1 2 Milan Holicky and Ton Vrouwenvelder
1Klokner Institute, Czech Technical University in Prague, Czech Republic 2Delft University of Technology, TNO BOUW, The Netherlands
Uncertainties affecting structural performance can never be entirely eliminated and must be taken into account when designing any construction work. Various design methods and operational techniques for verification of structural reliability have been developed and worldwide accepted in the past. The most advanced operational method of partial factors is based on probabilistic concepts of structural reliability and available experience. General principles of structural reliability can be used to specify and further calibrate partial factors and other reliability elements. Moreover, developed calculation procedures and convenient software products can be used directly for verification of structural reliability using probabilistic concepts and available experimental data.
Basic concepts of structural reliability are codified in a number of national standards, in the new European document EN 1990  and the International Standard ISO 2394 . Additional information may be found in the background document developed by JCSS  and in recently published handbook to EN 1990 . Guidance for application of probabilistic methods of structural reliability may be found in working materials provided by JCSS  and in relevant literature listed in [4 and . Elementary methods of the theory of reliability are described in Chapter II and III in this Handbook 2.
General principles of structural reliability are described in both the international documents EN 1990  and ISO 2394 . Basic requirements on structures are specified in Section 2 of EN 1990 : a structure shall be designed and executed in such a way that it will, during its intended life, with appropriate degrees of reliability and in an economic way
- sustain all actions and influences likely to occur during execution and use;
- remain fit for the use for which it is required.
It should be noted that two aspects are explicitly mentioned: reliability and economy (see also Handbook 1). However, in this Handbook 2 we shall be primarily concern with reliability of structures, which include
- structural resistance;
Additional requirements may concern fire safety of structures (see Handbook 5) or other accidental design situations. In particular it is required by EN 1990  that in the case of fire, the structural resistance shall be adequate for the required period of time.
To verify all the aspects of structural reliability implied by the above-mentioned basic requirements, an appropriate design lifetime, design situations and limit states should be considered (as described in Handbook 1). Note that the basic lifetime for a common building is 50 years and that, in general, four design situations are identified: permanent, transient, accidental and seismic. Two types of limit states are normally verified: ultimate limit states and serviceability limit states. Detail guidance is provided in Handbook 1.
It is well recognised that construction works are complicated technical systems suffering from a number of significant uncertainties in all stages of execution and use. Depending on the nature of a structure, environmental conditions and applied actions, various types of uncertainties become more significant than the others. The following types of uncertainties can be identified in general:
- natural randomness of actions, material properties and geometric data;
- statistical uncertainties due to a limited size of available data;
- uncertainties of the resistance and load effect models due to simplifications of actual conditions;
- vagueness due to inaccurate definitions of performance requirements;
- gross errors in design, during execution and use;
- lack of knowledge concerning behaviour of new materials and actions in actual conditions.
The order of the listed uncertainties corresponds approximately to the decreasing level of current knowledge and available theoretical tools for their description and consideration in design (see following sections). It should be emphasized that most of the above listed uncertainties (randomness, statistical and model uncertainties) can never be eliminated absolutely and must be taken into account when designing any construction work.
Natural randomness and statistical uncertainties may be relatively well described by available methods provided by the theory of probability and mathematical statistics. In fact the EN 1990  gives some guidance on available techniques. However, lack of credible experimental data (e.g. for new materials, some actions including environmental influences and also for some geometrical properties) causes significant problems. In some cases the available data are inhomogeneous, obtained under different conditions (e.g. for material resistance, imposed loads, environmental influences, for inner dimensions of reinforced concrete cross-sections). Then it may be difficult, if not impossible, to analyse and use them in design.
The uncertainties of computational models may be to a certain extent assessed on the basis of theoretical and experimental research. EN 1990  and materials of JCSS  provide some guidance. The vaguenesses caused by inaccurate definitions (in particular of serviceability and other performance requirements) may be partially described by the means of the theory of fuzzy sets. However, these methods have a little practical significance, as suitable experimental data are rarely available. The knowledge of the behaviour of new materials and structures may be gradually increased through theoretical analyses verified by experimental research.
The lack of available theoretical tools is obvious in the case of gross errors and lack of knowledge, which are nevertheless often the decisive causes of structural failures. To limit gross errors due to human activity, a quality management system including the methods of statistical inspection and control may be effectively applied.
