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Citation: Kachouh, N.; El-Maaddawy,

T.; El-Hassan, H.; El-Ariss, B.

Numerical Modeling of Concrete

Deep Beams Made with Recycled

Aggregates and Steel Fibers.

Buildings 2022, 12, 529. https://

doi.org/10.3390/buildings12050529

Academic Editor: Elena Ferretti

Received: 26 March 2022

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Article

Numerical Modeling of Concrete Deep Beams Made withRecycled Aggregates and Steel FibersNancy Kachouh, Tamer El-Maaddawy * , Hilal El-Hassan and Bilal El-Ariss

Department of Civil and Environmental Engineering, College of Engineering, Al Ain Campus, United ArabEmirates University, Al Ain P.O. Box 15551, United Arab Emirates; [email protected] (N.K.);[email protected] (H.E.-H.); [email protected] (B.E.-A.)* Correspondence: [email protected]

Abstract: A bilinear tensile softening law that can describe the post-cracking behavior of concretemade with recycled concrete aggregates (RCAs) and steel fibers was developed based on an inverseanalysis of characterization test data. Numerical simulation models were developed for large-scaleconcrete deep beams. The tensile softening laws along with characterization test results were used asinput data in the analysis. The numerical deep beam models were validated through a comparativeanalysis with published experimental results. A parametric study was conducted to investigate theeffect of varying the shear span-to-depth (a/h) ratio, steel fiber volume fraction (vf), and the presenceof a web opening on the shear response. Results of the parametric study indicated that the shearstrength gain caused by the addition of steel fibers at vf of 1 and 2% was higher in the deep beammodels with a lower a/h of 0.8, relative to that of their counterparts with a/h of 1.6. The effect of a/h onthe shear strength gain of the solid deep beam models diminished at the higher vf of 3%. The soliddeep beam models with a/h of 0.8 exhibited a shear strength gain of 78 to 108% due to the addition ofsteel fibers, whereas their counterparts with the web opening experienced a reduced shear strengthgain of 45 to 70%.

Keywords: numerical modeling; steel fibers; recycled concrete aggregates; deep beam; tensilesoftening; shear behavior; web openings

1. Introduction

Construction and demolition wastes (CDW) resulting from the demolition of agedconcrete structures are mostly disposed of in landfills, creating serious environmentalhazards [1]. Recycling wasted construction material in the form of recycled concreteaggregates (RCAs) is considered a sustainable solution to the problem of disposal of theCDW and to reduce the demand on natural aggregates (NAs). Despite their significantenvironmental benefits, the use of RCAs in the concrete industry is still limited to non-structural applications due to their inferior properties compared with those of NAs [2–4].The addition of steel fibers to cement-based mixtures improves the tensile strength, post-cracking tensile resistance, flexural strength, and energy absorption capacity [5,6]. Steelfibers have a potential to transform the mode of failure of cement-based mixtures fromquasi-brittle to pseudo-ductile through bridging micro- and macro-cracks caused by anapplied load [6]. As such, one of the solutions proposed in the literature to improvevarious properties of concrete with RCAs is the addition of steel fibers [7–10]. Steel fibersshould, however, be used with caution in the case of concrete structures in an aggressiveenvironment [11].

Reinforced concrete (RC) deep beams (i.e., a/h ≤ 2 [12]) are typically used as pre-caststructural walls, bent caps in concrete bridges, or transfer girders in high-rise buildings.Openings are typically installed in RC deep beams to create windows or provide passagefor electrical and mechanical ducts. Web openings produce discontinuity in the normal

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flow of stresses which reduces the beam shear capacity and stiffness [13–17]. Minimumamounts of transverse reinforcement are typically required in RC structural elements toensure adequate crack control and sufficient ductility [12,18]. The use of steel fibers insteadof conventional steel stirrups could compensate for the initial cost of steel fibers, eliminatecongestion of steel, and reduce the risk of defects and cracks [19–22]. Previous experimentalstudies verified the potential of steel fibers to improve the shear behavior of RC deep beamswith and without openings [23–28]. Results of these studies are, however, limited tospecific properties of the materials used, dimensions and geometry of the tested beams,and reinforcement details. Published analytical formulas for shear strength prediction ofsteel fiber-reinforced concrete deep beams [23,24,29–32] are not universally applicable andnot capable of predicting the complete structural response.

Finite element (FE) analysis is considered a powerful approach for an in-depth analysisof the non-linear behavior of complex concrete members, such as deep beams with andwithout openings made with RCAs and steel fibers. One of the key parameters neededfor modeling the response of steel-fiber reinforced concrete deep beams made of RCAs isthe tensile softening law (i.e., tension function). In principle, a uniaxial tensile test is usedto directly determine the tensile softening law of steel fiber-reinforce concrete [33]. Steelfibers remarkably enhanced the concrete post-cracking behavior through achieving higherductility, residual strength, and toughness [34]. However, due to the complexity of thistest, alternative indirect methods involving an inverse analysis have been proposed. Thesemethods require testing of concrete prisms, setting up parameters of a tension function,and iterations until the difference between the predicted response of the tested prisms andthat obtained from the experimental test becomes negligible [35–38].

Previous studies indicated that the tensile softening law of concrete reinforced with asingle type of fiber can be approximated with a bilinear law, whereas for concrete madewith the addition of different types of micro and macro steel fibers, a trilinear law maybe suggested [35,36]. A comparison between the predicted load-deflection response andthat of the tested prisms verified the validity of employing a bilinear softening curve inthe analysis [36]. Employing a trilinear softening curve also reasonably predicted theload-crack mouse opening displacement (CMOD) response of the tested prisms [36]. Thepredicted tensile stress-crack opening relationship employing the concept of the inverseanalysis of flexural test data was in good agreement with that recorded experimentally [37].The accuracy of the response predicted based on the inverse analysis conducted by [37]was even better than that obtained by adopting the procedure of the fib 2010 ModelCode [38]. Efforts on the development of FE models for large-scale steel fiber-reinforcedconcrete beams are continuously reported in the literature [39,40]. Numerical FE modelsof large-scale concrete elements reinforced with steel fibers employed a concrete tensilesoftening law derived from flexural tests on concrete prisms combined with the inverseanalysis [39,40]. Predictions of such FE models compared well with experimental data oftested deep beams with vf of 0.75% and 1% [39] and slender beams having vf of 0.5% [40].To the best knowledge of the authors, tensile softening laws necessary for FE modeling ofconcrete structures made with RCAs and steel fibers are not available in the literature.

