Digital Image Correlation and Reliability-Based Methods for the Design of Structural Beams Made from Recycled Concrete Using Aggregates from Precast Rejects

Abstract

The use of recycled aggregates in the manufacture of concrete is presented as a solution to reduce the consumption of resources and waste in the construction sector and contribute to a lower environmental impact. This work aims to explore the possibility of producing structural beams from recycled concrete using aggregates from precast concrete rejects and to improve their design using advanced characterisation techniques. To this end, the experimental data coming from mechanical test and the use of the digital image correlation approach are combined with a robust reliability-based method. The full-field data provided by the digital image correlation approach allow to determine the probabilistic density functions of the mechanical data. From these data, a predictive analysis of the maximum strength and deflection of flexural beams is carried out based on robust design techniques. This approach uses analytical theoretical models and a Monte Carlo-based simulation strategy that allows the prediction of the behaviour of the beams. This methodology was validated by manufacturing six beams with the previously analysed recycled concrete, HA-30, and testing them in the laboratory. All the beams showed behaviour within the predicted range: around 49.7 kN maximum load and just over 9.3 mm maximum deflection. These results demonstrate the robustness of the approach as well as the feasibility of using precast rejects for the manufacture of structural elements.

1. Introduction

Sustainability in construction is one of the Sustainable Development Goals (SDGs) established by the United Nations (UN) [1]. In order to prevent the negative environmental impacts of this industry, it is necessary to address the problem both from the point of view of building sustainable infrastructures and cities and from a vision of responsible production and consumption. In general, construction uses non-renewable materials, requiring a high consumption of natural resources for its production and generates a large amount of waste at the end of its useful life [2]. In Europe, construction and demolition waste (CDW) accounts for between 25% and 30% of all waste generated in the European Union, amounting to 800 million tonnes per year [3]. In Spain, 14,341,183 tonnes of waste were produced annually on average in the period of 2018–2020 from construction [4]. The main component of CDW is concrete [5], responsible for 4–8% of the world’s CO₂ emissions [6]. In turn, waste concrete infrastructure comes mainly from Public Works, which is supervised by the transport minister. Supervision is the responsibility of government administration and amounts to 6,140,009 tonnes on average per year, representing 39.76% of total production. Therefore, one of the main solutions proposed is the use of recycled materials in the production of concrete. This solution reduces the use of virgin aggregates and eliminates the presence of waste in landfills [7].

European guidelines indicate the need for a sustainable use of resources, as well as the need to ensure recycling of at least 70% of CDW [8]. Nevertheless, country regulations for the manufacture of structural concrete significantly restrict the use of recycled aggregates, limiting it to a portion of coarse aggregates and preventing the use of fine recycled aggregates [9]. Consequently, most studies for the re-use of CDW have focused on applications requiring lower mechanical performance with non-structural concretes [10,11,12,13]. Nevertheless, exhaustive research into the properties of recycled concrete aggregates [14] has demonstrated their viability for use in higher performance applications when properly treated [15] or when sourced from higher quality elements such as precast structural elements [16]. This type of element generally presents a characteristic concrete strength of more than 40 MPa, so its waste allows obtaining recycled aggregates of higher quality to be used in structural concretes [17]. The use of these recycled aggregates makes it possible to achieve strength and bonding properties comparable or superior to those of other structural concretes [18] whose minimum strength value is 20 MPa for mass concretes and 25 MPa for reinforced or prestressed concretes [19]. In addition, the use of precast concrete rejects as recycled aggregate results in up to 15% less energy consumption and carbon dioxide emissions compared to other recycled materials [20].

Due to regulatory constraints, studies in structural concrete are not very common. Nevertheless, research such as that of Perez et al. [21] affirms the potential of these rejects from precast elements even for self-consumption in the same precast plants. Zengfeng et al. [22] validate this theory by manufacturing new building blocks from precast block rejects on an industrial scale. Soares et al. [23] agree that this type of recycled aggregates have a higher quality than average, allowing them to achieve similar characteristics to conventional concrete and propose the elimination of restrictions on their use. Regarding the amount of aggregates to be used, Thomas et al. [24] highlight in their research that substitution percentages up to 20% do not affect the properties of the concrete, and higher substitution percentages cause an increase in the amount of cement. Nevertheless, according to the optimisation analysis of the mix design of these recycled concretes carried out in [25], a minimum substitution of 60% of the aggregates is recommended to achieve a balance between concrete strength and environmental impact.

In any case, recent industrial applications have been limited to the replacement of coarse aggregates, either in the manufacture of beams [26] or precast concrete plants [27]. In this sense, the replacement of all natural aggregates by recycled aggregates can be interesting from the point of view of a greater amount of resources and waste saved. In the same way, it would allow to improve the operational process if the aggregates granulometry allows to use them as all-in-one without the need to distinguish between coarse and fine aggregates. Nevertheless, this solution poses a challenge in terms of characterisation and definition due to the variation in their behaviour, especially in properties related to deformability or modulus of elasticity [24]. The wide variability in the type of aggregates, their origin or mode of use are some of the factors that lead to this heterogeneity in the behaviour of recycled concrete [28].

