The design of structures is concerned with the generation of solutions and the verification of safety. Potential design solutions are a result of a process of problem formulation, definition of objective functions, conceptual design and optimization. To control this process, many strategies have been developed over the years. An example of a practical approach is the mix of creative and systematic methods proposed by Cross [1]. The safety of the selected design solution is mostly verified with use of design rules. For engineering practice it is essential that these rules are generally accepted. Codes and standards can be seen as such state-of-the-art guidelines. In this chapter philosophies of safe design are discussed and a proposal is made for the development of design rules for structural adhesive bonded joints.

In the old days the experience of ancient builders guaranteed the safety of structures. The Gothic cathedral of Amiens in France built during the period 1220-1280 is generally seen as one of the best examples of the ultimate knowledge of medieval builders. They made use of the principles described by the Roman master builder Vitruvius approximately 30 BC [2]. A well-known handbook is the 'Vitruvius-Teutsch' [3], which can be regarded as a medieval design guideline. These design methods were used for many centuries. A turning point in the guarantee of the safety of structures is the industrial revolution in the early 19th century. The use of new materials and other applications forced modern builders to change their design philosophy.

Since the introduction of materials as cast iron used in e.g. bridges and industrial buildings, engineers have developed new techniques to design by calculation. One of the pioneers in developing the theory of mechanics is Navier [4]. As a professor on the École des Ponts et Chaussées in Paris, he formulated the basic principles of the theory of elasticity. But probably of greater importance is that he tried to implement these principles in practical engineering, e.g. in making new designs for suspension bridges. Further developments in design are initiated by the use of mild steel, reinforced concrete, aluminium and fibre reinforced materials in the 20th century. Related to these new calculation methods, is the introduction of the safety factor.

The safety factor used in design takes into account the uncertainties about loads and strengths. In a comprehensive overview Beeby [5] shows the major developments in the application of safety factors over the years. The early development of ideas on safety shows that these are strongly related to the method of design, as is properly documented in a paper by Pugsley [6]. In the early 19th century the ultimate load design was introduced for cast iron structures. In this approach the load is increased by a factor of safety of 4 to 6, and this value is compared with experimental determined resistance of cast iron beams and columns. During the second half of that century, however, the theory of elasticity was well understood and a direct link was made with permissible stresses. In this approach the stress in the material is limited to some fraction of its failure stress, and this value is compared with the calculated stress under a specified loading. The permissible stress design approach was widely accepted by the end of the 19th century and is extensively used in the 20th century. The specification loadings and permissible stresses were mainly based on engineering practice, and were strongly depending on the field of application. The resulting inconsistencies and the unknown probability of failure reached, were the reasons for a growing dissatisfaction amongst leading structural engineers in several fields about these approaches around the Second World War. During this period more consistent safety concepts based on a statistical approach were discussed.

The possibilities of designing a structure directly for a specified probability of failure during lifetime were first investigated in the aeronautical industry during the Second World War. Serious attempts to apply those probabilistic concepts to design were made a decade later. From that time on probabilistic reliability methods have been developed and implemented in design practice, see e.g. [7]. Exact methods take the statistical distribution of all variables into account, while more widely used approximate methods make some simplifications. A disadvantage of all these probabilistic reliability methods is that the calculations can only be performed by experts and that they are too extensive for daily design practice.

An other more consistent safety concept developed simultaneously, known as the partial safety factor approach, combines the ultimate load design and permissible stress design approaches. This approach has it roots in composite materials such as reinforced concrete. The general principles were e.g. adopted by the Comité Européen du Béton (CEB) [8] in 1964. The aim of the approach is "to achieve a more uniform safety over the whole structure than that given by traditional methods". This could be reached by using specific safety factors for each type of loading and material dependent safety factors. This flexible system has the advantage that it can deal with different levels of uncertainties for both loadings as well as materials. The use of statistical means led CEB to describe the partial safety factor approach as 'semi-probabilistic'. This indicates that there is a direct relation between this partial safety factor approach and probabilistic methods. Modern structural design codes make use of these structural reliability methods, and are based on the so-called limit state concept.

Discussions about the use of new theories to quantify the safety of a structure are going back to the years before the Second World War. E.g. Van den Broek [9] developed during this period attempts to modify the permissible stress approach, by taking the post-elastic behaviour of structural components into account. Freudenthal [10] on the other hand discussed the possibilities of using statistical techniques to quantify the safety factor within the scope of the generally used permissible stress approach. The common issue within all these papers, see also the discussion of Pugsley [6] in 1951 about future trends, is the application of the term "probability of failure". These developments and discussions can be seen as the first steps in developing a new concept of design, nowadays known as the limit state concept.

