Safety format

Safety format is the procedure that allow to perform the verification of a structure with respect to a particular limit state. This verification consists on a probability-based method and fib MC 2010 [2] proposes the following safety formats:

- Probabilistic safety format, also called fully probabilistic design method, is a method that permits to quantify the reliability requirements in terms of reliability index 𝛽 and reference period. This method is more suited for the assessment of existing structures, in particular for the computation of residual service life

- Partial safety factor format, is a simplified verification concept, based on past experiences and calibrated in such a way that general reliability requirements are satisfied. Usually adopted for verifying structural design - Global resistance format, as the name foretells, it consists on a global

resistance verification with partial safety factors. It is suited for design based on non-linear analysis, where numerical simulations are adopted to perform the verification of limit states

55

- Deemed-to-satisfy approach, consists on a set of rules for dimensioning, material and product selection and execution procedures, given in standard, aiming at maintaining the target reliability lower than the relevant limit state. Used for verifying service life design of new structures

- Design by avoidance, applicable both for verification of traditional structural design and design for service life, consists on avoiding or reducing harmful effects (e.g. protecting the structure from certain loads such as wind, wave loads, impacts and so on)

In the following, more details are given for the first three safety formats.

2.6.1 Probabilistic safety format

The probabilistic safety format consists on a probabilistic assessment of the safety of the structure, by means of an estimation of the failure probability 𝑃 , or, mutually, of the reliability index 𝛽.

The procedure is the same discussed in the previous section, where to estimate the failure probability, the verification is:

𝑃 = 𝑃[𝑔(𝑋 ) ≤ 0] ≤ 𝑃_, 𝑖 = 1,2, … , 𝑁 (2.42) where 𝑔(𝑋 ) is the performance function (or limit state function), the 𝑔(𝑋 ) ≤ 0 represents the failure or unsafe condition and 𝑃_, are the target probability of failure indexes, here reported in section 2.5, according to Table 2.2.

The relation between the reliability index 𝛽 and the failure probability 𝑃 is reported in this chapter, in sub-section 2.3.2. Moreover, for the methodology to be adopted for the evaluation of the basic variables 𝑋 as well as the failure probability 𝑃 it is suggested to look at the previous sections, in particular section 2.4.

2.6.2 Partial safety factor format

The idea behind this format is to separate the treatment of uncertainties from various cases by means of design values assigned to variables. According to fib MC 2010, this consists on selecting the representative values of variables and the partial safety factors so to meet reliability requirements in terms of 𝛽 index.

56

Distinction should be made between basic and other variables. The formers are actions (𝐹), material or product properties (𝑋), some geometrical quantities (𝑎), variables which account for the model uncertainties (𝜗). For these, design values include reliability margins. For the other variables, whose dispersion may be neglected or is covered by a set of partial factors, their most likely values are assumed.

The requirement consists on the following expression:

𝑔(𝐹 , 𝑋 , 𝑎 , 𝜗 , 𝐶) ≥ 0 (2.43)

with 𝐶 representing serviceability constraints.

2.6.2.1 Design values of basic variables

Design variables of basic values are expressed as following:

- Design values of actions:

𝐹 = 𝛾 𝐹 (2.44)

where 𝐹 is the representative value of actions and 𝛾 is a partial safety factor

- Design values of material or product property:

𝑓 = 𝑓 /𝛾 (2.45)

or if uncertainty in the design model is considered:

𝑓 = 𝑓 𝛾⁄ = 𝑓 (𝛾⁄ ∗ 𝛾 ) (2.46)

where 𝑓 is the characteristic value of the resistance, 𝛾 is a partial safety factor for a material property, 𝛾 is a partial safety factor related to uncertainty of resistance model plus geometric deviations if they are included in the model, 𝛾 = 𝛾 ∗ 𝛾 is a partial safety factor for a material property accounting for the model uncertainties

- Design values of geometrical property are usually taken equal to their design values 𝑎

57 2.6.2.2 Determination of partial safety factors

Partial safety factors for basic variables are expressed as following:

- Materials:

𝛾 = 𝛾 𝛾 (2.47)

𝛾 = 𝛾 𝛾 (2.48)

where 𝛾 is a partial safety factor accounting for model uncertainty set equal to 1.05 and 𝛾 is a partial safety factor accounting for geometrical uncertainty equal to 1.05. Assuming normal distribution for material uncertainties the value of 𝛾 is equal to 1.5 for concrete cylinder compressive strength with a coefficient of variation equal to 0.15, while for bar reinforcements 𝛾 is equal to 1.15 with a coefficient of variation equal to 0.05. Finally, the related target of reliability is defined by 𝛽 = 3.8 according to Table 2.2.

- Permanent actions (G) and variable loads (Q):

𝛾 = 𝛾 𝛾 and 𝛾 = 𝛾 𝛾 (2.49)

where 𝛾 is a partial safety factor accounting for model uncertainty and set equal to 1.05 while 𝛾 and 𝛾 are partial safety factors for permanent and variable actions respectively, described in Section 2.4.3, Eq. (2.39).

2.6.3 Global resistance format

The uncertainties of the structural behaviour are integrated in a global design resistance and can also be expressed by a global safety factor. Again, these values should be selected in order to meet the requirements for the reliability index 𝛽.

The representative variable for the global resistance is the structural resistance 𝑅. The uncertainty is expressed by the following values of resistance:

- 𝑅 mean value of resistance

- 𝑅 characteristic value of resistance (corresponding to a probability of failure of 5%)

- 𝑅 design value of resistance

58

The value of action F is considered in the same way as in the partial safety factor method, sub-section 2.6.2.

The safety condition is met when:

𝐹 ≤ 𝑅 , 𝑅 = 𝑅 /𝛾^∗ 𝛾 (2.50)

where 𝐹 is the design external action defined according to the partial factor format; 𝛾^∗ is denoted as the global resistance safety factor, which accounts for material aleatory uncertainties; 𝛾 represents the resistance model uncertainty safety factor, which accounts for the resistance model uncertainty.

The value of the model uncertainty factor depends on the quality of formulation of resistance model, recommended values are:

- 𝛾 = 1.0 for no uncertainties - 𝛾 = 1.06 for low uncertainties - 𝛾 = 1.1 for high uncertainties

It is important to underline the differences between global and partial safety factors. The former refer to the global structural response evaluated by means of mean values of material properties, instead, partial safety factors refer just to each material property (i.e. concrete compressive strength, reinforcement yielding strength) evaluated with its characteristic value for local verification of structural members [22].

59

3 RC building: design characteristics

This work of thesis is about the evaluation of reliability for robustness of a real building designed in seismic area. Previous works of thesis had already investigated the robustness, but not in probabilistic way, of the same building and they are the following: “Robustezza Strutturale di edifici intelaiati in calcestruzzo armato:

analisi parametrica e nuove proposte progettuali” by Fortunato Mauro [26] and

“Robustezza Strutturale di costruzioni multipiano in calcestruzzo armato: analisi parametrica di telai 2D per mezzo di modelli globali e locali” by Luca Capri [27].

In the following, the building is described in terms of design characteristics according to code rules (i.e. capacity design) and in the end, the frame, object of this work of thesis, is presented, as a result of the conclusions made in previous studies.