2. Research Objectives

International codes and standards do not offer a closed form solution to predict theshear behavior of steel fiber-reinforced recycled aggregates concrete deep beams. Pub-lished analytical formulas are largely empirical and not capable of predicting the completestructural response. This research aims to fill this gap through development of numericalsimulation deep beam models and conducting a parametric study to examine the influenceof key parameters on the shear behavior of RC deep beams made with RCAs and steelfibers. The specific objectives of the work are:

• Establish tensile softening laws (tension functions) that can describe the post-crackingbehavior of concrete made with RCAs and different steel fiber volume fractions.

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• Develop numerical simulation models capable of predicting the shear behavior of steelfiber-reinforced recycled aggregates concrete deep beams.

• Validate the prediction of the numerical models through a comparative analysis withpublished experimental data.

• Investigate the effect of varying a/h, vf, and presence of a web opening on the shearcapacity and strength gain caused by the addition of steel fibers.

3. Methodology and Material Characteristics

Previous investigations [40–48] verified the validity of the software ATENA 3D [49] topredict and simulate the behavior of different structural elements with good accuracy, andhence, was adopted in the current study. Although the software has built-in constitutivelaws for conventional concrete, such laws are not available for concrete made with RCAsand steel fibers. These constitutive laws are developed by the users based on materialcharacterization test data.

The uniaxial constitutive laws of the concrete in compression and tension start witha linear part with a slope value equal to the concrete modulus of elasticity (Ec). Thecompressive plastic behavior and the tensile softening response are then representedby a multilinear function as shown in Figure 1a,b, respectively. The compressive plasticresponse starts at a compressive stress ratio (σc/f’c) of 0.25, where f’c = cylinder compressivestrength. The following change in slope occurs at σc/f’c = 0.8 with a correspondingplastic compressive strain of 0.5εcp, where εcp is the plastic concrete strain at compressivestrength generated by the software based on the concrete cube strength (fcu). The post-peakcompressive stress-strain law is linearly descending. The value of the concrete plastic strainat a complete release of stress (εd) is generated by the software. For steel-fiber reinforcedconcrete, the software manual recommends adopting a value of κεd for the concrete plasticstrain at zero stress, where κ is a magnification factor equal to 100 [49]. It should be notedthat adopting a magnification factor of 50 rather than 100 had no effect on numerical results,whereas deactivating the magnification factor reduced the shear capacity by less than 4%.

Buildings 2022, 12, 529 3 of 28

• Develop numerical simulation models capable of predicting the shear behavior of steel fiber-reinforced recycled aggregates concrete deep beams.

• Validate the prediction of the numerical models through a comparative analysis with published experimental data.

• Investigate the effect of varying a/h, vf, and presence of a web opening on the shear capacity and strength gain caused by the addition of steel fibers.

3. Methodology and Material Characteristics Previous investigations [40–48] verified the validity of the software ATENA 3D [49]

to predict and simulate the behavior of different structural elements with good accuracy, and hence, was adopted in the current study. Although the software has built-in constitu-tive laws for conventional concrete, such laws are not available for concrete made with RCAs and steel fibers. These constitutive laws are developed by the users based on mate-rial characterization test data.

The uniaxial constitutive laws of the concrete in compression and tension start with a linear part with a slope value equal to the concrete modulus of elasticity (Ec). The com-pressive plastic behavior and the tensile softening response are then represented by a mul-tilinear function as shown in Figure 1a,b, respectively. The compressive plastic response starts at a compressive stress ratio (σc/f’c) of 0.25, where f’c = cylinder compressive strength. The following change in slope occurs at σc/f’c = 0.8 with a corresponding plastic compres-sive strain of 0.5εcp, where εcp is the plastic concrete strain at compressive strength gener-ated by the software based on the concrete cube strength (fcu). The post-peak compressive stress-strain law is linearly descending. The value of the concrete plastic strain at a com-plete release of stress (εd) is generated by the software. For steel-fiber reinforced concrete, the software manual recommends adopting a value of κεd for the concrete plastic strain at zero stress, where κ is a magnification factor equal to 100 [49]. It should be noted that adopting a magnification factor of 50 rather than 100 had no effect on numerical results, whereas deactivating the magnification factor reduced the shear capacity by less than 4%.

(a) (b)

Figure 1. Details of the user concrete material model: (a) compressive hardening–softening behav-ior; (b) tensile softening behavior.

As shown in Figure 1b, the post-peak response of plain concrete in tension is bilinear. The first part is linearly descending until the tensile stress reaches a tensile stress ratio (σt/ft) of 0.25 at a plastic tensile strain of 0.15εtpmax, where ft = tensile strength, and εtpmax = maximum plastic fracture strain at zero stress. The second part of the post-peak response is linearly descending but at a slighter slope until a complete release of stress occurs at a plastic strain value of εtpmax. The user-defined model allows users to account for the tension stiffening effect in heavily reinforced concrete by adopting a limiting minimum value of a reduced tensile strength ≤ 0.4ft on the tensile softening branch. In such cases, the tensile strength cannot drop below the pre-specified limiting tensile strength taken as 0.25 ft in the current study for the models without steel fibers. Steel fiber-reinforced concrete is characterized by a significant post-peak residual tensile stress–strain response. The typical

Figure 1. Details of the user concrete material model: (a) compressive hardening–softening behavior;(b) tensile softening behavior.

As shown in Figure 1b, the post-peak response of plain concrete in tension is bilin-ear. The first part is linearly descending until the tensile stress reaches a tensile stressratio (σt/ft) of 0.25 at a plastic tensile strain of 0.15εtpmax, where ft = tensile strength, andεtpmax = maximum plastic fracture strain at zero stress. The second part of the post-peakresponse is linearly descending but at a slighter slope until a complete release of stressoccurs at a plastic strain value of εtpmax. The user-defined model allows users to account forthe tension stiffening effect in heavily reinforced concrete by adopting a limiting minimumvalue of a reduced tensile strength ≤0.4ft on the tensile softening branch. In such cases,the tensile strength cannot drop below the pre-specified limiting tensile strength takenas 0.25 ft in the current study for the models without steel fibers. Steel fiber-reinforcedconcrete is characterized by a significant post-peak residual tensile stress–strain response.