For this reason, the mechanical characterisation and obtaining of the properties of these materials requires the use of techniques adapted to their behaviour, in this case heterogeneous. Similarly, the design of structural elements requires the uncertainty in their properties to be taken into account. In this context, the displacement and strain measurement techniques traditionally used, such as strain gauges or LVDT, have a local nature [29], obtaining punctual measurements that do not fully represent the real behaviour of the samples. In contrast to traditional techniques, new full-field techniques allow monitoring the complete behaviour of samples even without contact. Digital image correlation (DIC) [30] is the technique with the greatest potential due to its non-invasive and non-destructive characteristics, the accuracy and the adaptability to different test typologies through its two-dimensional and three-dimensional approach [31]. It is a technique that allows a complete field of displacements and strains to be obtained by capturing images in different load states during the testing process. DIC is a well-proven technique that has been used to characterise materials with variability in their properties such as wood [32], composites [33] or reinforced cementitious materials [34]. Regarding concrete, numerous studies can be found for the measurement of surface displacements or strains using 2D-DIC and 3D-DIC [35,36,37,38].

Based on the above-mentioned heterogeneity of the material and the possibility of full-field monitoring of DIC, the use of this technique is proposed for the mechanical characterisation of concrete manufactured by aggregates from precast concrete rejects. Whether in tensile, compression or elastic tests using cubic or cylindrical specimens, DIC can be used in its two-dimensional or three-dimensional variants for the measurement of displacements and strains. In all specimens, their surface can be monitored and values can be obtained that more adequately represent their actual behaviour, regardless of whether cracking is present or in which zone the failure occurs. In addition, the obtained full-field data facilitate a statistical treatment to estimate not only a mean value for the properties, but also a dispersion and distribution that represents the heterogeneity. This type of characterisation is proposed in order to achieve a more detailed definition of the material to introduce their uncertainty and heterogeneity using robust design techniques without the need for excessive over-dimensioning. In contrast to deterministic methods, reliability-based analysis is considered to be the most robust strategy for the design of concrete structures [39,40]. This type of strategy ensures safety by introducing the random uncertainties that occur in practical engineering with new materials, such as reinforced beams [41,42]. The main objective of these approaches is to determine the failure probability of a mechanical system under the influence of different uncertainties, such as loads or material properties, among others [33]. This can be achieved using approximation methods such as the First-Order Reliability (FORM) and Second-Order Reliability (SORM) methods or even simulation strategies such as the Monte Carlo Sampling (MCS) method [39]. The MCS is characterised by its simple and straightforward implementation to carry out a large number of simulations and to mathematically replicate the behaviour. Furthermore, when these simulation strategies are combined with one or more design objectives or cost functions, it is possible to achieve Reliability-Based Design Optimisations (RBDOs) [43].

While most previous works have focused on lower performance applications or partial replacement of aggregates, this work aimed to advance the understanding of the mechanical characterisation of structural concretes made from precast concrete rejects for the manufacture of structural elements such as beams. In addition, it incorporated advanced characterisation techniques and reliability-based design methods, adding a level of rigour and precision to the assessment of the uncertainties and variability inherent in recycled materials. In pursuit of this objective, a characterisation of the recycled aggregates was initially carried out in order to define an optimised concrete dosage. Then, tensile, compression and elasticity mechanical characterisation tests were carried out using the DIC technique to obtain their properties. Experimentally obtained full-field data from DIC were used along with robust design techniques to predict the behaviour of bending beams. Subsequently, prototypes of structural beams were manufactured using the same concrete. Finally, these beams were subjected to experimental bending tests to compare the results of these tests with the prediction made according to the properties obtained previously.

Following this introduction, Section 2 describes the materials and methodology used in the tests of this research. In Section 3, the experimental results of the mechanical characterisation tests are shown, as well as a discussion of these results. An analysis of the validation tests performed on the manufactured beams is also presented. Finally, Section 4 draws the main conclusions on the suitability of the introduction of these recycled aggregates for the fabrication of structural elements and discusses future lines of research.

2. Materials and Methods

2.1. Materials and Mix Proportions

For reinforced concrete with steel reinforcement, the recommended minimum strength depends on the durability requirements according to the general exposure classes for corrosion of the reinforcement, with a minimum of 25 MPa. Nevertheless, for bending elements such as beams, this requirement is increased to 30 or 35 MPa. These specifications have been evaluated with good quality aggregates in accordance with the usual durability considerations.

On the other hand, this research evaluates the case of low-quality aggregates, such as those obtained from recycling, in this case referring to those obtained from the crushing of rejected elements in the manufacture of prefabricated with siliceous aggregates from the Duero river basin (Figure 1). Specifically, the analysed aggregates come from the factory of the company Prefabricados Duero located in the town of Toro (Zamora, Spain), produced specifically for this research.

This research used prefabricated elements that failed quality control or were stored for long periods due to restrictions on the movement of goods during the SARS-CoV-2 virus pandemic, COVID-19. As a result, these elements were exposed to high levels of environmental degradation.