The limit state is defined as the condition in which the structure is no longer capable to fulfil its function under given actions. In practice this means that the structure collapses or that the structure can not be used normally. A mathematical presentation is given by the limit state function defined as the difference between the resistance (R) and the action effect (S):

(1)

As long as Z > 0 no failure will occur, while for Z < 0 the structure fails; the limit state is reached in case Z = 0. Both the resistance and the action effects are regarded as stochastic variables, which can be represented by their probability density functions f_R(r) and f_S(s) respectively. In case the resistance and the action effects are statistically independent, their combined probability function is defined as f_R(r)·f_S(s). This function can be graphically presented by contours in its R-S plane, as indicated in figure 1. The probability of failure is equal to the capacity of the combined probability function for which Z < 0. Mathematically this probability of failure is given by:

(2)

This integral can be equally applied to time-dependent limit state functions. To solve it, several methods have been developed, see e.g. [7]. Exact probabilistic methods like those based on a direct numerical solution of the above given integral and the Monto Carlo simulation, take the probability density functions of all variables into account, while approximate methods like the first-order reliability method (FORM) and second order reliability method (SORM,) simplify the problem. These exact and approximate methods are also known as level III and level II methods respectively. The essence of the level II methods is that these linearise the limit state function Z around a point of the limit state Z = 0 with the highest probability. This point is known as the design point. The advantages of the level II methods over the level III methods is that these are easier to use for practical applications, and that these give additional information about the contribution of each variable to the probability of failure. With use of these probabilistic reliability methods it is now possible to quantify the safety of a structure, and perhaps more importantly, the results can be used to calibrate safe design rules for daily engineering practice.

Figure 1 - Statistical presentation of the limit state concept

(3)

The most convenient structural reliability method for daily design practice is the partial safety factor approach, also known as the level I method. The reliability of a structure or component with respect to failure is checked on basis of the limit state function in combination with design values for the action and the resistance. These design values follow from so-called characteristic values combined with matching partial safety factors. The details necessary to understand the essence of this approach entirely are reviewed in the following section.

Within the partial safety factor approach the safety of a structure or component has to be validated by comparing the so-called characteristic values for the action S_k and the resistance R_k:

(4)

(5)

where k is the constant of the normal probability distribution. In case of a probability less than 5% the value of k is equal to 1.64. The values of the partial safety factors on the other hand have to be determined by calibration. In the second recommendations of CEB [13], it is stated that these values are only intended to take account of those aspects not yet amenable to statistical treatment. This means that the safety is based on engineering judgement. But developments in probabilistic techniques since 1970, open new ways to calibrate partial safety factors.

(6a)

(6b)

(7a)

(7b)

The relation between level I and II methods can now be determined with use of equation (4):

(8a)

(8b)

The above given simplified presentation of the procedure of determining partial safety factors can be seen as the fundamental method of calibrating design rules with use of probabilistic techniques.

Figure 2 - Location of the design point on the failure boundary Z = 0

From a practical point of view the partial safety factors take into account the effects of the stochastic nature of respectively the action and the resistance. The partial safety factor of the action, also known as the load factor, covers:

- the uncertainty in the action model;

- the uncertainty in the assessment of effect of the actions.

The partial safety factor of the resistance on the other hand, also known as the material factor, covers:

The above mentioned aspects have to be considered while calibrating partial safety factors.

In principle the partial safety factor approach according to equation 4 defines the action S and the resistance R in general terms. For practical purpose their values can represent the applied load and the strength of the component respectively. But in some cases it is more convenient to determine the action effect and to compare this value with a failure criterion. E.g. for an adhesive bonded joint the stress state within the bondline due to the applied load is calculated, and compared with a proper yield criterion. Design rules mostly represent the action S and the resistance R in a formulation, which is the most convenient one for the considered situation.

The partial safety factor approach opens the possibility of drafting a coherent set of design rules. Instead of determining the partial safety factors for each possible application according to a procedure described in the preceding section, researchers and code-writers have harmonised design rules for wider fields of applications. They found with extended probabilistic studies rather consistent values for the partial safety factors. The advantage for daily design practice is that a consistent set of design rules with only a limited number of partial safety factors can be used.

Nowadays the partial safety factor approach has been applied in many design codes. The Eurocodes primarily used for building applications, [15] to [23], give characteristic values of both actions and resistances together with the matching safety factors. Also the recently published Eurocomp Design Code and Handbook [24] for polymer composites is based on this approach. Following the discussions of the Delft IABSE Colloquium in 1996 [25] it becomes clear that each of these codes has its own format due to a different interpretation of the partial safety factor approach. To avoid the possibility of drafting inconsistent and probably unsafe design rules, it is necessary to define a coherent system of calculating action effects and resistances with matching characteristic values and partial safety factors.

The safety of most of the current design codes is guaranteed by experience of many decades of engineering practice. Probabilistic techniques have been used for final tuning of safety factors. E.g. in case of the Dutch building codes extensive calibration studies were performed during the 80's [14], and different part of the Eurocode 3 [17] and Eurocode 4 [18] have been considered in this connection. Additional to the tuning of existing design rules, probabilistic techniques has the possibility of calibrating design rules for new applications like adhesive bonded joints.