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The typical post-peak response of this concrete is linearly descending until it reaches a stressvalue of Csfft, where Csf = coefficient determined from the inverse analysis, and then theresponse continues to decrease, linearly or multilinearly at slighter slopes, until it reaches amaximum plastic fracture strain of εtpmax determined from the inverse analysis. The steelbars used in the large-scale RC deep beam models included No. 25 (25 mm diameter)deformed bars as flexural reinforcement and No. 5 (5 mm diameter) smooth bars as shearreinforcement. Based on testing of three replicate steel samples, the average measuredyield strengths of the No. 25 and No. 5 bars were 539 and 505 MPa, respectively, whereastheir ultimate tensile strengths were 649 and 543 MPa, respectively. The elastic modulus ofthe reinforcing steel bars was 200 GPa. The reinforcing steel bars were modeled using abilinear stress–strain relationship with a post-yield strain hardening of 1%.

3.1. Properties of Concrete Mixtures

Characterization tests were conducted on five concrete mixtures adopted in the currentstudy. The mixes are labeled RX-SFY, where X denotes the percentage of RCAs and Yindicates the value of vf (Table 1). The control mix R0-SF0 was designed to developf’c of 30 MPa, based on the American Concrete Institute (ACI) 211.1 [50]. The cementwas ordinary Portland cement (OPC), while the water–cement ratio (w/c) was 0.49. Thenatural coarse aggregates were crushed dolomitic limestone with a nominal maximumparticle size (NMS) of 19 mm, whereas the recycled coarse aggregates were collectedfrom a local concrete recycling plant with an NMS of 25 mm. The concrete recyclingplant crushes old concrete structures with an unknown compressive strength. The fineaggregates used were dune sand. The concrete mix proportions by weight were as follows(cement:fine aggregates:coarse aggregates:w/c; 1:1.21:2.4:0.49). The steel fibers used in thisstudy, Dramix®3D 65/35BG, were end-hooked with a tensile strength of 1345 N/mm2,Young’s modulus of 210 GPa, mean diameter (df) of 0.55 mm, mean length (lf) of 35 mm,and an aspect ratio (lf/df) of 65 [51]. A photograph of the steel fibers is presented in Figure 2.Mechanical properties of the concrete mixtures, based on data of three replicate specimens,are reported in Table 1. The addition of steel fibers insignificantly affected the compressivestrength and modulus of elasticity of the concrete, but remarkably improved the modulusof rupture (fr). An increase of 71% in fr was recorded at vf of 1%. Higher values of vfresulted in more than a 2-fold increase fr.

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post-peak response of this concrete is linearly descending until it reaches a stress value of Csfft, where Csf = coefficient determined from the inverse analysis, and then the response continues to decrease, linearly or multilinearly at slighter slopes, until it reaches a maxi-mum plastic fracture strain of εtpmax determined from the inverse analysis. The steel bars used in the large-scale RC deep beam models included No. 25 (25 mm diameter) deformed bars as flexural reinforcement and No. 5 (5 mm diameter) smooth bars as shear reinforce-ment. Based on testing of three replicate steel samples, the average measured yield strengths of the No. 25 and No. 5 bars were 539 and 505 MPa, respectively, whereas their ultimate tensile strengths were 649 and 543 MPa, respectively. The elastic modulus of the reinforcing steel bars was 200 GPa. The reinforcing steel bars were modeled using a bilin-ear stress–strain relationship with a post-yield strain hardening of 1%.

3.1. Properties of Concrete Mixtures Characterization tests were conducted on five concrete mixtures adopted in the cur-

rent study. The mixes are labeled RX-SFY, where X denotes the percentage of RCAs and Y indicates the value of vf (Table 1). The control mix R0-SF0 was designed to develop f’c of 30 MPa, based on the American Concrete Institute (ACI) 211.1 [50]. The cement was ordi-nary Portland cement (OPC), while the water–cement ratio (w/c) was 0.49. The natural coarse aggregates were crushed dolomitic limestone with a nominal maximum particle size (NMS) of 19 mm, whereas the recycled coarse aggregates were collected from a local concrete recycling plant with an NMS of 25 mm. The concrete recycling plant crushes old concrete structures with an unknown compressive strength. The fine aggregates used were dune sand. The concrete mix proportions by weight were as follows (cement:fine aggregates:coarse aggregates:w/c; 1:1.21:2.4:0.49). The steel fibers used in this study, Dramix® 3D 65/35BG, were end-hooked with a tensile strength of 1345 N/mm2, Young's modulus of 210 GPa, mean diameter (df) of 0.55 mm, mean length (lf) of 35 mm, and an aspect ratio (lf/df) of 65 [51]. A photograph of the steel fibers is presented in Figure 2. Me-chanical properties of the concrete mixtures, based on data of three replicate specimens, are reported in Table 1. The addition of steel fibers insignificantly affected the compressive strength and modulus of elasticity of the concrete, but remarkably improved the modulus of rupture (fr). An increase of 71% in fr was recorded at vf of 1%. Higher values of vf resulted in more than a 2-fold increase fr.

Figure 2. Steel fibers used in the current study.

Figure 2. Steel fibers used in the current study.

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Table 1. Mechanical properties of concrete mixtures.

Property StandardSample Type andDimensions (mm)

Characterization Test Results *

R0-SF0 R100-SF0 R100-SF1 R100-SF2 R100-SF3

f’c (MPa) ASTM C39 [52] Cylinder,150 × 300 36.4 (1.5) 23.6 (0.4) 25.8 (0.3) 25.6 (1.4) 25.0 (0.8)

fcu (MPa) BS 12390-3 [53] Cube, 150 40.5 (1.8) 24.7 (1.0) 32.2 (0.6) 30.0 (1.0) 28.3 (0.7)

fr (MPa) ASTM C1609 [54] Prism,100 × 100 × 500 3.4 (0.2) 2.4 (0.5) 4.1 (0.4) 5.4 (0.4) 6.1 (0.7)

Ec (GPa) ASTM C469 [55] Cylinder,150 × 300 34.7 (1.8) 19.8 (2.4) 21.5 (2.9) 20.7 (2.5) 21.1 (1.6)

* Values between parentheses represent the standard deviation.

