Section Basis Of Design Section Basis Of Design

2.1 Requirements C21 Requirements

Eurocode 2, Section 2, Part 1.1 states that concrete structures should be designed in accordance with the general rules of EN 1990 and with actions defined in EN 1991. EN 1992 has some additional requirements.

In particular, the basic requirements of EN 1990 Section 2 are deemed to be satisfied for all concrete structures if limit state design is carried out with the partial factor method in accordance with EN1990, and if actions are defined in accordance with EN1991, and if combinations of actions in accordance with EN1990, and finally if resistance, durability and serviceability are dealt with in accordance with EN1992.

2.1.2 Reliability management C2.1.2 Reliability management

EC2 points 2.1.2 and 2.1.3 refer to EN1990 section 2 for all rules in relation to reliability, design working life and quality management measures. These rules, as well as the basic concepts of structural reliability, will be referred to later.

By the definition given by EN1990, structural reliability is the ability of a structure or a structural member to fulfil the specified requirements for which it has been designed; it includes structural safety, serviceability and durability.

Given the random nature of quantities involved in structural design (actions, geometry, restraints, strength of materials, etc.), the assessment of structural reliability cannot be set up by deterministic methods, but requires a probabilistic analysis. The objective of safety verification is therefore to keep failure probability, i.e. probability that a certain danger condition is attained or exceeded, below a fixed value. This value is determined as a function of type of construction, influence on safety of people and damage to goods.

Every situation which is dangerous for a construction is referred to as a "limit state". Once a construction has attained this condition, it is no longer able to fulfil the functions for which it has been designed. Limit states are of two types: ultimate limit states and serviceability limit states -depending on the gravity of their consequences. Exceeding the first causes collapse of the whole structure or of part of it, exceeding the second causes limited damage that makes the structure unfit for the requirements of the project. Exceeding serviceability limit states can be reversible or irreversible: in the first case, no consequences of actions exceeding the specified service requirements will remain once those actions are removed; in the second, case, some consequences will remain. For example, a crack width limit state with limited width is a reversible limit state, whereas one defined by a high width is irreversible (in fact, if the crack width is high, once the actions are removed the cracks cannot close).

For a given limit state, let us define S and R as two random variables representing respectively stress and strength. We recall that by 'stress' we mean any effect produced in the structural members by actions applied or by any other effect such as strain, cracking, increase of reinforcing steel corrosion. 'Strength', on the other hand, means the capacity of a structure to respond to a given stress. A rigorous assessment of structural safety against a relevant limit state can be carried out by first introducing a safety factor FS, defined as the ratio between strength R and stress S, or alternatively by a safety margin Ms, defined as the difference between R and S:

Both these factors are random variables like R and S. The distribution of Fs or Ms is then determined on the basis of the statistical distribution of actions, strengths and geometrical dimensions of the structure, also taking account of the randomness of the structural scheme. Finally, the probability of failure is related to a fixed reference period of time T through one of the following expressions:

Pf=P {R/S < 1} = P{FS < 1} or Pf = P {R-S < 0} = P {MS < 0}.

Pf represents the probability that failure arises, i.e. that the considered limit state is attained or exceeded at least once during T.

This analysis (known as level 3 method) is very complex. Because of the difficulty of calculation and of the limitation of available data (data which often fail to give the probabilistic distributions necessary for calculation), this method is of limited applicability to the design practice.

Alternatively, if only the first and second order moments (averages and standard deviations) of the random variables R and S, but not their statistical distributions, are known, the probability of failure can be estimated based on a p index, called the "reliability index". Assuming that Ms is linear, it was first defined by Cornell as the ratio between the average value ^m of Ms and its standard deviation CTm:

In circumstances where R and S are not correlated (note that in case of normal distributions non-correlation is equivalent to statistical independence), ß is expressed as follows:

, where pR ,pS,aR ,aS are the averages and standard deviations of R and S.

This method (known as "level 2" method or "P-method") does not generally allow assessment of the probability of failure, with the exception of the particular case where the relation between Ms and the random variables of the problem is linear and the variables have normal distribution. The probability of failure, i.e. the probability that the safety margin Ms assumes non-positive values, is given by the distribution function ^m of Ms calculated in 0:

Introducing m as the normalized variable of the safety margin Ms, m_ MS - MM

the result: Ms = mM + m • ctM substituted in the expression of Pf gives:

Pf = P{Ms <0} = P{Mm + m• *0} = P{m< -pM/aM} = P{m <-ß} = Om(-ß) = 1-Om(ß) where Om indicates the distribution function of m.

The reliability index may be expressed in geometrical terms. In fact, if we introduce the normalized strength and stress variables [r = (R-pR)/aR and s = (S-pS)/cts], the limit condition (Ms = 0) is represented in the r - s plane by a line that divides the plane into a safe region and an unsafe region (Figure 2.1). The distance from the origin of the axis of this line equals the reliability index (in circumstances where R and S are not correlated), so the verification of safety is carried out by assigning a given value to this distance.

UNSAFE ppREGION

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