## Reliability differentiation Chapter Iii Reliability Differentiation

Milan Holicky1-' and Jana Markova1-1

1)Czech Technical University in Prague, Czech Republic

### Summary

Basic reliability elements specified in current standards for structural design commonly include failure probability related to a certain reference period T. Required reliability level of buildings and other civil engineering works is usually specified by the design (target) failure probability pd or by appropriate reliability index Jd corresponding to a specified design working life Td (for example 50 years). In reliability verification the design values Jd and Td are sometimes replaced with an alternative reliability index Ja derived from the design values Jd and Td for a convenient reference period Ta (for example 1 year).

Submitted study clarifies relationships between the alternative elements Ja, Ta and design values JJ, Td, and indicates relevant procedures for reliability verification when alternative reference period Ta is considered. It is emphasised that verification based on Ja, Ta should be distinguished from verification of temporary or auxiliary structures when the design working life Td itself is short. Theoretical consideration and numerical examples show that characteristic values and partial factors of basic variables describing material properties and self-weight are significantly dependent on the relevant reference period.

1 INTRODUCTION

### 1.1 Background documents

Recent documents [1], national [2], [3] and international documents ([4] to [7]) provide general principles and guidance for application of probabilistic methods to structural designs. The latest European document [5] and international standards [6] and [7] also indicate a theoretical basis of the so called "partial factor method" and procedures for determination of partial factors of material properties and actions using probabilistic principles.

The basic reliability elements considered in these procedures include probability of failure p (or equivalent reliability index J) corresponding to a certain reference period T used in verification of structural reliability. The reference period T used in verification may or may not coincide with the design working life Td, which is the time period during which a structure is required to perform adequately. When the reference period used in reliability verification is different from Td then it is called an alternative period and denoted in this study Ta.

### 1.2 General Principles

Basic probabilistic methods are used to analyse principles of reliability differentiation. Similarly as in Chapter I in this Handbook two essentially different cases are distinguished in the following:

- an alternative reference period Ta (for example 1 or 5 years), which is different from the design working life Td (for example 50 years), is considered; this case is applicable when probabilistic models related to the period Ta are more credible than those related to Td;

- the design working life Td itself is short (for example 2, 5 or 10 years); this is the case of temporary or auxiliary structures and structures under a transient design situation (during execution or repair).

In the following the principles of reliability differentiation specified in current international documents [5,6,7] and related procedures for determining reliability measures to be applied in verification cases considering various design-working lives are discussed. Appropriate reliability elements (characteristic values and partial factors) are derived for material properties, self-weight and climatic actions (temperature, snow and wind) taking into account time dependence of failure probability and the reliability index.

### 2 BASIC RELIABILITY ELEMENTS

The basic reliability measures include the probability of failure and reliability index as introduced in Chapter I and II in this Handbook. The probability of structural failure Pf can be generally defined as

The limit state (performance) function Z(X) is formulated in such a way that the reliable (safe) domain of a vector of basic variables X = X1, X2, ... , Xn corresponds to the inequality Z(X) > 0 while the failure domain to the complementary inequality Z(X) < 0. A simple example of Z(X) describes the basic relationship between the resulting load effect E and resistance R

The random variable Z in equation (2) is often called the reliability (safety) margin; its mean /Z, standard deviation <Z and skewness o>Z may be derived from corresponding characteristics of resulting variables R and E as indicated in Chapter II.

Instead of the failure probability Pf, the reliability index fi is frequently used in reliability consideration as an equivalent quantity to Pf. The reliability index fi is related to the failure probability Pf as already indicated in Chapter I

In this equation, 0( ) denotes the distribution function of standardised normal distribution. Note that, if the safety margin Z has normal distribution, then the reliability index may be determined simply as the ratio of /Z and <Z, thus fi = /Z /<z (in this case fi denotes the distance of the mean /Z from the origin taking the standard deviation <Z as a unit). Chapter I shows the numerical relationship of both quantities. It should be emphasized that the failure probability Pf and the reliability index fi represents fully the equivalent reliability measures with one to one mutual correspondence given by equation (3).

In the recent European document [5] a design working life for common structures is considered as Td = 50 years, the reliability index for ultimate limit states fid = 3,8 corresponds to the design failure probability Pd = 7,2 x 10-5, for serviceability limit states fid = 1,5 and pd = 6,7 x 10-2 (a more appropriate term is the "target probabilities" used in ISO documents [6] and [7]). These quantities are recommended as reasonable minimum requirements and it is emphasized that Pd and fid are formal conventional quantities only and may not correspond to actual frequency of failures.

In design analysis of a structure it is generally required that

or equivalently in terms of reliability index

where pd denotes specified design (target) failure probability corresponding to the target reliability index pd.

Conditions (4) or (5) have to be used by designers when probabilistic methods are applied for verification of structural reliability. Indicative target values pd and pd are declared in some national standards (e.g. [2] and [3]) and recently also specified in international documents (e.g. [4] to [7]) for various design conditions (limit states, failure consequences and economic aspects).

### 3 DESIGN WORKING LIFE AND RELIABILITY

Design working life Td is an assumed period of time for which a structure or part of it is to be used for its intended purpose with anticipated maintenance but without major repair being necessary. In recent documents of CEN [5] and ISO [6] indicative values of Td are provided for five categories of structures as shown in Chapter I of this Handbook.