3.2. Tensile Softening Laws of Concrete Mixtures

Four-point bending tests were conducted on concrete prisms (100 × 100 × 500 mm)made of RCAs and steel fibers following the ASTM C1609 [54]. Figure 3a shows thefour-point bending test setup, while Figure 3b shows a photograph of a test in progress.The inverse analysis technique started by developing FE models for the tested prismsand setting up specific tensile parameters in a user-defined tensile softening law of theconcrete. The value of ft was kept constant at 0.6fr [56]. The other input parameters of thepost-cracking tension function were modified, and several iterations were considered untilthe difference between the numerical and experimental load–displacement curves of theprisms became negligible.

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Table 1. Mechanical properties of concrete mixtures.

Property Standard

Sample Type and

Dimensions (mm)

Characterization Test Results *

R0-SF0 R100-SF0 R100-SF1 R100-SF2 R100-SF3

f’c (MPa) ASTM C39

[52] Cylinder, 150 x 300 36.4 (1.5) 23.6 (0.4) 25.8 (0.3) 25.6 (1.4) 25.0 (0.8)

fcu (MPa) BS 12390-3

[53] Cube, 150 40.5 (1.8) 24.7 (1.0) 32.2 (0.6) 30.0 (1.0) 28.3 (0.7)

fr (MPa) ASTM C1609

[54] Prism,

100 x 100 x 500 3.4 (0.2) 2.4 (0.5) 4.1 (0.4) 5.4 (0.4) 6.1 (0.7)

Ec (GPa) ASTM C469

[55] Cylinder, 150 x 300 34.7 (1.8) 19.8 (2.4) 21.5 (2.9) 20.7 (2.5) 21.1 (1.6)

* Values between parentheses represent the standard deviation.

3.2. Tensile Softening Laws of Concrete Mixtures Four-point bending tests were conducted on concrete prisms (100 × 100 × 500 mm)

made of RCAs and steel fibers following the ASTM C1609 [54]. Figure 3a shows the four-point bending test setup, while Figure 3b shows a photograph of a test in progress. The inverse analysis technique started by developing FE models for the tested prisms and set-ting up specific tensile parameters in a user-defined tensile softening law of the concrete. The value of ft was kept constant at 0.6fr [56]. The other input parameters of the post-cracking tension function were modified, and several iterations were considered until the difference between the numerical and experimental load–displacement curves of the prisms became negligible.

Figure 4a shows the experimental load–deflection curves of three replicate prisms for each concrete mix along with the corresponding predicted response, whereas Figure 4b shows the tension functions implemented in the analysis. Figure 4a shows that the flexural load capacity and toughness increased with an increase in the steel fiber volume fraction. Figure 4b indicates that the tension function of the concrete with steel fibers is approxi-mated with a bilinear law. When the concrete reaches its tensile strength, ft, the tensile stress drops to a value of approximately 0.6ft, then decreases linearly with an increase in the tensile strain. The tension function of the concrete with vf of 2 and 3% exhibited a slightly reduced rate of degradation in the tensile stress after cracking relative to that of the concrete with vf of 1%. The agreement between the predicted and experimental load–deflection responses of the prisms verified the validity of the tension function developed from the inverse analysis.

(a) (b)

Figure 3. Four-point bending test setup: (a) schematic representation (dimensions are in mm); (b) test in progress.

Figure 3. Four-point bending test setup: (a) schematic representation (dimensions are in mm); (b) testin progress.

Figure 4a shows the experimental load–deflection curves of three replicate prisms foreach concrete mix along with the corresponding predicted response, whereas Figure 4bshows the tension functions implemented in the analysis. Figure 4a shows that the flexuralload capacity and toughness increased with an increase in the steel fiber volume fraction.Figure 4b indicates that the tension function of the concrete with steel fibers is approximatedwith a bilinear law. When the concrete reaches its tensile strength, ft, the tensile stress dropsto a value of approximately 0.6ft, then decreases linearly with an increase in the tensilestrain. The tension function of the concrete with vf of 2 and 3% exhibited a slightly reducedrate of degradation in the tensile stress after cracking relative to that of the concrete with vfof 1%. The agreement between the predicted and experimental load–deflection responses ofthe prisms verified the validity of the tension function developed from the inverse analysis.

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R100-SF1

R100-SF2 R100-SF3 (a)

R100-SF1

R100-SF2

R100-SF3

(b)

Figure 4. Four-point bending test data: (a) load–deflection response; (b) tension functions.

3.3. Large-Scale Deep Beam Numerical Models Fourteen FE models were developed for large-scale RC deep beams tested previously

by the authors [27,28]. A summary of the deep beam models is given in Table 2. The mod-els were divided into two groups, S and N, where S refers to solid beams, and N refers to beams with a circular web opening in the middle of each shear span with an opening height-to-beam depth ratio (h0/h) of 0.3. The deep beam models of group S had a/h of 1.6, whereas those of group N had a/h of 0.8. The variables in each group included the type of aggregates (NAs and RCAs), steel fiber volume fraction (1%, 2%, and 3%), and presence of minimum steel stirrups recommended by provisions of the American Concrete Institute (ACI) 318-14 [12] for deep beams.

A quarter FE model was built for each beam to take advantage of the planes of sym-metry and reduce the overall computational time. Figures 5 and 6 show the geometry, boundary condition, details of reinforcement, and location of monitoring points of typical deep beam models of groups S and N, respectively. The deep beam models were rein-forced with 4 No. 25 (25 mm in diameter) steel bars at the tension side and 2 No. 25 (25 mm in diameter) at the compression side. The concrete cover to the center of the steel reinforcement was 50 mm, rendering an effective depth (d) of 450 mm. The steel stirrups, when employed, consisted of two curtains of No. 5 (5 mm in diameter) bars spaced at 80 mm in both vertical and horizontal directions. Steel plates (150 × 150 × 20 mm) were placed at the load and support points. The concrete and steel plates were modeled as solid 3D tetrahedron macro-elements with a mesh size of 15 mm. The steel bars were modeled as one-dimensional discrete elements embedded in the concrete macro-element.