Various design methods and operational techniques, which take these uncertainties into account, have been developed and worldwide used. The theory of structural reliability provides background concept techniques and theoretical bases for description and analysis of the above-mentioned uncertainties concerning structural reliability.
The term "reliability" is often used very vaguely and deserves some clarification. Often the concept of reliability is conceived in an absolute (black and white) way - the structure either is or isn't reliable. In accordance with this approach the positive statement is understood in the sense that "a failure of the structure will never occur". This interpretation is unfortunately an oversimplification. Although it may be unpleasant and for many people perhaps unacceptable, the hypothetical area of "absolute reliability" for most structures (apart from exceptional cases) simply does not exist. Generally speaking, any structure may fail (although with a small or negligible probability) even when it is declared as reliable.
The interpretation of the complementary (negative) statement is usually understood more correctly: failures are accepted as a part of the real world and the probability or frequency of their occurrence is then discussed. In fact in the design it is necessary to admit a certain small probability that a failure may occur within the intended life of the structure. Otherwise designing of civil structures would not be possible at all. What is then the correct interpretation of the keyword "reliability" and what sense does the generally used statement "the structure is reliable or safe" have?
A number of definitions of the term "reliability" are used in literature and in national and international documents. ISO 2394  provides a definition of reliability, which is similar to the approach of national standards used in some European countries: reliability is the ability of a structure to comply with given requirements under specified conditions during the intended life, for which it was designed. In quantitative sense reliability may be defined as the complement of the probability of failure.
Note that the above definition of reliability includes four important elements:
- given (performance) requirements - definition of the structural failure,
- time period - assessment of the required service-life T,
- reliability level - assessment of the probability of failure Pf,
- conditions of use - limiting input uncertainties.
An accurate determination of performance requirements and thus an accurate specification of the term failure are of uttermost importance. In many cases, when considering the requirements for stability and collapse of a structure, the specification of the failure is not very complicated. In many other cases, in particular when dealing with various requirements of occupants' comfort, appearance and characteristics of the environment, the appropriate definitions of failure are dependent on several vaguenesses and inaccuracies. The transformation of these occupants' requirements into appropriate technical quantities and precise criteria is very hard and often leads to considerably different conditions.
In the following the term failure is being used in a very general sense denoting simply any undesirable state of a structure (e.g. collapse or excessive deformation), which is unambiguously given by structural conditions.
The same definition as in ISO 2394 is provided in Eurocode EN 1990  including note that the reliability covers the load-bearing capacity, serviceability as well as the durability of a structure. Fundamental requirements include the statement (as already mentioned) that "a structure shall be designed and executed in such a way that it will, during its intended life with appropriate degrees of reliability and in an economic way sustain all actions and influences likely to occur during execution and use, and remain fit for the use for which it is required". Generally a different level of reliability for load-bearing capacity and for serviceability may be accepted for a structure or its parts. In the documents  and  the probability of failure Pf (and reliability index ( are indicated with regard to failure consequences (see Handbook 1).
The most important term used above (and in the theory of structural reliability) is evidently the probability of failure Pf. In order to defined Pf properly it is assumed that structural behaviour may be described by a set of basic variables X = [X1, X2, ... , Xn] characterizing actions, mechanical properties, geometrical data and model uncertainties. Furthermore it is assumed that the limit state (ultimate, serviceability, durability or fatigue) of a structure is defined by the limit state function (or the performance function), usually written in an implicit form as
The limit state function Z(X) should be defined in such a way that for a favourable (safe) state of a structure the function is positive, Z(X) > 0, and for a unfavourable state (failure) of the structure the limit state function is negative, Z(X) < 0 (a more detailed explanation is given in the following Chapters of this Handbook 2).
For most limit states (ultimate, serviceability, durability and fatigue) the probability of failure can be expressed as
The failure probability Pf can be assessed if basic variables X = [X1, X2, ... , Xn] are described by appropriate probabilistic (numerical or analytical) models. Assuming that the basic variables X = [X1, X2, ... , Xn] are described by time independent joint probability density function ^Xx) then the probability Pf can be determined using the integral
More complicated procedures need to be used when some of the basic variables are time-dependent. Some details concerning theoretical models for time-dependent quantities (mainly actions) and their use for the structural reliability analysis are given in other Chapters of this Handbook 2. However, in many cases the problem may be transformed to a time-independent one, for example by considering in equation (2) or (3) a minimum of the function Z(X) over the reference period T.
