Experimental and Statistical Capacity of Rubberized Continuous Deep Beams under Monotonic and Repeated Load

Continuous Deep Beams (CDBs) are highly exposed to repeated loads especially at the bridges during vehicle loads and adding rubber into it support the ductility to overcome concrete brittleness. Six Beams were designed in considering the strut tie method (STM) and tested under monotonic and repeated loads. Mathematical model for such problem has been investigated. It can be concluded that, the CDBs lose some of its ultimate capacity due to the dropping of concrete compressive strength after aggregate-rubber replacement, while an increment in the deflection could be seen due to the higher deformability of the rubcrete mix i.e., the concrete gained some flexibility after replacing. Also, the ultimate strength of the beams decreases when comparing with the static loads by 14%, 8% and 9% for conventional beam, gravel and sand replacement beams respectively. Fuzzy regression system offers a polynomial from the third degree to represent the Fuzzy data of the CDBs.


Introduction
Concrete continuous deep beams are highly exposed to repeated loads on bridges and girders [1][2][3].Sustainable rubberized concrete beams may use at such places to supply the concrete by some ductility, where the general behaviour of conventional concrete is brittle.Rubber found in the nature in amounts causing sufferer pollution.So, scraped tires rubber were added to concrete to develop its ductility as well as to get rid of its pollution.Adding rubber into mixes may be as a volumetric or mass replacement as well as additives.The replacement minimizes the mechanical properties but increases the dynamics [4][5][6][7][8].Rubber as an aggregate can be classified into four categories in accordance with its size.Which are: shredded/chips (2-20mm), crumb (0.4-4.75mm), ground (passing through sieve 0.4mm) and shredded fibers [4,9].Tires cleaned then crashed into small particles in special machines (figure 1) from Abraj alkut factory at Al-Diwahniya for tire recycling).Different sizes of rubber products.Essentially, rubber crashed till reaches the general sizes of sand and gravel particle sizes.Recycling factories also product rubberized floors for electrical isolation (figure 2).The study focus on the capacity of Rubberized Continuous Deep Beams (RCDBs) which exposed to repeated and static loads.Also, to estimate a suitable mathematical model for the data using Fuzzy nonparametric regression system.

Experimental program
Six continuous deep beams were cast and test experimentally, but divided into two groups.The first one is tested under monotonic load to estimate the overall RCDBs capacities.While the second group is tests to estimate the behaviour of beams under repeated loads.Both sand and gravel replacement were investigated in 10% volumetric replacement as investigated in article [7].Mixes properties shown in details at reference [7], and as a summery in Table 1.Concrete beams were cased in dimensions of 1.5*0.2*0.1 m (and as explain in figure 3) and tested by using static press of 1000 kN capacity with a load distributer as shown in figure 4. A continuous deep beam was prepared to the Digital video Image Correlation (DIC), in which it required speckling the beam by regular black points and videoing the beam under the stepped loading from near distance with a high accuracy camera.

Static loads
After applying monotonic loads which mentioned previously, CDBs results were recorded and listed in figure 5 to figure 8.It can be concluded that, sustainable rubberized CDBs results supported the literature conclusions especially which involved to the Strut -Tie Method (STM), where the experimental load is larger than the theoretical one which will fixed that the STM is a lower bound theorem [1,10,11].As well as, it can be found that, The CDBs lose some of its ultimate capacity due to the dropping of concrete compressive strength after aggregate-rubber replacement, while an increment in the deflection could be seen due to the higher deformability of the rubcrete mix i.e. the concrete gained some flexibility after replacing as well as figure 8.It can be also noting that, 39%, of the concrete conventional beam capacity was decreases after replacing ST.S10, and 30.1% for ST.G10.Experimental beam results were collected in table 2. The failure load mechanism was pure shear for all specimens.The cracks start to form in bending zones then the behaviour convert towards the shear failure as observed previously in the literature researches [3], the failure mode of specimens were shown in figure 9, figure 10 and figure 11.

Experimental repeated load results
The repeated load series which applied experimentally on the two CDBs results experimentally were illustrated at figure 13.Due to the repeated cycles of loading, the ultimate strength of the beams decreases when comparing with the static loads by 8% and 9% for gravel and sand replacement respectively.Ultimate load capacity of R.G10 larger than R.S10 due to the higher compressive strength of the first, while deflection values for sand replacing beam were slightly larger than the R.G10 due the higher flexural capacity of S.R.10 itself as investigated at reference [7].It can be noting that, figure 13 and table 4, the residual deflection at the end of each cycle increases gradually due to crack developing after each cycle.Beams after failure were shown in figure 14.