A line support was placed at the middle of the bottom surface of the support steel plate to restrict its movement in the vertical and transverse directions. The surfaces of the planes of symmetry were restrained from movement in a direction perpendicular to the other symmetrical part of the beam through surface supports. A displacement-controlled applied load was induced at the middle of the top steel plate at a rate of 0.1 mm per step. The standard Newton–Raphson iterative solution method implemented in ATENA [49] was adopted. The iteration had to satisfy a tolerance limit of convergence criteria of 1%.

0

4

8

12

16

20

0 1 2 3 4 5 6 7

Load

(kN

)

Deflection (mm)

R100-SF1-1R100-SF1-2R100-SF1-3FE Prediction

0

4

8

12

16

20

0 1 2 3 4 5 6 7

Load

(kN

)Deflection (mm)


0

4

8

12

16

20

0 1 2 3 4 5 6 7

Load

(kN

)

Deflection (mm)


0.0

0.3

0.6

0.9

1.2

0 0.05 0.1 0.15 0.2

Stre

ss/f t

Strain

0.0

0.3

0.6

0.9

1.2

0 0.05 0.1 0.15 0.2

Stre

ss/f t

Strain

0.0

0.3

0.6

0.9

1.2

0 0.05 0.1 0.15 0.2

Stre

ss/f t

Strain

Figure 4. Four-point bending test data: (a) load–deflection response; (b) tension functions.

3.3. Large-Scale Deep Beam Numerical Models

Fourteen FE models were developed for large-scale RC deep beams tested previouslyby the authors [27,28]. A summary of the deep beam models is given in Table 2. Themodels were divided into two groups, S and N, where S refers to solid beams, and N refersto beams with a circular web opening in the middle of each shear span with an openingheight-to-beam depth ratio (h0/h) of 0.3. The deep beam models of group S had a/h of 1.6,whereas those of group N had a/h of 0.8. The variables in each group included the type ofaggregates (NAs and RCAs), steel fiber volume fraction (1%, 2%, and 3%), and presence ofminimum steel stirrups recommended by provisions of the American Concrete Institute(ACI) 318-14 [12] for deep beams.

A quarter FE model was built for each beam to take advantage of the planes ofsymmetry and reduce the overall computational time. Figures 5 and 6 show the geometry,boundary condition, details of reinforcement, and location of monitoring points of typicaldeep beam models of groups S and N, respectively. The deep beam models were reinforcedwith 4 No. 25 (25 mm in diameter) steel bars at the tension side and 2 No. 25 (25 mm indiameter) at the compression side. The concrete cover to the center of the steel reinforcementwas 50 mm, rendering an effective depth (d) of 450 mm. The steel stirrups, when employed,consisted of two curtains of No. 5 (5 mm in diameter) bars spaced at 80 mm in both verticaland horizontal directions. Steel plates (150 × 150 × 20 mm) were placed at the load andsupport points. The concrete and steel plates were modeled as solid 3D tetrahedron macro-elements with a mesh size of 15 mm. The steel bars were modeled as one-dimensionaldiscrete elements embedded in the concrete macro-element.

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Table 2. Summary of deep beam models.

Group ModelDesignation a/h RCA

(%)vf

(%)

Presence ofTraditional

Shear Reinforcement

Presence ofOpenings

S

SR0-SF0 1.6 - - - -

SR0-SF0-S 1.6 - -√

-

SR100-SF0 1.6 100 - - -

SR100-SF0-S 1.6 100 -√

-

SR100-SF1 1.6 100 1 - -

SR100-SF2 1.6 100 2 - -

SR100-SF3 1.6 100 3 - -

N

NR0-SF0 0.8 - - -√

NR0-SF0-S 0.8 - -√ √

NR100-SF0 0.8 100 - -√

NR100-SF0-S 0.8 100 -√ √

NR100-SF1 0.8 100 1 -√

NR100-SF2 0.8 100 2 -√

NR100-SF3 0.8 100 3 -√

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Table 2. Summary of deep beam models.

Group Model Designation a/h

RCA (%)

vf (%)

Presence of Traditional

Shear Reinforcement

Presence of Openings

S

SR0-SF0 1.6 - - - - SR0-SF0-S 1.6 - - √ - SR100-SF0 1.6 100 - - -

SR100-SF0-S 1.6 100 - √ - SR100-SF1 1.6 100 1 - - SR100-SF2 1.6 100 2 - - SR100-SF3 1.6 100 3 - -

N

NR0-SF0 0.8 - - - √ NR0-SF0-S 0.8 - - √ √ NR100-SF0 0.8 100 - - √

NR100-SF0-S 0.8 100 - √ √ NR100-SF1 0.8 100 1 - √ NR100-SF2 0.8 100 2 - √ NR100-SF3 0.8 100 3 - √

(a)

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Figure 5. A typical deep beam model of group S (dimensions are in mm): (a) beam geometry con-figuration and boundary conditions; (b) reinforcement details and monitoring points. Figure 5. A typical deep beam model of group S (dimensions are in mm): (a) beam geometryconfiguration and boundary conditions; (b) reinforcement details and monitoring points.


(a)

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Figure 6. A typical deep beam model of group N (dimensions are in mm): (a) beam geometry con-figuration and boundary conditions; (b) reinforcement details and monitoring points.

4. Numerical Results and Validation of FE Deep Beam Models Predictions of the FE deep beam models are compared with those obtained from ex-

perimental tests conducted previously by the authors [27,28] for validation.

4.1. Solid Deep Beam Models 4.1.1. Shear Load–Deflection Response

The shear load–deflection curves of the solid RC deep beam models are illustrated in Figure 7. The response of the deep beam models made with 100% RCAs without steel fibers was inferior to that of their counterparts made with NAs (Figure 7a). The addition of steel fibers remarkably improved the shear capacity and stiffness of the RCA-based models (Figure 7b). The stiffness of model SR100-SF2, with vf = 2 %, coincided with that of SR100-SF0-S having steel stirrups, while that of model SR100-SF3, with vf = 3%, was supe-rior to that of SR100-SF0-S. These findings are in agreement with those obtained from ex-perimental tests [27,28].

