A Strain model for uPVC tube-confined concrete

Abstract The brittle response of concrete largely influences concrete deformation under axial load in the post-peak descending branch of the stress-strain curve. In this study, unplasticized polyvinyl chloride (uPVC) plastic tube used to confine concrete cylinders and improve the post-peak performance of specimens subjected to axial compression loads. Three experimental parameters, tube diameter, tube thickness, and H/D ratio, used to examine the mechanical behavior of eighteen uPVC tube confined short concrete columns. Test results show that when the outer diameter to thickness ratio (D/t) changed from 22 to 12.9, local buckling of the polymeric tube decreased, and the strain at peak stress was increased. So far, there is no model for predicting the strain of plastic tube confined concrete. The present test data was combined with more than 160 test data collected from the published literature to compile an experimental database. The number of test data for the strain of uPVC tube confined concrete reported in the literature is limited and less than those for strength. However, the assembled database was abundant enough to develop an ultimate condition model for the maximum axial strain accurately, based on the database. The developed model’s accuracy was verified by comparing model predictions with twenty existing FRP-confined concrete models using three statistical indices. The distinguishing feature of the strain model includes several parameters on the concrete confined by the uPVC tube.


PUBLIC INTEREST STATEMENT
The uPVC tube is a universal thermoplastic polymer commercially available for sewage and several other construction industry applications. The tube can be used for encasing concrete with durability, environmental and economic advantages. The tube exhibits acceptable behavior under applied load with a ductile response and sufficient deformations before the ultimate failure. The encasing tube is of particular interest for structural or architectural columns in several civil applications such as low rise industrial and residential buildings. Other potential uses of plastic tubes are as a pier and pile in bridge infrastructure applications found in hostile environments-the thin-walled tubular functions as a barrier to prevent the ingress of harmful materials into the encased concrete. The plastic could be strengthened externally using fiber reinforced polymers as strips or full wraps for carrying additional loads.
Furthermore, a 50% enhancement in fracture energy was achieved when the tube thickness was increased by 30%. Engineering plastics were used to jacket concrete cylinders (Abdulla, 2020c). The two components' deformations were monitored by introducing a small gap of 1.2 mm between the concrete and the polymeric tube. Test results show a ductile post-peak response with compression strain-softening and the formation of two peak strains. In another study (Woldemariam et al., 2020), Test results have illustrated that the uPVC confinement of concrete enhanced the ductility and energy absorption capacity by 1. 84-15.3 and 11-24.3 times over that unconfined specimens. An artificial neural network (ANN) technique was employed to predict the axial strain of concrete-filled plastic tubular specimens tested under direct compression load (Abdulla, 2020e). Statistical evaluations demonstrated the soft computing technique to be more effective and accurate than several existing strength models for fiber-reinforced polymer (FRP)confined concrete.
For mass concrete structures, the embedded water tube cooling system is considered a useful tool to monitor hydration temperature during construction (Hong et al., 2019). Due to its improved constructability and durability, Polymeric uPVC can be employed as an alternative to traditional steel or wood formworks (Abdulla, 2020e;Michel Murillo et al., 2019;Abdulla, 2020c). The seismic behavior of embedded PVC tube confined reinforced high-strength concrete columns was explored (Chen et al., 2020). The effects of PVC tube diameter, axial compression ratio, and concrete strength on seismic behavior indexes were analyzed.
PVC tube was used to encase cementitious composite materials. The compressive behavior of compound concrete-filled reinforced PVC tubes was compared to PVC tubes without steel reinforcement (Babu & Paulose, 2019). In marine and offshore infrastructure construction, foundation piles' durability needs to be accounted for due to the corrosion from the salt in the seabed soil (Song et al., 2008). Intending to treat the corrosive foundation, stabilized soil confined by a PVC tube was used (Shiyang et al., 2018). The axial strain of specimens contained with the tube was 6.45-25.09 times the unconfined ones, respectively. PVC tube was employed for the beam-column joint construction. The deformation of joint core reinforced with a ring beam for connection of PVC fiber-reinforced polymer confined concrete column, and a reinforced concrete beam was investigated (Bu et al., 2020).
Recent research (Jiequn et al., 2019) has shown that PVC tubes' lifting effect strengthened with carbon fiber wraps not to be significant in recycled aggregate concrete (RAC). However, the CFRP wraps introduce a positive influence on incrementing the peak strain of uPVC-confined recycled aggregate concrete. The authors concluded that the plastic tube confinement could considerably enhance the strain at peak stress of RAC specimens. The effect of uPVC tube confinement on RAC's peak strain is more significant than that of normal aggregate concrete. Compared with the elastic modulus of unconfined RAC, the PVC-RAC's elastic modulus was improved by 12.9%. The improvement was increased to 27.3% when 50% of the aggregate was replaced by RA (Jiequn et al., 2019). This ratio was further increased to 29.8% when 100% RA was used. Generally, the low modulus plastic tube restrains concrete core to some extent from dilating laterally, enhancing its strength and ductility capacities (Chen et al., 2020;Michel Murillo et al., 2019;Abdulla, 2020c).
So far, there is no model for predicting the strain of uPVC tube confined concrete, which is one of the present study's objectives. Another goal is to check the applicability of FRP-confined concrete strain models for predicting the strain of uPVC confined concrete specimens. Furthermore, the current work collects all the published data on a novel technique of concrete confinement using commercially available PVC or uPVC tubes with material and labor cost savings since the tube is lightweight and no skilled labor is needed. The number of test data for the strain of uPVC-concrete specimens reported in the literature is limited and is less than those for strength. However, the assembled database was abundant enough to accurately develop a maximum condition model for the peak axial strain.