Specifically, the aggregate was obtained from the total crushing of prefabricated parts, without any dosing system in the sense that no aggregate fraction is added or removed; the economic viability, and therefore the use of recycled materials, involves processing the material as a whole aggregate of continuous granulometry, regardless of dosing in different sizes.

This research did not evaluate basic properties of the aggregates that initially form the precast elements used in the crushing process for their subsequent recovery as recycled aggregates, as it is assumed that characteristics such as chemical composition or other properties such as shape, resistance to fragmentation or cleanliness are evaluated in the quality systems for the reception of materials in any factory that has to present the mandatory CE marking for construction products in the European Union.

In this study, for practical purposes, the aggregate from the crushing of precast concrete products was considered as precast construction and demolition waste (CDWPrec). To check the suitability of the aggregate for the particle size conditions required for reinforced concrete, the Fuller method [44], was used (Figure 2), which uses the Gessner parabola as a reference curve, representing a continuous particle size.

Figure 2 shows the lack of continuity of the precast aggregate curve, especially in the intermediate sizes, with the largest deviations of 14% in the 2 mm sieve and 6% in the 1 mm sieve. In any case, as shown later, the intermediate sizes are the least representative in terms of obtaining the strength and compactness of the concrete, so these aggregates have not been ruled out for the production of structural concrete. As repeatedly stated, the advantages of these CDWPrec aggregates must be considered in the context of a sustainable future with Industry 4.0, moving towards zero waste models and maximum sustainability.

In order to evaluate the performance of CDWPrec for concrete production, its granulometric curve was also compared with that of the Bolomey method [45] (Figure 3) for bulk concrete dosing. In this case, the reference curve is the Gessner parabola modified by a coefficient depending on the consistency and type of aggregate.

Table 1. Aggregate type and particle size distribution characteristics of CDWPrec.

Material	D10 (mm)	D30 (mm)	D50 (mm)	D60 (mm)	Cu	Cc	% Fines	% Sand Size	% (4.75–9.5) mm	% (9.5–40.0) mm
CDWRCer	0.25	1.0	3.0	5.0	20	0.8	3.4	60	16	24

shows the particle size distribution. In CDWPrec, values of uniformity coefficient, Cu = 20.0, and curvature coefficient, Cc = 0.8, were obtained. The high value of the uniformity coefficient shows the high size variation obtained in the unclassified crushing. On the other hand, the low value of the curvature coefficient Cc = 0.8 does not allow it to be classified as well graded, 1.0 ≤ Cc ≤ 3.0.

The high absorption associated with the porosity of the recycled aggregates is reflected in the high water–cement ratio W/C of 0.52, 2% higher than that recommended for resistant concretes with natural or artificial aggregates.

In this case, it can be seen in Figure 3 that there is a lack of fines above 1 mm, with the greatest difference being 14% in the 0.063 mm sieve, although it should be noted that this is later corrected when calculating the composite curve for this dosage method, calculating that the cement is an additional aggregate and therefore increasing the percentage of fines.

Finally, the dosage for one cubic metre of concrete was 1400 kg of CDWPrec, 410 kg of cement (C), 210 litres of water (W) and 5 litres of admixture, resulting in a ratio W/C = 0.51. The blinder used for manufacturing the concrete was cement type BL II/B-LL 42.5 R. This cement has the following components: (i) a clinker content comprising between 65 and 79%; (ii) a limestone content of 21–35%; (iii) a chloride content of≤ 0.10; (iv) a sulphate content of≤ 4.0; and (v) a soluble toilet chromium VI content of≤ 0.0002%. It has a beginning setting of ≥60 min and an end setting of ≤720 min. The expansion is lesser than 10 mm. Resistance at 2 days is ≥20 MPa and resistance at 28 days is in the interval 42.5 ≤ R ≤ 62.5 MPa. A white cement with a whiteness of ≥85% was chosen as an additional characteristic in the search for the best termination.

2.2. Mechanical Characterisation of the Concrete

2.2.1. Test Set up and Specimens

The proposed recycled concrete dosage was characterised by mechanical tests in order to obtain its properties and to define the material behaviour for the subsequent design of structural elements. Specifically, tests were carried out on tensile [46], compression [47] and the determination of the modulus of elasticity [48] of the concrete.

For this purpose, cylindrical and cubic specimens were manufactured in accordance with the Spanish standard UNE-EN 12390-2 [49]. The dimensions were 150 mm in diameter and 300 mm in height for the cylindrical specimens and 150 mm on each side for the cubic specimens. For the first 24 h after manufacture, the specimens were kept in their moulds at a controlled temperature of 20 ± 5 °C and then a humid chamber was used to store the specimens for the 28-day curing period. This chamber allowed controlled temperature conditions of 20 ± 2 °C and a relative humidity of more than 9% to be maintained.

The so-called Brazilian indirect tensile test was used to determine the tensile strength. During this test, a cylindrical specimen is subjected to a compressive load applied in a thin strip along its entire length. The resulting orthogonal tensile load causes the specimen to break in tensile. For this tensile test, a loading rate of 0.05 MPa/s was set.

