PF) values over a wide range of load and in the range of safety index fl = 3 to 4, whilst others (e.g. L N / L N ) provide very much smaller values of P F and are highly sensitive to the relative position of the S and R distributions. For material such as timber which has a large coefficient of variation in strength (VR) it is shown (Fig. 5) that E 1 / N and N / N are virtually identical due to the fact that the PF distribution lies mainly within the body of S and therefore the shape of the tail region is unimportant. This observation would not necessarily be true for materials with a small VR (e.g. steel). When the 3-parameter Weibull is used for resistance, the situation is complicated by the fact that the models E l / W 3 and N / W 3 are extremely sensitive to the value of the location parameter (Fig. 9), which in turn is strongly related to the size of the sample from which the parameters are estimated (Fig. 7). On the evidence of 21 samples analysed here it would appear that for fairly large sample sizes (n greater than about 250) the location parameter (a) is generally small and the resulting pv is only marginally greater than that obtained from the N / N , E 1 / N and N / W 2 models. However, for small samples the true population value of a is subject to very large error in estimation, and consequently if the estimated a is large the corresponding PF calculated from the E~/W3 or N / W 3 models will be ridiculously small. The analysis shows that it would be unwise to use sample sizes less than about 250 for predicting reliability estimates of populations, and that if large samples are available there is little need to assume an extreme type 1 distribution for loading--a normal distribution provides results that are quite accurate enough. Apart from this advantage of simplicity there is an added bonus that in design situations where (very commonly) the total load is the sum of dead and imposed loads and may therefore also be assumed normal, the results shown in Figs. 3, 6 and 10 are all applicable to this more general case. The effect of a Madison type duration of load curve on PF has also been investigated (Fig. 6). This shows that (apart from the small sample E l / W 3 and N / W 3 models and L N / L N ) the models are fairly insensitive in terms of the safety index 13 to changes in duration of load. It is envisaged that the PF values given by the simpler models (e.g. N / N , N / W 2 or N / W 3 (n > 250)) would not be much affected by a change in duration of load assumption such as might be provided by fracture mechanics or damage accumulation theories. However, the timber beam designed for a long-term loading situation does have a reserve of strength in its early life that gives it an advantage when compared with other materials. The work on sampling also raises the question as to whether the current practice of estimating 5th percentile characteristic stresses of populations from small samples using the 3-parameter Weibull distribution is satisfactory: the true population value of a is likely to be very small and therefore either the 3-parameter Weibull or the 3-parameter Weibull with an arbitrarily low value of a may be more applicable. Ideally more research is required to compare the efficiency of these models in estimating 5th percentiles from small samples. It is evident that a better specification of the loading distributions (particularly imposed loads) and better knowledge of the effects of long-term loading and position of defects on the strength of a beam would contribute significantly to the accuracy of any reliability calculation. However, it is possible that the simpler models described here (e.g. N / N , N / W 2 , N / W 3 (n > 250)) could

65

be used in limited exercises to assist in rationalising the safety of particular elements. These methods must not be used indiscriminately or on a large scale, but they could be useful in comparing the safety levels of existing and proposed design methods as long as the comparisons are confined within particular element types of the same material so that the same set of assumptions is valid throughout.

REFERENCES 1 Rationalisation of safety and serviceability factors in structural codes, CIRIA Report 63, 1976. 2 Wind in Western Europe. Results of an enquiry, Proc. 34/13/4047, Laboratorio Nacional de Engenharia Civil, February 1973. 3 D.E. Allen, Structural Safety, Canadian Building Digest, National Research Council of Canada, March 1972. 4 P. Thoft-Christensen and M.J. Baker, Structural reliability theory and its applications, Springer Verlag, Berlin, 1982. 5 C.B. Pierce, The Weibull distribution and the determination of its parameters for application to timber strength data, BRE Current Paper CP 26/76, Building Research Establishment, 1976. 6 L.W. Wood, Relation of strength of wood to duration of load, Report No 1916, U.S. Forest Products Laboratory, Madison, WI, 1951. 7 B. Madsen and J.D. Barrett, Time-strength relationships for timber, University of British Columbia, Struct. Res. Series Report No. 13, 1976. 8 A.R. Fewell and C.B. Pierce, Considerations for the development of a U.K. limit state design code for timber, BRE Occasional Paper, Building Research Establishment, 1985. 9 C.B. Pierce, The analysis of load distribution between rafters in a traditional timber roof structure: Part 2 Simulations under concentrated load, BRE Current Paper CP 3/82, Building Research Establishment, 1982. 10 CP 3, Chapter V, Loading. Part 1, 1967, Dead and imposed loading; Part 2, 1972, Wind loads; British Standards Institution. 11 BS 6399, Part 1, 1984, Design loading for buildings. Code of Practice for dead and imposed loads, British Standards Institution. 12 BS 648, 1964, Schedule of weights of building materials, British Standards Institution. 13 G.R. Mitchell and R.W. Woodgate, Floor loading in domestic premises--the results of a survey, BRE Current Paper CP 2/77, Building Research Establishment, 1977. 14 A.R. Fewell, The determination of softwood strength properties for grades, strength classes and laminated timber for BS 5268, Part 2, Building Research Establishment Report, 1984.