Materials
Ordinary hydraulic Portland cement was used to cast concrete. The concrete was used for both the unconfined specimens and confined specimens made up of coarse river gravel with a 9.5 mm maximum size. The fine aggregate was river sand with a maximum size of 4.75 mm. All the specimens were cast from the same concrete, which was proportioned by weight. The mixture proportions were 1 (cement): 2 (sand): 2.75 (gravel). The water-cement ratio (w/c) was 0.5. The 28 days targeted cylindrical compressive strength was 40MPa.

Plastic tube
Grey color uPVC was cut to the required heights, and the ends ground and cleaned. Two average thicknesses of polymeric tubes were used in the present study, 5 and 7 mm, respectively. The plastic tubes were assembled on a stable timber base and used as a temporary mold for casting unconfined concrete specimens and as permanent formwork for casting the composite specimens. For tensile testing of uPVC, rods were cut from the tubes in the longitudinal direction, and the edges were finished for testing. The geometric details of the coupons are shown in Figure 1.

uPVC coupons
The coupons were assembled in the testing machine and hold vertically through the upper and lower grips Figure 2a. The coupons tested at a speed of 5 mm/min (as per ISO 6259-2PVC), and the average fracture strain was 42%. The specimen underwent linear elastic deformations up to the proportional limit, where the yield stage was initiated, resulting in the test specimen's necking under direct tensile load due to the material yielding. The coupon continued to resist load until the peak load was reached. Beyond maximum load, the specimen underwent softening with a smooth plateau until it was split into two parts Figure 2b.

uPVC hollow tubes
Hollow uPVC tubes were tested under axial compression load at a speed of 1 mm/min. Hollow uPVC tubes were vulnerable to local buckling under direct axial load, which reduced their global stiffness. Only when the tube was loaded continued to shrink and fold in the axial direction and expand in the lateral direction, Figure 3. Unlike in the tensile testing of uPVC coupons, where the split of the coupon terminated the test at the point of necking, the hollow tubes continued to resist load with substantial deformation reaching large ultimate strains exhibiting considerable buckling near the upper end. The test results for yield strength of coupons and hollows tubes were close.

uPVC-confined concrete
The uPVC tube confined concrete specimens were tested under axial compression in a universal testing machine, Figure 4. Three unconfined specimens were tested to failure for each corresponding H/D ratio. A displacement-controlled procedure at a loading rate of 0.3 mm/min was used to test the confined and unconfined short columns. The test continued beyond the ultimate load to capture the falling branch of the stress-strain curve. The testing machine was switched off following PVC's failure confined concrete or destruction of the unconfined specimens. Typical

Failure mode
Most of the confinement pressure that the tube can exert on concrete reached when the radial strain in the plastic tube reaches yield. The maximum confinement pressure is achieved when the compression stress reaches its ultimate or peak value. Beyond the peak stress, the tube continues to apply confinement pressure but on a smaller scale due to the low modulus of the tube. The maximum top strain in the tube is attained when the plastic tube's circumferential strain reaches its burst pressure and, ultimately, the tube rupture resulting in the failure of the damaged concrete cylinder. Low stiffness is one of the main disadvantages of the tube. This limits the use of plastic tubes in structural applications into lightly loaded industrial and residential buildings. Other more convenient uses for the plastic tube are for piers and underground piles in infrastructure  applications. The significance and the vital role of confinement in improving the lateral buckling of concrete columns is an essential pivoting factor when considering structural members' safety against collapse. To avoid brittle catastrophic failure, the strength, and deformation capacity of confining material influence concrete members' behavior and ensure a ductile flexural response, which was the case in the present study, and a further improvement in the function of the uPVC tube can be achieved using FRP wraps or strips.

uPVC coupons
The uPVC coupon exhibit no clear yielding, and the yield stress has to be approximated from 0.2% strain, a parallel line that offsets the abscissa at 0.2% strain and cuts the stress-strain curve.