The compressive strength test was carried out using both cylindrical and cubic specimens in order to determine the relationship between the different types of geometry and to calculate a conversion factor to extrapolate the results obtained for each test. In addition, the strain at the moment before failure, known as peak strain, was recorded in order to obtain the maximum strain. For this compressive test, a loading rate of 0.6 ± 0.2 MPa/s was set.

The modulus of elasticity was obtained from tests with compression cycles. Initially, three preload cycles were applied to stabilise the specimen, starting from an initial stress of 0.5 MPa up to a stress corresponding to 10% of the compressive strength, obtained from the previous tests. Then, three loading cycles are applied, from a lower stress corresponding to 10% of the compressive strength to a higher stress corresponding to one third of the compressive strength. The strain at the upper and lower peaks of each cycle is recorded in order to calculate the initial and stabilised secant modulus of elasticity. For both load application and load reduction, a loading rate of 0.6 ± 0.2 MPa/s was set.

Since the determination of the tensile strength was carried out by indirect methods, all tests were performed on a compression machine. In particular, a Servosis electromechanical testing machine PCD 1065W equipped with a 1500 kN load cell and corresponding compression platens was used.

2.2.2. Digital Image Correlation Fundaments and Prototype

The mechanical characterisation tests were complemented with the digital image correlation technique to measure the displacements and strains of the specimens. DIC allows these measurements to be calculated from images acquired during the testing process, comparing an initial reference image without strain with that corresponding to each of the loading states [30].

Taking into account the type of specimens used and the existence of flat and curved specimens, a different approach was used for each of them. A 2D-DIC approach was used to measure the in-plane displacements and strains of the cubic specimens, while a 3-DIC approach was used for the cylindrical specimens.

In both cases, it is necessary to carry out a preliminary preparation of the specimens. Firstly, the surface is primed with a matt white paint, and after that a speckle pattern is generated on it. In this case, the spray technique with matt black paint was used, which makes possible to generate a stable greyscale pattern on the surface whose speckle is random, unique and non-periodic [50]. In order to evaluate the quality of the generated pattern, the Mean Intensity Gradient (MIG) parameter [51] and the coverage factor were calculated. The value obtained for the MIG was higher than 30 for all samples, with a coverage factor between 45 and 55%, considered optimal according to Lecompte et al. [52].

The prototype used for image acquisition consisted of high-resolution cameras Manta G-917B 1” (Allied Vision, Stadtroda, Germany) Monochrome CCD equipped with 50 mm macro lens and neutral LED lights to enhance illumination and facilitate acquisition (Figure 4). The synchronisation of the image acquisition and the data provided by the test machine was achieved using a Quantum data acquisition system plus an Omron CP1H programmable logic controller (PLC) that allowed the synchronisation and programming of the triggering of the different cameras.

For the 2D-DIC approach (Figure 4a), a single camera was used and positioned perpendicular to the measurement plane using a micrometric ball joint. For the 3D-DIC approach (Figure 4b), two cameras with a stereo angle of 15° were used. In order to achieve the highest possible accuracy in the measurements, similar to a strain gauge [53], in both cases the device was placed at a distance of 1.25 m from the specimens, which allowed capturing the entire specimen surface with a GSD of 0.09 mm/px. A lens aperture of f8 and a shutter speed of 1/100 s were set to allow image acquisition with adequate illumination, depth of field and sharpness.

Once images were captured, image analysis was performed for subsets of pixels within a region of interest (ROI). DIC uses Zero mean Normalised Cross-Correlation (ZNCC) [54] to analyse and compare the subsets. This process is facilitated by the greyscale speckle pattern. The application of this criterion to all subsets of the ROI and successive images is performed by optimisation algorithms, which are combined with interpolations to achieve sub-pixel accuracy [55].

While for the 2D-DIC approach this process is carried out directly for flat images, in the 3D-DIC approach, it is necessary to carry out an orientation process prior to image acquisition that allows the conversion of image coordinates to three-dimensional coordinates. In this case, an approach similar to the one established by Solav et al. [56] was used, which combines the Bundle Adjustment (BA) algorithm with the Direct Liner Transformation (DLT) algorithm. This methodology allows obtaining the distortion and the internal and external parameters of the cameras, which are used for the three-dimensional reconstruction of the ROI points for which the displacements and strains were calculated using the procedure described above.

2.3. Structural Beam Prototype

2.3.1. Manufacture of Prototypes and Test

In order to study the feasibility of using this type of recycled concrete in structural elements, the fabrication and testing of structural beams was proposed, using a prototyping approach and test setup similar to that used by Visintin et al. [57]. Specifically, it was decided to fabricate beams with dimensions of 1200 mm, 100 mm and 125 mm for length, width and height, respectively. These beams were then subjected to four-point bending tests.

The concrete dosage used for the construction of the beams was the same as the one analysed in the mechanical tests, described in Section 2.1. In addition, the beams were reinforced trying to locate the failure in the centre and avoiding shear failure outside the central third. For this purpose, a reinforcement was designed (Figure 5) with two 10 mm ribbed bars raised to 25 mm and 6 mm wire stirrups with 40 mm spacing in the extreme thirds.