Note that a number of different methods  and software products [7, 8, 10] are available to calculate failure probability Pf defined by equation (2) or (3).
An equivalent term to the failure probability is the reliability index J, formally defined as a negative value of a standardized normal variable corresponding to the probability of failure Pf. Thus, the following relationship may be considered as a definition
Here o-1(pf) denotes the inverse standardised normal distribution function. At present the reliability index J defined by equation (4) is a commonly used measure of structural reliability in several international documents , , .
It should be emphasized that the failure probability Pf and the reliability index J represent fully equivalent reliability measures with one to one mutual correspondence given by equation (4) and numerically illustrated in Table 1.
_Table 1. Relationship between the failure probability Pf and the reliability index J.
In EN 1990  and ISO 2394  the basic recommendation concerning a required reliability level is often formulated in terms of the reliability index J related to a certain design working life.
When the vector of basic variables X = X1, X2, ... , Xm is time variant, then the failure probability p is also time variant and should be always 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 general reference period Tn = n T1 can be derived from the alternative probability pa = p1 corresponding to Ta = T1 (to simplify notation note that the previously used subscript "d" corresponds now to "n" and subscript "a" to "1"). Detail description of this transformation is provided in Chapter III.
4 DESIGN TARGETS
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 the recent documents CEN  and ISO  indicative values of Td are provided for five categories of structures as shown in Table 2.
A more detailed specification of structural categories and design working lives may be found in some national standards. In general the design working lives may be greater (in some cases by 100 %) than those given in Table 2. For example the design working life for temporary structures may be 15 years, for agricultural structures 50 years, for apartment and office buildings 100 years, and for railways structures, dams, tunnels and other underground engineering works 120 years or more.
Design working life Td (years)
10 to 25
Replaceable structural parts, bearings, girders
15 to 30
Agricultural and similar structures
Building structures and common structures
100 and more
Monumental building or civil structures, bridges
4.2 Target reliability level
Design failure probabilities pd are usually indicated in relation to the expected social and economical consequences. EN 1990  provides the classification of target reliability levels into three classes of consequences (high, normal, low) and indicates the adequate reliability indexes ( for two reference periods T (1 year and 50 years). No explicit link to the design working life Td. is given Similar (-values may be found also in some national standards and international standards ISO . Detail description of the target is given in Chapter III in this Handbook 2.
It should be underlined that the couple of ( values (a and (d) recommended in  for each reliability class (for 1 year and 50 years) correspond to the same reliability level. Practical application of these values depend on the reference 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 (d = 3,8 should be used in the verification of structural reliability. The same reliability level corresponding to the class 2 is achieved when the time period Ta = 1 years and (a = 4,7 are considered. Thus, various reference periods Ta, in general different from the design working life Td, may be used for achieving a certain reliability level.
During their historical development the design methods have been closely linked to the available empirical, experimental as well as theoretical knowledge of mechanics and the theory of probability. The development of various empirical methods for structural design gradually crystallized in the twentieth century in three generally used methods, which are, in various modifications, still applied in standards for structural design until today: the permissible stresses method, the global factor and partial factor methods. All these methods are often discussed and sometimes reviewed or updated.
The following short review of historical development illustrates general formats of above mentioned design methods, indicate relevant measures that are applied to take into account various uncertainties of basic variables and to control resulting structural reliability. In addition a short description of probabilistic methods of structural reliability and their role in further development of design procedures is provided. Detailed description of probabilistic methods of structural reliability is given in Chapter II, Chapter III and in Annex B of this Handbook 2.
The first of the worldwide-accepted design methods for structural design is the method of permissible stresses that is based on linear elasticity theory. The basic design condition of this method can be written in the form
The coefficient k (greater than 1) is the only explicit measure supposed to take into account all types of uncertainties (some implicit measures may be hidden). Moreover, only a local effect (a stress) omax is compared with the permissible stress oper and, therefore, a local (elastic) behaviour of a structure is used to guarantee its reliability. No proper way is provided for treating geometric non-linearity, stress distribution and ductility of structural materials and members. For that reasons the permissible stress method leads usually to conservative and uneconomical design.
