Structural Reliability Assessment Including Variability of Reinforcement Cover Based on Measurements on Selected Buildings

The problem of shaping building structures due to ensur their reliable and safe use has existed since the beginning of human construction. Despite a fairly widespread understanding of the importance of structural safety and the need to reasonably account for the random nature of the parameters, reliability analysis is not a widely accepted design practice. Unfortunately, in order to simplify calculations or reduce costs at the structure design stage, a thorough examination of the impact of the dispersion of random variables on a specific limit state of a structure is often omitted. It is assumed that safety is guaranteed by using quantile multipliers of random variables [1]. Structural failures are related, among other things, to the quality and durability of the construction materials used and the quality of workmanship, which is difficult to define [2, 3]. Statistics show that the main causes of catastrophes and failures not resulting from random events were the poor technical condition of the structures (about 29%) and faulty workmanship (about 20%) [4]. One such error is the lack of due care in the process of constructing the reinforcement of the elements, which most often results in increased reinforcement cover. According to PN-EN 13670, the permissible positive deviation of reinforcement cover depending on the height of the element and the tolerance class may not be greater than 10 mm for elements with h 150 mm and 15 mm for an element with h= 400 mm. However, as the measurements carried out on selected real objects have shown, the standard conditions are exceeded in most cases.

The effect of changing the reinforcement cover thickness on the reliability of the structure is not a very widely described issue. Information on such studies in the context of simple elements such as reinforced concrete beams can be found in the publication [5]. The effect of changing the thickness of reinforcement cover on meeting the requirements of the reliability classes defined in [1] and [6], has also been investigated [7]. The effect of fabrication errors on the deformation and cracking of reinforced concrete beams was also examined [8]. The analyses carried out showed a significant effect of changing the reinforcement cover thickness on the performance and reliability of reinforced concrete beams and, together with measurements on real objects, inspired further research and analysis.

The obtained reinforcement cover thickness distributions were used to determine the reliability of a floor slab with a simple static scheme – a simply supported beam. This simplification allows easy and clear application of the FORM analytical method for reliability assessment. More complex systems will be considered in subsequent stages of the work carried out using the Monte Carlo method and research models using the finite element method.

In order to determine the effect of cover variation on the reliability of slab elements with different heights, analyses were carried out for four slabs differing in span length and section height. Each of the slabs was designed for the same typical loads for residential buildings, i.e. permanent load of 1.4 kN/m² and an imposed load of 2.8 kN/m². Concrete class C25/30, steel with yield strength f_y=500 MPa, and reinforcement cover of 25 mm was used. Table 1 summarises the spans and slab thicknesses considered and the calculated moment values at the centre of the span and the selected reinforcement area A_s.

Reliability analyses were performed using the first-order FORM method considered to be one of the most efficient approximate methods for calculating reliability measures. This method allows the complex nature of random variables to be taken into account by using information about the probability distributions of the random variables considered. The values of failure probability and reliability index obtained by this method are sufficiently accurate for most engineering structures, with considerably less effort. Reliability analyses were carried out using the FREeT programme from Cervenka Consulting, which allows statistical analysis, sensitivity assessment and component reliability.

In order to estimate the reliability index using the FORM method, a margin of safety function was formulated in the form: (1) Δ=R−E \Delta = R - E where: Δ–margin of safety, R – resistance, E – load effect.

The bending resistance of the reinforced concrete slab was determined for a width of 1 m from the formula: (2) R=As⋅fy⋅h−c−ϕ2−As2⋅.fy22b⋅fc R = {A_s} \cdot {f_y} \cdot \left( {h - c - {\phi \over 2}} \right) - {{A_s^2 \cdot .f_y^2} \over {2b \cdot {f_c}}} where: A_s – cross sectional area of reinforcement, f_y – yield strength of reinforcement, f_c – compressive strength of concrete, h – height of section, c – reinforcement cover thickness, ϕ – diameter of a reinforcing bar, b – width of a cross-section.

The effect of the interaction is the moment in the middle of the span, which depends on the value of the permanent and variable load and the span: (3) E=0.125⋅l2⋅g+p E = 0.125 \cdot {l^2} \cdot \left( {g + p} \right) where: l – span, g– permanent load, p – variable load.

An important step in reliability analysis is to decide which parameters to take as deterministic and which as random variables. In the analyses presented here, the dimensions of the slab (l, h, b) and the diameter and cross-sectional area of the reinforcement were taken as deterministic values. The other parameters assumed as random variables were described by mean value, standard deviation and type of distribution, as shown in Table 2. The coefficients of variation were assumed on the basis of literature and standards [10] for permanent loads of 6%, for variable loads of 30% and for yield strength of reinforcement of 8%. For the compressive strength of concrete, the recommended standard deviation value σ_c=4.86 N/mm² was adopted.

The distribution parameters of the variable reinforcement cover thickness were selected on the basis of the previously discussed measurement results on real objects. The measured results of the reinforcement cover from the sites with a basic value of 25 mm were inserted into the FREeT software. On the basis of the histogram created from the inserted data, the mean value, standard deviation and asymmetry coefficient describing the best-fitting distribution – in this case log-normal [Fig. 4] – were determined.

Figure 4.

The distributions presented were used to generate 10⁶ samples using the Latin hypercube method, which were the input data for the formulated margin of safety reserve function. As a result of the calculations, distributions of the slab resistance R and the effect of actions E were obtained (Fig. 5).

Figure 5.

On the basis of the distributions obtained, the probability of failure P_f understood as the probability that R-E will be less than 0, was calculated. The value of the reliability index β related to the probability of failure of an element was also determined with the following relation: (4) Pf=ϕ−β {P_f} = \phi \left( { - \beta } \right) where: ϕ – Laplace function

The determined values for the probability of failure and reliability index are shown in Table 3.

The calculated values of the reliability index should meet the requirements that we can find in [1] and [6]. The required values of β depend on the assumed failure consequence class and take values in the range of 1.5 to 5.2. The diagram below (Fig. 6) compares the results obtained with the required values in [1] for a reference period of 50 years and for moderate and small consequences according to [6].

Figure 6.

Measurements on real objects showed that in many cases the deviation of the reinforcement cover exceeds the permissible values, which significantly affects the safety of the structure. Reliability analyses carried out taking into account the obtained distribution of variable cover thickness (based on measurements on real objects), randomly varying material characteristics and loads showed that for the random variables adopted in this way, the reliability indices do not meet the requirements for reliability class RC 2 and higher. It is obvious that the smaller the thickness of the slab, the greater the significance of any deviation causing a reduction in the useful section height, however, this is not explicitly taken into account in the standard recommendations (10 mm for elements with h ≥ 150 mm and 15 mm for an element with h = 400 mm). Taking into account the measurements of reinforcement cover on real objects and the analyses carried out on their basis, it was concluded that there is a need for further research and analyses to clarify the permissible execution deviations depending on the height of the element and to assess their influence on the reliability of the structure.