The beams were constructed using ad hoc formwork moulds manufactured from coated plywood panels. The reinforcements were then placed inside them and concrete spacers were used to ensure correct placement. Subsequently, the concrete mix manufactured in a concrete mixer was poured into the formwork, applying the corresponding vibration and compaction to ensure uniform distribution throughout the beam. The curing process was similar to that described for the specimens in Section 2.2.1, although in this case the moulds were removed two days after manufacture. The specimens were kept for 28 days in a humid chamber at a controlled temperature of 20 ± 2 °C and a relative humidity of more than 95%.

A platform was designed to carry out the bending tests in order to adapt the test press (Figure 6). In this case, an electromechanical test machine Servosis ME-405/50/5 was used with a 500 kN load cell. A specific bench performed by structural profiles was designed; for the specimen support, two mobile 45 mm rollers were placed at a distance of 1100 mm, working as punctual loads. In addition, a load plate was fabricated with two 13 mm rollers welded at one third of the span (366.67 mm) for the application of the load at the two corresponding points.

2.3.2. Prediction of Flexural Performance

Taking into account the mechanical characterisation of the recycled concrete carried out in this work, it is proposed to predict the flexural performance of the manufactured beams in order to subsequently validate the experimental tests. In particular, the flexural strength and deflection of the beams are studied. For this purpose, an approach based on Eurocode 2 for the design of concrete structures [58] is proposed.

According to the design configuration proposed for the beams and their reinforcement, the equilibrium equation that determines the ultimate moment of their section is the following Equation (1):

$$M_u=f_{ck}·\mathsf{\lambda }·\mathrm{x}·\mathrm{b}·\left({\displaystyle \frac{h}{2}}-\mathsf{\lambda }·{\displaystyle \frac{x}{2}}\right)-A_s·{\sigma }_s·\left({\displaystyle \frac{h}{2}}-\mathrm{d}\right)$$

(1)

where f_ck is the characteristic compressive strength of concrete; λ is the coefficient defining the effective depth of the compression zone (λ = 0.8 for f_ck ≤ 50 MPa); x is the depth of the neutral axis; b and h are the base and height of the beam section, respectively; A_s is the area corresponding to the reinforcement; and σ_s is the characteristic strength of concrete steel.

Based on the test and load configuration, the maximum total load and maximum deflection of the beam under ultimate moment conditions can be calculated using the following Equations (2) and (3):

$$F_{max}=2·{\displaystyle \frac{M_u}{a}}$$

(2)

$$y_{max}={\displaystyle \frac{M_u}{24·E·I_{ef}}}·\left(3·L^2-4·a^2\right)$$

(3)

where M_u is the ultimate moment; E is the elastic modulus of concrete; L is the beam length; a is the shear span and I_ef is the effective moment of inertia, defined by Equation (4):

$$I_{ef}={\displaystyle \frac{I_{cr}}{1-\left(1-\frac{I_{cr}}{I_g}\right)·{\left(\frac{M_{cr}}{M_u}\right)}^2}}$$

(4)

where I_cr is the cracked moment of inertia; I_b is the gross moment of inertia; and M_cr is the moment to cause cracking, which depends on the tensile strength of the concrete.

In order to provide a more accurate calculation, the usual design safety coefficients for concrete and steel strength are not used. Instead, robust design techniques are implemented to incorporate material uncertainty by adopting a probabilistic approach based on reliability. This approach takes advantage of the larger population of data obtained by the DIC technique to calculate a significant mean and covariance associated with each of the parameters. In this way, a probabilistic density function (PDF) of the main mechanical properties can be extracted.

The simulation strategy is based on using a Monte Carlo Sampling (MCS) method to generate a large set of random numbers that match the previously calculated PDFs. The proposed ultimate moment and maximum load equations are then evaluated for each of these parameters to obtain a probability density function for the corresponding results. Considering the reliability analysis, the probability of failure can be evaluated according to the following Equation (5):

$$P_f=Prob\left[G\left(Y\right)\le 0\right]=\underset{G\left(Y\right)\le 0}{\int }{p}_y\left(Y\right)dY$$

(5)

where P_f is the probability of failure; G is the performance function; Y is the variable’s vector and p_y is the joint probability density function.

Taking into account that the analytical theoretical models include several variables and that a probabilistic analysis is chosen, it is important to know the influence of each input on the final result. To this end, the MCS strategy allows an adequate sampling of the parameters used in the model, thus generating equiprobable situations to carry out a sensitivity analysis. The strategy used is based on the estimation of so-called Sobol indices [59], which assume that the sum of the variances of the input parameters represents the variance of the model output, Equation (6). Sobol indices of different orders (from 1 to 2n − 1) can be obtained by considering the normalisation of each variance with respect to the total variance, Equation (7). The sum of these indices is the overall Sobol index. Likewise, the sum of all their values is equal to 1.

$$V\left(Y\right)=\sum _i{V}_i+\sum _i\sum _{j>i}{V}_{ij}+\sum _i\sum _{j>i}\sum _{k>j}{V}_{ijk}+\dots V_{123..N}$$

(6)

where V(Y) is the variance of the model; V_i = V(E(Y|X_i)) is the first-order partial variance; V_ij = V(E(Y|X_i,X_j)) is the second-order partial variance, etc.

$$S_i={\displaystyle \frac{V_i}{V\left(Y\right)}},S_{ij}={\displaystyle \frac{V_{ij}}{V\left(Y\right)}}$$

(7)

where S_i is the first-order Sobol index and S_ij is the second-order Sobol indices.