The shear load–deflection responses predicted numerically are compared with those obtained from the tests in Figure 8. The response of the deep beam models SR0-SF0 and SR100-SF0 exhibited a drop in load at the onset of the initiation of shear cracking. Then, the deflection continued to increase but at a higher rate (Figure 8a,c). Another drop in load was observed prior to failure due to the initiation of another shear crack (Figure 8a,c). The

Figure 6. A typical deep beam model of group N (dimensions are in mm): (a) beam geometryconfiguration and boundary conditions; (b) reinforcement details and monitoring points.

A line support was placed at the middle of the bottom surface of the support steelplate to restrict its movement in the vertical and transverse directions. The surfaces of theplanes of symmetry were restrained from movement in a direction perpendicular to theother symmetrical part of the beam through surface supports. A displacement-controlledapplied load was induced at the middle of the top steel plate at a rate of 0.1 mm per step.The standard Newton–Raphson iterative solution method implemented in ATENA [49]was adopted. The iteration had to satisfy a tolerance limit of convergence criteria of 1%.

4. Numerical Results and Validation of FE Deep Beam Models

Predictions of the FE deep beam models are compared with those obtained fromexperimental tests conducted previously by the authors [27,28] for validation.

4.1. Solid Deep Beam Models4.1.1. Shear Load–Deflection Response

The shear load–deflection curves of the solid RC deep beam models are illustratedin Figure 7. The response of the deep beam models made with 100% RCAs without steelfibers was inferior to that of their counterparts made with NAs (Figure 7a). The additionof steel fibers remarkably improved the shear capacity and stiffness of the RCA-basedmodels (Figure 7b). The stiffness of model SR100-SF2, with vf = 2 %, coincided with thatof SR100-SF0-S having steel stirrups, while that of model SR100-SF3, with vf = 3%, was

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superior to that of SR100-SF0-S. These findings are in agreement with those obtained fromexperimental tests [27,28].

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deep beam models with steel stirrups (Figure 8b,d) or steel fibers (Figure 8e–g) exhibited a quasilinear load–deflection response until the shear capacity was reached. The stiffness of the numerical models was insignificantly higher than that obtained from the tests. Ac-tual beams tested in the laboratory were vulnerable to microcracking due to shrinkage, which would reduce the actual stiffness of the tested beams.

The experimental and numerical shear capacities, VEXP and VFE, respectively, along with the deflections at failure, ΔEXP and ΔFE, are compared in Table 3. The difference be-tween the numerical and experimental shear capacities was within 10% only. The ratio of VFE/VEXP was on average 0.98 with a corresponding standard deviation and coefficient of variation of 0.07% and 7%, respectively. The FE models tended to underestimate the de-flections at failure. The ratio of ΔFE/ΔEXP was on average 0.83 with a standard deviation of 0.10 and a coefficient of variation of 12%. Generally, the difference between numerical and experimental results could be due to a variation between the actual properties of materials of the tested large-scale deep beams and those obtained from characterization test samples used as input data in the analysis. The difference between numerical and experimental results can be considered within the expected margin of error.

(a) (b)

Figure 7. Numerical shear load–deflection response of group S: (a) beam models without steel fibers; (b) beam models with RCAs/steel fibers.

(a) (b)

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Figure 7. Numerical shear load–deflection response of group S: (a) beam models without steel fibers;(b) beam models with RCAs/steel fibers.

The shear load–deflection responses predicted numerically are compared with thoseobtained from the tests in Figure 8. The response of the deep beam models SR0-SF0 andSR100-SF0 exhibited a drop in load at the onset of the initiation of shear cracking. Then, thedeflection continued to increase but at a higher rate (Figure 8a,c). Another drop in loadwas observed prior to failure due to the initiation of another shear crack (Figure 8a,c). Thedeep beam models with steel stirrups (Figure 8b,d) or steel fibers (Figure 8e–g) exhibited aquasilinear load–deflection response until the shear capacity was reached. The stiffness ofthe numerical models was insignificantly higher than that obtained from the tests. Actualbeams tested in the laboratory were vulnerable to microcracking due to shrinkage, whichwould reduce the actual stiffness of the tested beams.

The experimental and numerical shear capacities, VEXP and VFE, respectively, alongwith the deflections at failure, ∆EXP and ∆FE, are compared in Table 3. The differencebetween the numerical and experimental shear capacities was within 10% only. The ratioof VFE/VEXP was on average 0.98 with a corresponding standard deviation and coefficientof variation of 0.07% and 7%, respectively. The FE models tended to underestimate thedeflections at failure. The ratio of ∆FE/∆EXP was on average 0.83 with a standard deviationof 0.10 and a coefficient of variation of 12%. Generally, the difference between numericaland experimental results could be due to a variation between the actual properties ofmaterials of the tested large-scale deep beams and those obtained from characterizationtest samples used as input data in the analysis. The difference between numerical andexperimental results can be considered within the expected margin of error.

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deep beam models with steel stirrups (Figure 8b,d) or steel fibers (Figure 8e–g) exhibited a quasilinear load–deflection response until the shear capacity was reached. The stiffness of the numerical models was insignificantly higher than that obtained from the tests. Ac-tual beams tested in the laboratory were vulnerable to microcracking due to shrinkage, which would reduce the actual stiffness of the tested beams.

The experimental and numerical shear capacities, VEXP and VFE, respectively, along with the deflections at failure, ΔEXP and ΔFE, are compared in Table 3. The difference be-tween the numerical and experimental shear capacities was within 10% only. The ratio of VFE/VEXP was on average 0.98 with a corresponding standard deviation and coefficient of variation of 0.07% and 7%, respectively. The FE models tended to underestimate the de-flections at failure. The ratio of ΔFE/ΔEXP was on average 0.83 with a standard deviation of 0.10 and a coefficient of variation of 12%. Generally, the difference between numerical and experimental results could be due to a variation between the actual properties of materials of the tested large-scale deep beams and those obtained from characterization test samples used as input data in the analysis. The difference between numerical and experimental results can be considered within the expected margin of error.

(a) (b)

Figure 7. Numerical shear load–deflection response of group S: (a) beam models without steel fibers; (b) beam models with RCAs/steel fibers.