Repeated load statistic modelling
Usually, analysis regression used for modelling the relation between dependent (y , deflection) and independent (x, Applied load) variables in order to find a mathematical model for the CDBs load versus deflection phenomena.Also, for estimating the regression function for the beams.Classical methods of estimation may not give an efficient estimation due to the data uncertainty (i.e.Fuzzy data).It cannot be identifying the certain applied load on the beam actually in the nature, nor the deflection, so it has to be considered as a Fuzzy system data [12-14].
As a first step, the data has to be tested to identify its distribution, if it was normal or not, the Easy Fit software has been considered.In which, Kolmogorov, Anderson and Chi-squared tests were investigated for testing the normal distribution or not for the three beams samples as shown in table 5.
It can be noting that, all tested values results is larger than 0.05, so the null hypotheses must be rejected and the alternative hypotheses will be considered.The impossibility of identify the applied load and the deflection result due it, also the impossibility of identify the relation between them, leads to the necessity of using Fuzzy nonparametric regression model for represent the realistic data and finding a mathematical model for each beam using two criterions, ISE and AMSE.The Fuzzy nonparametric regression model formula is shown in equation 1.
Regression equations for the three beams were listed in equations 1,2 and 3, while the Fuzziness results are listed in Figure 15, figure 16 and figure 17, as well as table 6. table 7 and table 8 showing the fuzziness independent and dependent variables respectively.R.G10 beam was listed only for curtailment due to the large tables.
= 0.951 3 − 0.295 2 + 0.0648 + 0.2426 (4)  ..10 = −1.298 3 + 1.772 2 − 0.175 + 0.2836 (5)  ..10 = 0.4533 3 − 0.607 2 + 0.2005 + 0.2624 The results showed that, a polynomial from the 3rd degree was suitable for reference beam data estimation better than the two other beams (as explain in table 6 ).Which means that, the presence of rubber caused some randomness in the data which make it difficult to be estimating.The CDBs lose some of its ultimate capacity due to the dropping of concrete compressive strength after aggregate-rubber replacement, while an increment in the deflection could be seen due to the higher deformability of the rubcrete mix i.e. the concrete gained some flexibility after replacing.39% and 30.1% respectively are the losing percentage after sand and gravel replacement.2. The ultimate strength of the beams decreases when comparing with the static loads by 14%, 8% and 9% for conventional beam, gravel and sand replacement beams respectively.3. Ultimate load capacity of R.G10 larger than R.S10 due to the higher compressive strength of the first, while deflection values for sand replacing beam were slightly larger than the R.G10 due the higher flexural capacity of S.R.10 itself.

Figure 4 .
Figure 4. Static load test for CDBs (all dimensions are in mm)

Figure 13 .
Figure 13.Repeated experimental load-deflection curves for both specimens

Table 2 .
Ultimate load capacities and displacements for statically loaded beams

Table 3 .
LVDT and DIC deflections for every load step

Table 4 .
Residual deflection at the end of each cycle (all values in mm)

Table 5 .
Easy fit software result

Table 6 .
Results of estimating for the three groups Represent estimate the regression function to R 10 Figure 17.

Table 8 .
Represent fuzziness the dependent variables (Deflection R.G.10) 4. The residual deflection at the end of each cycle increases gradually due to crack developing after each cycle.Beiram A A H and Al-Mutairee H M K 2022 The effect of chip rubber on the properties of concrete Mater.Today Proc.[9] .S K G 2016 Fundamental Properties of Self-Compacting Concrete Utilizing Waste Rubber Tires-a Review Int.J. Res.Eng.Technol.05 254-61 [10] Ali Abdulameer Kadhim and Hayder M.K. Al-Mutairee 2021 Experimental Investigation of Rubberized Reinforced Concrete Continuous Deep Beams J. King Saud Univ.[11] Zhang N and Tan K H 2007 Direct strut-and-tie model for single span and continuous deep beams Eng.Struct.29 2987-3001 [12] Liu J M, Chen R and Yao Q 2010 Nonparametric Regression with Trapezoidal Fuzzy Data J. Econom.157 151-64 [13] YILDIZ M and MEMMEDLİ M 2021 Comparative Analysis for Fuzzy Nonparametric Regression Models Eskişehir Tech.Univ.J. Sci.Technol.A -Appl.Sci.Eng.22 353-65 [14] Pasha E, Razzaghnia T, Allahviranloo T and Mostafaei H R 2007 Fuzzy Linear Regression Models with Fuzzy Entropy Math.Teach. 1 1715-24