3. Experimental Results

3.1. Mechanical Properties of the Recycled Concrete Evaluated

A total of 20 specimens were tested for indirect tensile, compression and elasticity according to UNE-EN 12390-6 [46], UNE-EN 12390-3 [47] and UNE-EN 12390-13 [48], respectively. Additionally, the compressive strength tests were duplicated as they were performed for both cylindrical and cubic specimens. Finally, to determine the workability of the concrete, two slump tests were carried out according to UNE-EN 12350-2 [60], with a result of 16 cm, which corresponds to a fluid consistency.

The DIC technique described in Section 2.2.2 was used to obtain displacements and strain values. A 2D-DIC approach was used for cubic specimens using the open source software Ncorr (Version 1.2) [55] and a 3D-DIC approach was used for cylindrical specimens using the open source software MultiDIC (Version 1.0) [56]. The processing parameters were similar in both cases, defining an ROI covering the entire specimen and using a subset size of 20 × 20 pixels with a step of 7 pixels, which ensured an overlap of 65%. In addition, a checkerboard calibration target was used to remove image distortion and perform 3D reconstruction using the DLT algorithm.

To obtain the longitudinal strains, several virtual extensometers were placed in order to analyse the entire surface of the region and exploit the potential of DIC. Nevertheless, due to regulatory restrictions, the strain cannot be measured over the entire surface, but it has to be considered in the central third of the specimen. Taking into account the dimensions of the specimens, a total of 10 virtual extensometers with a spacing of 15 mm were placed on each specimen (Figure 7) to analyse their behaviour and spatial influence.

For the compression tests, the peak longitudinal strain at maximum load prior to failure was recorded on virtual extensometers placed over the entire surface. For the elasticity tests, the strain at the maximum and minimum of the applied load and unloading cycles was obtained, as well as over the entire surface.

Considering all the test types and specimens, as well as the virtual extensometers placed on them, the results of the mechanical characterisation are shown in

Table 2. Results obtained from the mechanical characterization.

Property	Mean	STD	CoV (%)	Lower Bound	Upper Bound
fc-cyl	36.5	4.3	11.8	29.1	39.4
fc-cub	40.5	2.2	5.4	37.8	43.5
fti	2.8	0.3	11.2	2.5	3.1
E	27.62	4.56	16.5	21.18	36.82
εp-cyl	0.0021	0.0002	11.6	0.0017	0.0026
εp-cub	0.0034	0.0006	18.3	0.0023	0.0048

f_c-cyl = Compressive strength of cylindrical specimens (MPa). f_c-cub = Compressive strength of cubic specimens (MPa). f_ti = Indirect tensile strength (MPa). E = Secant modulus of elasticity in compression (GPa). ε_p-cyl = Peak strain at time of maximum stress of cylindrical specimens. ε_p-cub = Peak strain at time of maximum stress of cubic specimens.

. Along with the mean values, the main statistical parameters such as standard deviation (STD), coefficient of variation (CoV) and lower and upper bounds are also presented, which allow the behaviour and heterogeneity of these properties to be analysed.

The dosed concrete offered a compressive strength, related to cylindrical specimens, of f_c-cyl = 36.5 MPa. On the other hand, a compressive strength of f_c-cub = 40.5 MPa was obtained from tests carried out on 15 cm cubic specimens. This implies a conversion coefficient from cubic to cylindrical specimens, γ = 0.90, which is similar to the coefficient usually used for conventional concretes with a strength value UCS < 60 MPa. It is worth noting that the results for the cubic specimens showed less variation, with a CoV of 5.4% compared to a CoV of 11.8% for the cylindrical specimens. The larger contact surface for the cubes may be one of the reasons for the greater homogeneity of these results. In any case, very high compressive strength values were obtained, with only one specimen below 30 MPa, allowing the sustainable concrete studied to be classified as HA-30 according to the Spanish Structural Code [19].

The high minimum resistance of 30 MPa obtained makes it possible to meet the durability requirements for all environmental exposure classes and, in all cases, to comply with the dosage parameters for the minimum cement content, with a high cement content C > 400 kg/m³, thus obtaining a concrete with reduced permeability.

Tensile strength f_ct is usually obtained from indirect tensile strength f_ci, which for concretes made with CDWPrec has an average value of f_ti = 2.8 MPa, i.e., tensile strength f_tk = 2.5 MPa.

The modulus of elasticity showed an outcome of 27.6 GPa, which is very similar to the experimental value obtained when strains in concretes made by conventional aggregates is calculated.

The peak strain at the time of maximum stress was 0.0021 for cylindrical specimens and 0.0034 for cubic specimens. In this case, the conversion coefficient between the two types of specimens is approximately 0.6. While for cylindrical specimens the value is similar to the 2‰ generally considered for conventional concrete, for cubic specimens the strain is higher, also associated with the higher strength that this type of specimen achieves due to its geometry.