uPVC tube confined concrete
The thin-walled plastic tube was subjected to biaxial stress due to the applied direct compression load and the transfer of the axial loads from the core concrete to the external confining device. This was evident in plastic tubes having a thickness of 5 mm, where the tube underwent considerable outward buckling deformations, Figure 5(a) resulting in a bowel type failure. When the tube's thickness was increased to 7 mm, less axial load was transferred from the concrete core. The tube exhibited less deformations due to the improvement in its geometrical properties, Figure 6. Generally, simultaneous axial loading of both concrete core and confinement tube yields enhancement in the strength of PVC-confined concrete due to interfacial bonding between the two components, which improves the global stiffness of test specimens. When the hollow tube is under direct load, it undergoes deformations in axial and lateral directions resulting in local buckling. When the uPVC-confined concrete is tested, the encasing tube's inward deformations were ceased Figure 7. In the stress-strain curve of PVC-confined concrete, Figure 7, an initial linear portion of the stress-strain curve is maintained up to about 70-80% of the ultimate load. With further increase in the maximum load, the curve becomes non-linear, and a non-sharp peak is reached. Beyond the peak, the stress-strain relationship undergoes compression strain-softening with a noticeable drop in stress (25-35% of peak load) and an increase in the corresponding strain. After that, the curve stabilizes, and small declines in stress being registered with corresponding large increments in strain. All the test results for the ultimate stress and strain of uPVC-confined concrete columns were summarized in Table 1. The ability of uPVC-confined concrete to absorb mechanical energy continues with a large increase in strain and a continuous marginal drop in stress up to the point of ultimate strain-the toughness increases due to the rise in the area under the stress-strain curve. The load capacity of confined specimens showed a slightly increasing trend with the increase in tube thickness. When the outer diameter to thickness ratio (D/t) changed from 22 to 12.9, local buckling of the polymeric tube decreased, and the strain at peak stress was increased. The increment in H/D ratio had adversely influenced the strength.
Engineers must seek alternative confining materials to improve structural members' performance in displaying sufficient deformations and strain softening past peak load. The confinement device was efficient in dissipating the internal strain energy and develops adequate ductility. However, for additional load capacity achievement, fiber-reinforced polymers (FRP) could be externally applied to the tubular shell for further lateral confinement, leading to improved overall tube toughness. Recent research (Jiequn et al., 2019) has shown that the PVC tube confinement can enhance the elastic modulus of plastic tube-confined recycled concrete.

Experimental database
An extensive review of published test data on PVC-confined concrete was conducted. Only the studies that report the stress and strain values were considered. The compiled database covers the test results of PVC-confined concrete columns of circular cross-sections under direct axial compression load. The final database covered the following details for each specimen: geometric properties of the uPVC tube, diameter (D), thickness (t), and height (H); material properties of the uPVC tube, tensile yield strength (fyp), elastic modulus (E), confined strength of uPVC-confined concrete at the peak of the stress-strain relationship (f'cc) and the strain (Ԑ'cu) at the peak stress; unconfined concrete cylinder strength (f'co) and strain (Ԑ'co). The final database meets the following criteria's: 1-Confinement methods; full jacketing, where only the confined specimens with no gaps at top and bottom were included.
2-The specimens included in the database failed due to uPVC tube yielding.
3-To consider the size effect, small-sized and large-sized specimens were considered. 4-Specimens with internal steel reinforcement were excluded.

Methods of evaluation
Three statistical indexes were employed to evaluate the applicability of the existing twenty strain models of FRP-confined concrete for predicting the strain of PVC-confined concrete. The statistical indexes refer to important additional details of the predicted data.
1-The average absolute error (AAE) given by the following equation: The AAE close to zero gives the best model with the highest precision. The second index was the average ratio (AR), which is equal to: The perfect AR value, which equals one, displays the boundary between the underestimation (AR<1), where the predicted value is less than the experimental value, and overestimation (AR>1), where the predicted value is more than the observed value. A third statistical indicator used to assess the performance of the models given by: Where the model prediction is expressed by (model. i ), the experimental value represented by (exp.i), N is the total number of data.

Strain models
Twenty existing FRP-confined concrete strain models Benzaid et al., 2010;Binici, 2008;Ciupala et al., 2007;Fardis & Khalili, 1982;Gao et al., 2019;Ilki et al., 2002Ilki et al., , 2004Jiang & Teng, 2007;Mandal et al., 2005;Miyauchi et al., , 1999Saadatmanesh et al., 1994;Saafi & Li, 1999;Samaan et al., 1998;Shehata et al., 2002;Toutanji, 1999;Wu et al., 2009;Youssef et al., 2007) were used to examine these models' applicability to predict the strain of PVC tube-confined concrete short columns subjected to axial compression loads. Although the FRP-confined concrete displays a bilinear response with the ascending branch, a descending branch is also possible depending on the confinement level, which is the case with PVC-confined concrete, which usually exhibit compression softening with a descending branch of the stress-strain curve. Several test parameters used in the strain models of FRPconfined concrete can evaluate the strain of the uPVC tube confined concrete. However, one parameter is different: the fiber rupture strain ε h,rup, and it was replaced with the strain at peak stress Ԑ'cu.