More detailed specification of structural categories and design working lives is available in the ISO documents [6, 7]. In general the design working lives indicated in [2] are greater (in some cases by 100 %) than those given in Chapter I. For example the design working life for temporary structures indicated in [2] is 15 years, for agricultural structures 50 years, for apartment and office buildings 100 years, and for railway structures, dams, tunnels and other underground engineering works 120 years.

Design failure probabilities pd are usually indicated in relation to the expected social and economical consequences. Table 1 shows classification of target reliability levels provided in EN 1990 [5]. Reliability indexes P are given for two reference periods T (1 year and 50 years) only, without any explicit link to the design working life Td. Similar P-values as in Table 1 are given in [3] for the ultimate limit states, for which, however, the design working life Td = 80 years (for building structures) is considered.

It should be underlined that a couple of P values (pa and pd) specified in Table 1 for each reliability class (for 1 year and 50 years) correspond to the same reliability level. Practical application of these values depends on the time period Ta considered in the verification, which may be connected with available information concerning time variant vector of basic variables X = X1, X2, ..., Xn. For example, if the reliability class 2 and 50 years design working period is considered, then the reliability index pd = 3,8 should be used in the verification of structural reliability. The same reliability level corresponding to class 2 is achieved when the time period Ta = 1 year and pa = 4,7 is used. Thus, various reference periods Ta, in general different from the design working life Td, may be used for achieving a certain reliability level.

 Table 1. Re iability classification in accordance with CEN [5] Reliability classes Consequences for loss of human life, economical, social and environmental consequences Reliability index ß Examples of buildings and civil engineering works Pa for Ta= 1 year ßd for Td= 50 years 3 - high High 5,2 4,3 Bridges, public buildings 2 - normal Medium 4,7 3,8 Residential and office buildings 1 - low Low 4,2 3,3 Agricultural buildings, greenhouses

Similar target Pd values are provided in ISO 2394 [6] for the design working life Td (called in ISO "life time") without specification of any particular value of Td. As indicated in Table 2, two factors are considered for reliability differentiation in [6]: relative costs of safety measures and consequences of failure.

 Relative costs of safety Consequences of failure measures small some moderate great High 0 1,5 2,3 3,1 Moderate 1,3 2,3 3,1 3,8 Low 2,3 3,1 3,8 4,3

It appears that available documents do not provide an explicit guidance on how to take into account the design working life Td. Both international documents CEN [5] and ISO [6] give the target value Pd for specific reference periods T, however, no explicit rule is offered for adjustment of target value Pd to different working design lives Td recommended for various types of construction works.

Nevertheless, some indication is provided in another ISO document [7] for assessment of existing structures where it is recommended that reliability levels for any residual lifetime could be similar to those considered for the design working life Td in the case of a new structure. Consequently, similar reliability levels (expressed in terms of probability pd or reliability index Pd) may be considered when designing structures for different design working lives Td, for example for Td = 50 and Td = 25 years.

### 4 VARIATION OF FAILURE PROBABILITY WITH TIME

When the vector of basic variables X = X1, X2, ... , Xm is time variant, then failure probability p is also time variant and should always be related to a certain reference period T, which may be generally different from the design working life Td. Considering a structure of a given reliability level, the design failure probability pd = pn related to a reference period Tn = n T\ can be derived from the alternative probability pa = pi corresponding to Ta = T\ (to simplify notation note that previously used subscript "d" corresponds now to "n" and subscript "a" to "1") using approximate relationship given in [6], [7]

For very small probabilities, this relationship could be further simplified as pn = p1 Tn / T1. Time periods T1 and Tn may have an arbitrary length and n = Tn / T1 may not be an integer; T1 is, however, often one year. Probability pn increases (almost linearly) with Tn.

It follows from equation (6) that reliability indexes P1 = Pa and Pn = Pd, given in accordance to equation (3) as p1 = ©(-P0 and pn = ©(-P) are related as follows [5]

Here ©(.) denotes the distribution function of standardised normal distribution. Figure 1 shows variation of Pn with P1 for n = 5, 25, 50 and 100. Note that, if the reference period T1 is one year, then n indicates the number of years of the reference period Tn (n = Tn).

Figure 1 confirms data indicated in Table 1. For example, if the target reliability level of a structure is specified by p50 = 3,8 for the design working life Td = Tn = 50 years, then it could be verified using reference period Ta = T1 =1 year and pa = p1 =4,7. When, however, the same reliability index 3,8 is specified for a structure having a design working life Tn = 25 years only, thus p25 = 3,8, then the reliability of this structure could be verified using an alternative reference period T1 = 1 year and reliability index p1 = 4,5, similarly when p5 = 3,8 then p1 = 4,2 (see Figure 1).

Figure 1 confirms data indicated in Table 1. For example, if the target reliability level of a structure is specified by p50 = 3,8 for the design working life Td = Tn = 50 years, then it could be verified using reference period Ta = T1 =1 year and pa = p1 =4,7. When, however, the same reliability index 3,8 is specified for a structure having a design working life Tn = 25 years only, thus p25 = 3,8, then the reliability of this structure could be verified using an alternative reference period T1 = 1 year and reliability index p1 = 4,5, similarly when p5 = 3,8 then p1 = 4,2 (see Figure 1).