Chapter 1
Measures of Structural Reliability

1.1 Introduction

The manner in which an engineering structure will respond to loading depends on the type and magnitude of the applied load and the structural strength and stiffness. Whether the response is considered satisfactory depends on the requirements that must be satisfied. These include safety of the structure against collapse, limitations on damage, or on deflections or other criteria. Each such requirement may be termed a limit state. The 'violation' of a limit state can then be defined as the attainment of an undesirable condition for the structure. Some typical limit states are given in Table 1.1.

Table 1.1 Typical limit states for structures

From observation it is known that very few structures collapse, or require major repairs, etc., so that the violation of the most serious limit states is a relatively rare occurrence. When violation of a limit state does occur, the consequences may be extreme, as exemplified by the spectacular collapses of structures such as the Tay Bridge (wind loading), Ronan Point Flats (gas explosion), Kielland Offshore Platform (local strength problems), Kobe earthquake (ductility), etc.

The study of structural reliability is concerned with the calculation and prediction of the probability of limit state violation for an engineered structural system at any stage during its life. In particular, the study of structural safety is concerned with the violation of the ultimate or safety limit states for the structure. More generally, the study of structural reliability is concerned with the violation of performance measures (of which ultimate or safety limit states are a subset). This broader definition allows the scope of application to move from structural criteria as specified in traditional design codes (Chapter 9) to broader-based performance requirements for structures, such as might be used in design optimization processes (Chapter 11).

In the simplest case, the probability of occurrence of an event such as limit state violation is a numerical measure of the chance of its occurrence. This measure either may be obtained from measurements of the long-term frequency of occurrence of the event for generally similar structures, or it may be simply a subjective estimate of the numerical value. In practice it is seldom possible to observe for a sufficiently long period of time, and a combination of subjective estimates and frequency observations for structural components and properties may be used to predict the probability of limit state violation for the structure.

In probabilistic assessments any uncertainty about a variable (expressed, as will be seen, in terms of its probability density function) is taken into account explicitly. This is not the case in traditional ways of measuring safety, such as the 'factor of safety' or 'load factor'. These are 'deterministic' measures, since the variables describing the structure, its strength and the applied loads are assumed to take on known (if conservative) values about which there is assumed to be no uncertainty. Precisely because of their traditional and really quite central position in structural engineering, it is appropriate to review the deterministic safety measures prior to developing probabilistic safety measures.

1.2 Deterministic Measures of Limit State Violation

1.2.1 Factor of Safety

The traditional method to define structural safety is through a 'factor of safety', usually associated with elastic stress analysis and which requires that:

where si(?) is the i th applied stress component calculated to act at the generic point ? in the structure, and spi is the permissible stress for the i th stress component.

The permissible stresses spi are usually defined in structural design codes. They are derived from material strengths (ultimate moment, yield point moment, squash load, etc.), expressed in stress terms sui but reduced through a factor F:

where F is the 'factor of safety'. The factor F may be selected on the basis of experimental observations, previous practical experience, economic and, perhaps, political considerations. Usually, its selection is the responsibility of a code committee.

According to (1.1), failure of the structure should occur when any stressed part of it reaches the local permissible stress. Whether failure actually does occur depends entirely on how well si(?) represents the actual stress in the real structure at ? and how well spi represents actual material failure. It is well known that observed stresses do not always correspond well to the stresses calculated by linear elastic structural analysis (as commonly used in design). Stress redistribution, stress concentration and changes due to boundary effects and the physical size effect of members all contribute to the discrepancies.

Similarly, the permissible stresses that, commonly, are associated with linear elastic stress analysis are not infrequently obtained by linear scaling down, from well beyond the linear region, of the ultimate strengths obtained from tests. From the point of view of structural safety, this does not matter very much, provided that the designer recognizes that his calculations may well be quite fictitious and provided that (1.1) is a conservative safety measure.

By combining expressions (1.1) and (1.2) the condition of 'limit state violation' can be written as

Expressions (1.3) are 'limit state equations' when the inequality sign is replaced by an equality. These equations can be given also in terms of stress resultants, obtained by appropriate integration:

where Ri is the i th resistance at location ? and Si is the i th stress resultant (internal action). In general, the stress resultant Si are made up of the effects of one or more applied loads Qj; typically

where D is the dead load, L is the live load and W is the wind load.

The term 'safety factor' also has been used in another sense, namely in relation to overturning, sliding, etc., of structures as a whole, or as in geomechanics (dam failure, embankment slip, etc.). In this application, expressions (1.3) are still valid provided that the stresses sui and si are interpreted appropriately.

1.2.2 Load Factor

The 'load factor' ? is a special kind of safety factor developed for use originally in the plastic theory of structures. It is the theoretical factor by which a set of loads acting on the structure must be multiplied, just enough to cause the structure to collapse. Commonly, the loads are taken as those acting on the structure during service load conditions. The strength of the structure is determined from the idealized plastic material strength properties for structural members [Heyman, 1971].

For a given collapse mode (i.e. for a given ultimate 'limit state'), the structure is considered to have 'failed' or collapsed when the plastic resistances Rpi are related to the factored loads ?Qj by

where RP is the vector of all plastic resistances (e.g. plastic moments) and Q is the vector of all applied loads. Also, WR( ) is the internal work function and WQ( ) the external work function, both described by the plastic collapse mode being considered.

If proportional loading is assumed, as is usual, the load factor can be taken out of parentheses. Also the loads Qj usually consist of several components, such as dead, live, wind, etc. Thus (1.5) may be written in the form of a limit state equation:

with 'failure' denoted by the left-hand side being less than unity.

Clearly there is much similarity in formulation between the factor of safety and the load factor as measures of structural safety. What is different is the reference level at which the two measures operate: the first at the level of working loads and at the 'member' level; the second at the level of collapse loads and at the 'structure level'.

1.2.3 Partial Factor ('Limit State Design')

A development of the above two measures of safety is the so-called 'partial factor' approach. For limit state i it can be expressed at the level of stress resultants (i.e. member design level)...