Assessment of the concrete strength in existing buildings using a finite population approach
Introduction
In the safety assessment of existing buildings, quantifying the “as-built” material properties is of the utmost importance due to the impact that it has on the subsequent application of safety assessment methods. In the case of reinforced concrete (RC) buildings, the concrete compressive strength is a material property that requires careful consideration [1] due to its inherent variability. This fact leads to the usual consideration of the concrete strength as being a random variable that has a certain (unknown) level of aleatory uncertainty [2]. This aleatory uncertainty is related to the inherent variability of the hardened concrete strength in existing structures [3] which can reach large values [4], [5], often exceeding a coefficient of variation (CoV) of 20% [6]. Among other factors, this variability is associated with mix, casting and curing operations, which require a significant level of workmanship. Several studies (e.g. see [3], [7], [8]) have analyzed the impact of workmanship on the strength of hardened concrete and found that it can induce several types of variability depending on the structural system being analyzed. Primarily, expected variations can be associated to batch-to-batch variability, involving the randomness related mainly with the construction management and planning and with quality control. Likewise, member-to-member variability can occur due to the influence of workmanship in casting operations. Variations of the concrete strength can also be expected within each structural member due to the previously mentioned factors. Moreover, a recent study [9] also described cracking, damage and the selection of the testing positions within the length of a structural element as sources of potential variability.
In addition to the aleatory uncertainty associated with the concrete strength, epistemic uncertainty will also be generated due to the lack of knowledge associated with non-surveyed structural elements. Since survey plans only comprise tests on a few structural members in order to minimize the damage and the cost of inspection operations, the selection of a given set of elements to be tested instead of another will generate uncertainty. This uncertainty is even more important due to the low number of material tests that are generally carried out in existing buildings, a trend partially supported by existing norms (e.g. [10], [11], [12], [13]). Often, standards regulating the assessment of existing buildings require a limited number of tests/inspections to be performed at each storey and for each type of primary component that is part of the building in order to obtain estimates of the mean values of the material properties. Nonetheless, as referred in [14], current building codes do not address the uncertainty level in the survey results and neglect the impact that sampling may have on the estimate of the dispersion of concrete strength (specifically on the estimate of the CoV) and on the corresponding estimate of the mean value. Therefore, controlling the epistemic uncertainty about the CoV of the concrete strength is a key component of a survey framework since it will affect the variability of the estimate (i.e. its precision), especially when it is based on a reduced number of tests. Moreover, this uncertainty is also seen to depend on the relation between the number of structural elements that are not tested during survey operations and the total number of structural elements of the population.
To control the extent of this uncertainty in survey operations and its impact on the estimate of the mean value of the concrete compressive strength in existing buildings, a method based on finite population statistics is proposed herein. The proposed approach will enable to effectively control the uncertainty in the estimates of the variability and of the mean value of the concrete strength in a population to improve their reliability. By accounting for the number of structural elements that are not tested during survey operations, the proposed method overcomes limitations of current standard methods and enables the development of more consistent survey frameworks to assess concrete strength in existing buildings.
Section snippets
Assessing statistical parameters in finite populations
In statistics, a population is said to be finite when it is possible to count all its elements. Statistical parameters characterizing these populations have specific features which are associated to finite size conditions. To evaluate the exact value of these parameters, knowledge about all the N independent elements of the population is required. If all the N elements are observed, the population mean is then:where U represents the population, N is the finite population size and
Discretizing the concrete strength and disaggregating its variability
By depending on both n and N, finite population statistics enable to control the epistemic uncertainty about the estimates of the mean and of the variability of a population using data provided by a ratio of n/N elements. This approach is somehow similar to the uncertainty reduction principle that underlines the procedures in current standards (e.g. see [10]) where it is implicit that an increase in the number of structural elements that are tested during survey operations will lead to a
An alternative method to estimate the finite population CoV of concrete strength
An alternative approach is proposed herein to estimate the variability (i.e. the CoV) of a finite population of concrete strength values using auxiliary information obtained from non-destructive tests (NDTs). These tests are often used in survey campaigns since they induce limited levels of damage to the structural components and can be used in a larger number of elements usually at a lower cost. An example of this kind of methods is the surface hardness determination test using the rebound
Validation of the proposed procedure using experimental data
To assess the validity of the proposed finite population approximations defined by Eqs. (14) and (15), five additional datasets of RN and core strength values were considered. Datasets C1–C4 correspond to pairs of data extracted from multi-storey RC buildings constructed in the mid-1990 s that were surveyed within the present study. Each pair has a core strength value evaluated in a structural element and a RN value from the same location. Since dataset C4 presented a wide range of concrete
Analysis of the ratios
Fig. 5 presents the ECDFs of the ratios obtained using the different strategies defined in the previous Section (i.e. RMP1, RMP2, RM1, RM2 and RM3). As mentioned before, all the computed ECDFs are conditioned to a sample size corresponding to n/N = 0.30. Hence, the presented ECDFs reflect the sampling uncertainty associated with the selection of different test locations for the rebound hammer test within a given finite population.
The results show that the RMP1 and RMP2 models lead to data
Conclusions
A finite population statistics-based approach that uses auxiliary information for the assessment of concrete strength in existing RC buildings has been presented in this study. The proposed approach effectively controls the uncertainty in the estimate of the variability of the concrete strength in a population as well as the uncertainty in the estimate of the mean value of the concrete strength. The approach relies on a discretization of the concrete strength distribution within the building
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