However, the main insufficiency of the permissible stress method is lack of possibility to consider uncertainties of individual basic variables and computational models used to assess load effects and structural resistances. Consequently, reliability level of structures exposed to different actions and made of different material may be not only conservative (uneconomical) but also considerably different.
The second widespread method of structural design is the method of global safety factor. Essentially it is based on a condition relating the standard or nominal values of the structural resistance R and load effect E. It may be written as
Thus the calculated safety factor s must be greater than its specified value s0 (for example so = 1,9 is commonly required for bending resistance of reinforced concrete members). The global safety factor method attempts to take into account realistic assumptions concerning structural behaviour of members and their cross-sections, geometric non-linearity, stress distribution and ductility; in particular through the resulting quantities of structural resistance R and action effect E.
However, as in the case of the permissible stresses method the main insufficiency of this method remains a lack of possibility to consider the uncertainties of particular basic quantities and theoretical models. The probability of failure can, again, be controlled by one explicit quantity only, by the global safety factor s. Obviously harmonisation of reliability degree of different structural members made of different materials is limited.
At present, the most advanced operational method of structural design [1, 2] accepts the partial factor format (sometimes incorrectly called the limit states method) usually applied in conjunction with the concept of limit states (ultimate, serviceability or fatigue). This method can be generally characterised by the inequality
where the design values of action effect Ed and structural resistance Rd are assessed considering the design values of basic variables describing the actions Fd = y yF Fk, material properties fd = fk Ym, dimensions ad + Aa and model uncertainties 9d. The design values of these quantities are determined (taking into account various uncertainties) using their characteristic values (Fk,fk, ak, 66), partial factors y, reduction factors yand other measures of reliability [1, 2, 3, 4], Thus the whole system of partial factors and other reliability elements may be used to control the level of structural reliability. Detailed description of the partial factor methods used in Eurocodes method is provided in Handbook 1.
Compared with previous design methods the partial factor format obviously offers the greatest possibility to harmonise reliability of various types of structures made of different materials. Note, however, that in any of the above listed design methods the failure probability is not applied directly. Consequently, the failure probability of different structures made of different materials may still considerably vary even though sophisticated calibration procedures were applied. Further desired calibrations of reliability elements on probabilistic bases are needed; it can be done using the guidance provided in the International standard ISO 2394  and European document EN 1990 .
The probabilistic design methods introduced in the International Standard  are based on a requirement that during the service life of a structure T the probability of failure Pf does not exceed the design valuepd or the reliability index fi is greater than its design value fid
In EN 1990  the basic recommended 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 corresponds to Pd = 6,7 x 10- . These values are related to the design working life of 50 years that is considered for building structures and common structures. In general greater fi - values should be used when a short reference period (one or five years) will be used for verification of structural reliability.
It should be mentioned that probabilistic methods are not yet commonly used in design praxis. However, the developed calculation procedures and software products (for example [7, 8] and ) already enable the direct verification of structural reliability using probabilistic concepts and available experimental data. Recently developed software product CodeCal  is primarily intended for calibration of codes based on the partial factor method.
In Chapter II of this Handbook 2 numerical examples will be presented to illustrate the methods discussed above.
In some cases there is a need to base the design on a combination of tests and calculations, for instance if no adequate calculation model is available. The tests may vary from wind tunnel tests to prototype testing of new structural materials, elements or assemblies. Tests may also be carried out during or after execution to confirm the design assumptions. The extreme example is a proof load. For design by testing the following types of tests can be distinguished:
a) tests to establish directly the resistance for given loading conditions b) tests to obtain specific material properties c) tests to reduce model uncertainties in loads, load effects or resistance models
Test should be set up and evaluated in such a way that the usual required level of reliability is achieved. The derivation of a characteristic or design value should take into account the scatter of test data, statistical uncertainty associated with the number of tests and prior statistical knowledge. If the response of the structure or structural member or the resistance of the material depends on influences not sufficiently covered by the tests such as duration or scale effects, corrections should be made.
When evaluating test results, the behaviour of test specimens and failure modes should be compared with theoretical predictions. When significant deviations from a prediction occur, an explanation should be sought: this might involve additional testing, perhaps under different conditions, or modification of the theoretical model.