The high CoV in the modulus of elasticity and ultimate strain results show the heterogeneous behaviour of these materials. Although heterogeneity is present in conventional concrete, particularly in the strains produced by the phase change between aggregates of different sizes, recycled aggregates accentuate this irregularity due to their own characteristics, such as in the case of greater disaggregation of aggregates and cement [61]. It should also be noted that the compressive and tensile strength data are unique to each specimen. Nevertheless, the ultimate strain and modulus of elasticity data also take into account the spatial variability of the material within the specimen itself. This is because DIC allows multiple measurements to be taken within a single specimen. Issues such as manufacturing heterogeneity, inaccuracies in the alignment of the press during the test, or edge effects can lead to higher variability and therefore increase the CoV of these data.

3.2. Beam Performance

The approach proposed in Section 2.3.2 for predicting the bending behaviour through reliability analysis allows the uncertainty in the behaviour to be quantified. To introduce this uncertainty, the populations of results obtained in the previous section were fitted to probabilistic distributions functions by performing goodness-of-fit tests. The candidate PDFs were selected based on physical considerations and other experimental campaigns with similar methodologies [33,62]. To obtain a larger population size, both cylindrical and cubic sample values were used, applying conversion factor γ = 0.90. Considering that most of the data accepted all the tests, a lognormal distribution was chosen, including the mean and the deviation for each of the parameters, as shown in Table 2. In particular, after goodness-of-fit tests, a lognormal distribution was selected since, in contrast to the normal distribution, these physical parameters cannot reach negative values.

Taking into account the model defined to obtain the maximum theoretical load, the variables related to the properties of the concrete were introduced as probabilistic variables, except for the tensile strength. In this case, due to its low value in relation to the steel used, the tensile strength of the concrete was not included in the calculation. In the case of the steel parameters, the properties corresponding to a B400S steel were used as a constant input. The ultimate strain of the recycled concrete was also used for the ultimate stress testing domain.

With regard to the model for obtaining the deflection, the experimentally obtained compressive and tensile strengths and the modulus of elasticity were taken into account according to Equations (3) and (4). Due to the difficulty of knowing exactly the effective moment of inertia, a probabilistic coefficient of uniform distribution in the range of 30 to 50% was also introduced in this case and applied to the gross moment of inertia. The uniform distribution was chosen in this case due to the complexity of knowing this parameter and to achieve an equiprobable and uniform sampling. The higher uncertainty of this parameter is a handicap in accurately predicting the deflection result.

In order to obtain a probabilistic density function for the maximum load and deflection supported by a beam manufactured with the recycled concrete studied, a total of 10⁶ simulations were carried out using MCS for the previously defined model parameters. The distribution of the simulation results is shown in Figure 8a for the maximum load and in Figure 8b for the deflection.

Both graphs show how the results are distributed according to a distribution that can be considered lognormal, with the modal value close to the mean. The most frequent value for the maximum load is 49.65 kN, while the most frequent value for the deflection is 9.32 mm.

In order to validate the model and obtain the real behaviour of the recycled concrete bending beams, the experimental tests described in Section 2.3.1 were carried out. A total of six beams made of the studied recycled concrete were tested with the proposed configuration. A sample of six beams was used to strike a balance between representativeness of results and practical constraints such as available resources. This sample size ensures a variety of representative conditions from which meaningful conclusions can be drawn. This representative sample allows the model to be validated so that it can be scaled up and generalised to larger populations through further statistical analysis, such as Monte Carlo simulations.

The results corresponding to the maximum load and deflection of each of the beams are also shown in Figure 8a and Figure 8b, respectively. The DIC technique was also used to calculate the deflection experimentally. Due to the span of the beam, only a central ROI was selected where the largest displacements occur. A set of 5 × 5 px under the midpoint was then taken to obtain the average deflection.

The behaviour of all tested beams is within the PDF obtained from the simulation. For the case of the maximum load, the experimental results are close to the modal value. Only beams B3 and B4 are slightly further away from this central value. In the case of deflection, the experimental results are further apart and in all cases exceed the modal value of deflection. One of the reasons for this greater dispersion is the greater number of probabilistic parameters involved in this simulation. In addition, the difficulty of predicting deflection in reinforced concrete is greater because parameters such as the effective moment of inertia or the moment to cause cracking are less accurately estimated. Even some factors such as creep or shrinkage are not taken into account in the theoretical analytical models for calculating deflection. In any case, all the results were within the range predicted by the reliability analysis, so it can be assumed that the model is valid and reliable.

From the reliability analysis and the probability of failure corresponding to each load and deflection, the percentile in which the behaviour of each beam is located was calculated. The results of the maximum load and deflection of each beam, together with their corresponding, are shown in

Table 3. Experimental test results for the bending beams tested.

Beam	Load (kN)	Pload (%)	Deflection (mm)	Pdef (%)
B1	49.02	58.2	11.70	73.2
B2	49.23	56.3	11.38	69.4
B3	46.83	77.1	14.11	92.0
B4	44.51	91.0	12.81	84.0
B5	50.72	42.8	13.31	87.6
B6	48.50	63.0	13.98	91.4

.