Data range
A 160 test data database for strain results of uPVC tube-confined concrete was compiled from the published literature combined with the eighteen test data from the present study. The range of geometric properties of uPVC tube used in the database was as follows; thickness of uPVC tube varies from 2.5 mm  to 8.5 mm (Wang & Yang, 2012), external tube diameter ranged from 68 mm (A.  to 168 mm (Fakharifar & Chen, 2016), height from 38 mm (Abdulla, 2014) to 600 mm . The ratio of εcc/εco ranged from 1.28 (Abdulla, 2014)

Predictive expression for strain at peak axial strength of PVC-confined concrete
The analysis shows that a more suitable strain model is needed, considering more experimental parameters. Several studies have pointed out the difficulty in modeling and predicting confined concrete strain (Yazici & Muhammad, 2012). Based on the comprehensive database, most of the FRP strain models were not suitable for predicting the strain of uPVC tube-confined concrete specimens. Only one model   Figure 8U, showed reasonable predictions, with AAE and AR values of 27.23% and 0.84, respectively. Therefore, an attempt was made in the present work to develop a model that includes most of the parameters given in the database as follows: Figure 8. Predicted strain ratio εcu/εco using FRP equations versus experimental εcu/εco.

Figure 8. (Continued).
The strain predictions using the proposed model were plotted versus the experimental values, Figure. 9. Compared with the FRP models, the proposed model's strain predictions were considerably discrete, with AAE and AR equal to 26.7% and 0.9, as shown in Fig. 9. The developed model captured the test data of the database with acceptable accuracy. Several studies pointed out the fact that the computed average absolute error values for Ԑ'cu /'co were more significant than that of fcc/fco estimates. An average absolute error of 30% is not unfamiliar for εcu/εco, and a number of studies on FRP-confined concrete studies have reported similar values (Yazici & Muhammad, 2012). In a similar finding, four existing FRP strain models were used to predicate the strain of FRPconfined concrete columns under cyclic axial loading based on a comprehensive experimental database (Fanaradelli & Rousakis, 2020), yielding predictions with AAE values close to or higher than 50%.

Conclusions
Three experimental parameters, tube diameter, tube thickness, and H/D ratio, were used to examine the mechanical behavior of eighteen short PVC tube confined concrete. Based on an experimental database of 160 test data combined with the 18 test data of the present study, the following conclusion can be drawn: 1-The thin-walled plastic tube was subjected to biaxial stress due to the applied direct compression load and the transfer of the axial loads from the core concrete to the external confining device. This was evident in plastic tubes having a thickness of 5 mm, where the tube underwent considerable outward buckling deformations resulting in a bowel type failure. When the tube's thickness was increased to 7 mm, less axial load was transferred from the concrete core. The tube exhibited less deformations due to the improvement in its geometrical properties.
2-The three test variables influenced the performance of uPVC-confined concrete, where the increase in tube thickness reduced buckling in the thin-walled plastic tube and increased the strain at peak stress of uPVC-confined concrete. The increase in the H/D ratio had adversely influenced the strength.
3-The PVC tube confinement device enhanced the strain at peak stress of concrete by 1.62-3.29 times compared with unconfined specimens. 4-Twenty of the existing strain models for FRP-confined concrete were used to predict the strain ratio εcu/εco for uPVC-confined concrete. All the models yielded AAE values above 30%, with only one model with an AAE value of 27.23%. 4-Based on a comprehensive database of 178 test data, a strain model was developed for predicting the strain at peak stress of PVC tube-confined concrete. The developed model's accuracy was verified by comparing model predictions with twenty existing FRP-confined concrete models using three statistical indexes. The distinguishing feature of the proposed strain model is that it includes the influence of several parameters on the concrete confined by the uPVC tube.

5-
The plastic tube's significance and vital role in improving the lateral buckling of concrete columns is a crucial pivoting factor when considering structural members' safety against the nonductile brittle collapse of the concrete core. A database of confined columns that undergo strain softening in the stress-strain curve's descending branch is more critical than the columns with second ascending branch FRP-confined columns. Therefore, it is imperative to predict PVC columns' deformations with reasonable precision at failure for structural applications. Hence more research and effort are needed to improve the developed models' performance, which requires more design variables, such as length effect and different loading conditions. Research is needed to establish an analytical technique to predict the stress-strain of PVC tube-confined concrete under axial load.