The evaluation of test results should be based on statistical methods. In Eurocodes both Bayesian and classical frequentistic methods are used. When frequentistic methods are used a confidence level has to be chosen. The level of the confidence interval may influence the final value. On the average, a confidence level of 0.75 leads to the same result as the Bayesian methods. For this reason 0.75 is chosen in most cases, however also other numbers are used (eg. 0.85 in EN 1995).
In Basis of Design the preference is given for Bayesian methods, which generally is believed to be more consistent with modern reliability theory then frequentistic methods. Moreover, Bayesian methods provide a formal framework for the use of prior knowledge, which is essential especially in the case of small samples and quality control methods. In most Eurocodes special rules for small samples are presented, but in general without formal background.
Rules for execution and evaluation of design by testing are presented in Annex D of EN 1990 Basis of Design. For detailed background and worked examples, the reader is referred to Annex A of this Handbook 2.
The basic concepts of the probabilistic theory of reliability are characterized by two equivalent terms, the probability of failure Pf and the reliability index J. Although they provide limited information on the actual frequency of failures, they remain the most important and commonly used measures of structural reliability. Using these measures the theory of structural reliability may be effectively applied for further harmonisation of reliability elements and for extensions of the general methodology for new, innovative structures and materials.
Historical review of the design methods worldwide accepted for verification of structural members indicates different approaches to considering uncertainties of basic variables and computational models. The permissible stresses method proves to be rather conservative (and uneconomical). The global safety factor and partial factor methods lead to similar results. Obviously, the partial factor method, accepted in the recent EN documents, represents the most advanced design format leading to a suitable reliability level that is relatively close to the level recommended in EN 1990 (J = 3,8). The most important advantage of the partial factor method is the possibility to take into account uncertainty of individual basic variables by adjusting (calibrating) the relevant partial factors and other reliability elements.
Various reliability measures (characteristic values, partial and reduction factors) in the new structural design codes using the partial factor format are partly based on probabilistic methods of structural reliability, partly (to a great extent) on past empirical experiences. Obviously the past experience depends on local conditions concerning climatic actions and traditionally used construction materials. These aspects may be considerably different in different countries. That is why a number of reliability elements and parameters in the present suite of European standards are open for national choice.
It appears that further harmonisation of current design methods will be based on calibration procedures, optimisation methods and other rational approaches including the use of methods of the theory of probability, mathematical statistics and the theory of reliability. The probabilistic methods of structural reliability provide the most important tool for gradual improvement and harmonisation of the partial factor method for various structures from different materials. Moreover, developed software products enable direct application of reliability methods for verification of structures using probabilistic concepts and available data.
Design assisted by testing may be used when there is a need to base the design on a combination of tests and calculations. The tests may vary from wind tunnel tests to prototype testing of new structural materials, elements or assemblies. Tests may also be carried out during or after execution to confirm the design assumptions. An operational technique recommended in  is described in Chapter IV of this Handbook 2.
 EN 1990 Eurocode - Basis of structural design. CEN 2002.
 ISO 2394 General principles on reliability for structures, ISO 1998.
 JCSS: Background documentation, Part 1 of EC 1 Basis of design, 1996.
 Gulvanessian, H. - Calgaro, J.-A. - Holicky, M.: Designer's Guide to EN 1990, Eurocode:
Basis of Structural Design; Thomas Telford, London, 2002, ISBN: 07277 3011 8, 192 pp.
 JCSS: Probabilistic model code. JCSS working materials, http://www.jcss.ethz.ch/, 2001.
 EN 1991-1-1 Eurocode 1 Actions on structures. Part 1-1 General actions. Densities, self-
weight, imposed loads for buildings, CEN 2002
 VaP, Variable Processor, version 1.6, ETH Zurich, 1997.
 COMREL, version 7.10, Reliability Consulting Programs, RCP MUNICH, 1999.
 ISO 13822. Basis for design of structures - Assessment of existing structures, ISO 2001.
 CodeCal, Excel sheet developed by JCSS, http://www.jcss.ethz.ch/.
1. MATHCAD sheet "Beta-Time.mcd"
Mathcad sheet "Beta-Time" is intended for transformation of probability and reliability index Beta" for different reference periods.
Attachment 1 - MATHCAD sheet "Beta-Time.mcd"
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