The following criteria were used to calculate the percentiles: higher maximum load values correspond to better behaviour and lower percentiles; higher deflection values correspond to worse behaviour and higher percentiles. The results obtained in calculating the percentiles allow us to quantify the above for the behaviour of the experimental tests. The experimental results for maximum load show percentiles close to the mean, except for specimen B4, which is in the last decile. The experimental deflection results show higher percentiles, mostly in the last quartile.

It should be noted that the analytical models used did not include safety coefficients, so it is understandable that the results are not in the first percentiles. Other factors other than the main properties obtained experimentally, such as creep or shrinkage deformation, which should be taken into account to improve the accuracy of the model, were also not included. In addition, all the tests were within an acceptable range, as none of them exceeded what could be considered a limit of 5%.

In order to understand which variables are most relevant in the theoretical analytical performance model of recycled concrete, a Sobol sensitivity analysis was carried out. As the prediction of deflection was the most complex model with the most variables and the most dispersed behaviour, the analysis was performed on this model. The same analytical model used for the reliability-based design analysis was employed, with input variables including the compressive strength, tensile strength, and modulus of elasticity of the recycled concrete, as well as the coefficient for calculating the effective moment of inertia. Similarly, the variability of these parameters was introduced by means of their corresponding PDF. In order to carry out this sensitivity analysis, a total of 10⁶ simulations were carried out using MCS. For each of these simulations, the input parameters were given a random value based on the distributions and ranges obtained in the experimental tests. The variances of the input parameters were analysed with respect to the variance of the model outputs, which allowed Sobol indices to be obtained. The results of this analysis are shown in Figure 9 for both the first-order and total indices. It can be seen that both indices are practically similar, so the second-order indices can be discarded.

According to these results, the most relevant parameter in the deflection model is the modulus of elasticity of the concrete, accounting for 47% of the total variance. On the other hand, the parameter with the least influence is the tensile strength of the concrete, with an index of 0.01 (1%), as it is only used for the calculation of the moment to cause cracking. It is also worth mentioning the high influence (39%) of the coefficient used to calculate the effective moment of inertia. Considering that this is a complex parameter to calculate experimentally, this may explain the lower accuracy of the model.

4. Conclusions

This work aimed to investigate the characterisation of structural recycled concrete for the production of beams. The concrete was produced by replacing all the natural aggregates with recycled aggregates from precast concrete rejects. These aggregates have been considered as all-in-one, which represents an advantage in terms of economy and operability for the manufacturing process, in addition to the already known environmental benefits due to the reduction of waste and the lower extraction of raw materials, as set out in the targets of “Goal 12: Responsible consumption and production” of the Sustainable Development Goals established by the United Nations.

Firstly, the analysis of the recycled aggregates showed the adaptability of their granulometric curve to the Fuller method, which allows them to be used in the production of concrete. Subsequently, a large number of cubic and cylindrical specimens were made in order to obtain the main mechanical properties of the recycled concrete. These tests allowed the definition of a HA-30 structural recycled concrete. In addition, the digital image correlation technique was used to obtain more accurate and full-filed strains. Thanks to this technique, the high heterogeneity of the material and the difference found between the results of the cylindrical and cubic specimens were analysed.

The results of the experimental characterisation were then used to define an analytical theoretical model to obtain the maximum load and deflection in bending beams. The data provided by DIC allowed a reliability-based probabilistic approach to be adopted, performing one million simulations by modifying the input parameters according to the distributions calculated from the experimental tests. This allowed to obtain a probabilistic density function for the maximum load and deflection of the recycled concrete beams studied.

Finally, the theoretical simulations were validated by experimental tests, in which six beams were manufactured and tested in the laboratory. In all cases, the results were in agreement with the model, and the predicted values were within the acceptable range. These results demonstrate the great potential of the proposed methodology to characterise and predict the behaviour of recycled concrete structural elements. Nevertheless, the strength prediction model was more accurate than the deflection model, so a sensitivity analysis of the parameters included in the model was carried out to better understand its behaviour. The results showed a higher influence of the modulus of elasticity (47%), followed by the arbitrary coefficient used to calculate the effective moment of inertia (39%). The difficulty in precisely determining this parameter may account for the model’s lower accuracy and represents one of its potential weaknesses. Furthermore, it should be noted that a simple analytical theoretical model was chosen for this study, which presents a limitation when incorporating certain indirect factors. Factors such as interfacial slippage, creep or shrinkage, although having less direct influence, can affect the result and limit the accuracy of the model if they are not taken into account.

One of the main future works will focus on improving the analytical theoretical model, trying to include some of these factors that could improve the accuracy of the results, e.g., creep or shrinkage deformation, and even using a Bayesian approach to adjust the results from the experimental data with a larger number of samples and variability in loading or experimental conditions such as temperature and humidity. In addition, adaptation of the model parameters to material properties and configurations from existing studies could help to validate its extrapolation so that it can be generalised and its use extended to the scientific community. The next step will be to apply this methodology to more complex models by carrying out simulations using the finite element method, in which a more thorough analysis of the behaviour of the elements can be carried out. The results of these full-field models are intended to be compared with full-field experimental measurements obtained through digital image correlation.