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Figure 8. Cont.


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Figure 8. Numerical and experimental responses of group S: (a) SR0-SF0; (b) SR0-SF0-S; (c) SR100-SF0; (d) SR100-SF0-S; (e) SR100-SF1; (f) SR100-SF2; and (g) SR100-SF3.

Table 3. Comparison between numerical and experimental results of group S.

Model Shear Capacity Deflection Capacity

VFE/VEXP ΔFE/ΔEXP

VEXP (kN) VFE (kN) ΔEXP (mm) ΔFE (mm) SR0-SF0 203 198 7.3 7.1 0.98 0.97

SR0-SF0-S 309 280 10.3 8.0 0.91 0.78 SR100-SF0 193 181 9.0 6.2 0.94 0.69

SR100-SF0-S 300 270 10.7 8.2 0.90 0.77 SR100-SF1 235 251 8.3 7.9 1.07 0.95 SR100-SF2 271 296 10.4 9.5 1.09 0.91 SR100-SF3 401 387 15.8 12.0 0.97 0.77 Average 0.98 0.83 Std Dev 0.07 0.10 COV (%) 7.03 11.95

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Figure 8. Numerical and experimental responses of group S: (a) SR0-SF0; (b) SR0-SF0-S; (c) SR100-SF0;(d) SR100-SF0-S; (e) SR100-SF1; (f) SR100-SF2; and (g) SR100-SF3.

Table 3. Comparison between numerical and experimental results of group S.

ModelShear Capacity Deflection Capacity

VFE/VEXP∆FE/∆EXP

VEXP (kN) VFE (kN) ∆EXP (mm) ∆FE (mm)

SR0-SF0 203 198 7.3 7.1 0.98 0.97

SR0-SF0-S 309 280 10.3 8.0 0.91 0.78

SR100-SF0 193 181 9.0 6.2 0.94 0.69

SR100-SF0-S 300 270 10.7 8.2 0.90 0.77

SR100-SF1 235 251 8.3 7.9 1.07 0.95

SR100-SF2 271 296 10.4 9.5 1.09 0.91

SR100-SF3 401 387 15.8 12.0 0.97 0.77

Average 0.98 0.83

Std Dev 0.07 0.10

COV (%) 7.03 11.95

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4.1.2. Crack Patterns

The crack patterns of the solid deep beams captured numerically are shown inFigure 9a. The deep beam models SR0-SF0 and SR100-SF0 exhibited initially an inclinedshear crack in the shear span. At failure, the models exhibited an additional shear crackin the shear span in addition to longitudinal splitting cracks parallel to the tension steel.The deep beam models with stirrups, SR0-SF0-S and SR100-SF0-S, showed a band of shearcracks in the shear span prior to failure. The steel fibers played a role similar to that of thesteel stirrups and prevented the formation of longitudinal splitting cracks. The deep beammodels with steel fibers failed in a diagonal compression model of failure.

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4.1.2. Crack Patterns The crack patterns of the solid deep beams captured numerically are shown in Figure

9a. The deep beam models SR0-SF0 and SR100-SF0 exhibited initially an inclined shear crack in the shear span. At failure, the models exhibited an additional shear crack in the shear span in addition to longitudinal splitting cracks parallel to the tension steel. The deep beam models with stirrups, SR0-SF0-S and SR100-SF0-S, showed a band of shear cracks in the shear span prior to failure. The steel fibers played a role similar to that of the steel stirrups and prevented the formation of longitudinal splitting cracks. The deep beam models with steel fibers failed in a diagonal compression model of failure.

The minimum principal concrete strain of the deep beam models prior to failure are shown in Figure 9b. The profile of the minimum principal concrete strain indicated the formation of a bottle-shaped strut in the shear span of the deep beam models with steel stirrups and a prismatic strut for the models with steel fibers. Photographs of the tested beams at failure are shown in Figure 9c. The crack patterns obtained from the tests, shown in Figure 9c, verified the formation of a band of shear cracks in the shear span of the deep beams with steel stirrups and concentration of inclined shear cracks along the natural load path of the beams with steel fibers.

It should be noted that the deep beam models without steel fibers exhibited similar values of minimum principal concrete strain of approximately −0.006 to –0.008 prior to failure. The models with steel stirrups failed, however, at a higher shear capacity than that of their counterparts without stirrups, signifying the role of stirrups in reducing the rate of increase in the minimum principal concrete strain. The deep beam models with steel fibers exhibited minimum principal concrete strain values of approximately −0.006 to −0.01 at peak load but failed at a higher load relative to that of their counterparts without stirrups. The increased shear capacity of the steel-fiber reinforced models signified the effectiveness of the steel fibers in reducing the rate of increase in the minimum principal concrete strains.

SR0

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SR10

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(a) (b) (c)

Figure 9. Crack patterns and principal strains of group S: (a) numerical crack patterns; (b) minimum principal strains; and (c) experimental crack patterns.

4.1.3. Stirrup Strains The numerical stirrup strain responses of the model SR0-SF0-S are compared with

those measured experimentally in the west and east shear spans in Figure 10a,b, respec-tively. Readings of SG-V2 were not captured experimentally in the east shear span due to malfunction of the SG. The locations of the stirrups monitoring points are shown in Figure 5. It should be noted that predicted strains of SG-V1 and SG-V2 were identical. Similarly, the rate of increase in the stirrup strain in the vertical stirrups recorded experimentally was insignificantly different, as shown in Figure 10a. Numerical data of the model SR0-SF0-S indicated failure of the beam without yielding stirrups. This behavior was in align-ment with the experimental data, except in one location (SG-V1) in the west shear span. Numerical results indicated that no strains were recorded in the stirrups in the pre-crack-ing stage. Following shear cracking, the stirrup strains increased almost linearly until the model reached its shear capacity. This behavior was in agreement with the stirrup strains measured in the west and east shear spans as shown in Figure 10a,b, respectively.

The numerical stirrup strain responses of the model SR100-SF0-S are compared with those measured experimentally in the west and east shear spans in Figure 11a,b, respec-tively. The vertical stirrups in the model SR100-SF0-S made with RCAs exhibited higher strains than those exhibited by the horizontal stirrups, indicating more contribution to the shear resistance. Similarly, the vertical stirrup strains obtained from the tests tended to be higher than those of the horizontal stirrups in the west and east shear spans as shown in Figure 11a,b, respectively. The strain in the vertical stirrup of model SR100-SF0-S pre-dicted numerically at peak load exceeded the yielding strain. Experimental results veri-fied yielding of SG-V1 and SG-V2 in the west shear span (Figure 11a) and SG-V2 in the east shear span (Figure 11b) prior to failure. The numerical results show that the vertical stirrup strains of model SR100-SF0-S increased at a higher rate than that of SR0-SF0-S, indicating less contribution of RCAs to the shear resistance. This behavior was less evident in the experimental results that showed no significant difference in the rate of increase in the stirrup strains of the two tested beams, SR0-SF0-S and SR100-SF0-S.

Figure 9. Crack patterns and principal strains of group S: (a) numerical crack patterns; (b) minimumprincipal strains; and (c) experimental crack patterns.

The minimum principal concrete strain of the deep beam models prior to failure areshown in Figure 9b. The profile of the minimum principal concrete strain indicated theformation of a bottle-shaped strut in the shear span of the deep beam models with steel

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stirrups and a prismatic strut for the models with steel fibers. Photographs of the testedbeams at failure are shown in Figure 9c. The crack patterns obtained from the tests, shownin Figure 9c, verified the formation of a band of shear cracks in the shear span of the deepbeams with steel stirrups and concentration of inclined shear cracks along the natural loadpath of the beams with steel fibers.

It should be noted that the deep beam models without steel fibers exhibited similarvalues of minimum principal concrete strain of approximately −0.006 to −0.008 prior tofailure. The models with steel stirrups failed, however, at a higher shear capacity than thatof their counterparts without stirrups, signifying the role of stirrups in reducing the rate ofincrease in the minimum principal concrete strain. The deep beam models with steel fibersexhibited minimum principal concrete strain values of approximately −0.006 to −0.01 atpeak load but failed at a higher load relative to that of their counterparts without stirrups.The increased shear capacity of the steel-fiber reinforced models signified the effectivenessof the steel fibers in reducing the rate of increase in the minimum principal concrete strains.

4.1.3. Stirrup Strains

The numerical stirrup strain responses of the model SR0-SF0-S are compared withthose measured experimentally in the west and east shear spans in Figure 10a,b, respec-tively. Readings of SG-V2 were not captured experimentally in the east shear span dueto malfunction of the SG. The locations of the stirrups monitoring points are shown inFigure 5. It should be noted that predicted strains of SG-V1 and SG-V2 were identical.Similarly, the rate of increase in the stirrup strain in the vertical stirrups recorded exper-imentally was insignificantly different, as shown in Figure 10a. Numerical data of themodel SR0-SF0-S indicated failure of the beam without yielding stirrups. This behaviorwas in alignment with the experimental data, except in one location (SG-V1) in the westshear span. Numerical results indicated that no strains were recorded in the stirrups in thepre-cracking stage. Following shear cracking, the stirrup strains increased almost linearlyuntil the model reached its shear capacity. This behavior was in agreement with the stirrupstrains measured in the west and east shear spans as shown in Figure 10a,b, respectively.

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(a) (b)

Figure 10. Numerical and experimental stirrup strain response of SR0-SF0-S: (a) west shear span; (b) east shear span.

(a) (b)


4.1.4. Tensile Steel Strains The numerical and experimental tensile steel strain responses of the solid RC deep

beam models are presented in Figure 12. The monitoring point of SG1 was located at a distance 125 mm away from the face of the support plate, whereas that of SG4 was under the load point. Other monitoring points, SG2 and SG3, were located in between SG1 and SG4 at a spacing of 200 mm (refer to Figure 5). The numerical and experimental steel strain responses showed a similar trend. The steel strain responses of the deep beam models SR0-SF0 and SR100-SF0 exhibited load decays due to development of shear cracks (Figure 12a,c). No load decays occurred in the deep beam models with steel stirrups (Figure 12b,d) or steel fibers (Figure 12e–g). The strains predicted numerically at SG1, which was close to the support, were lower than those recorded at other locations in all models. In agree-ment with the experimental results, the steel strains in other locations within the shear span were insignificantly different, verifying the arch action effect in all models. The ten-sile steel reinforcement did not reach the yielding strain in any of the models, which was verified experimentally.

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Figure 10. Numerical and experimental stirrup strain response of SR0-SF0-S: (a) west shear span;(b) east shear span.

The numerical stirrup strain responses of the model SR100-SF0-S are compared withthose measured experimentally in the west and east shear spans in Figure 11a,b, respectively.The vertical stirrups in the model SR100-SF0-S made with RCAs exhibited higher strainsthan those exhibited by the horizontal stirrups, indicating more contribution to the shearresistance. Similarly, the vertical stirrup strains obtained from the tests tended to behigher than those of the horizontal stirrups in the west and east shear spans as shown inFigure 11a,b, respectively. The strain in the vertical stirrup of model SR100-SF0-S predictednumerically at peak load exceeded the yielding strain. Experimental results verifiedyielding of SG-V1 and SG-V2 in the west shear span (Figure 11a) and SG-V2 in the eastshear span (Figure 11b) prior to failure. The numerical results show that the vertical stirrup

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strains of model SR100-SF0-S increased at a higher rate than that of SR0-SF0-S, indicatingless contribution of RCAs to the shear resistance. This behavior was less evident in theexperimental results that showed no significant difference in the rate of increase in thestirrup strains of the two tested beams, SR0-SF0-S and SR100-SF0-S.

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(a) (b)


(a) (b)


4.1.4. Tensile Steel Strains The numerical and experimental tensile steel strain responses of the solid RC deep

beam models are presented in Figure 12. The monitoring point of SG1 was located at a distance 125 mm away from the face of the support plate, whereas that of SG4 was under the load point. Other monitoring points, SG2 and SG3, were located in between SG1 and SG4 at a spacing of 200 mm (refer to Figure 5). The numerical and experimental steel strain responses showed a similar trend. The steel strain responses of the deep beam models SR0-SF0 and SR100-SF0 exhibited load decays due to development of shear cracks (Figure 12a,c). No load decays occurred in the deep beam models with steel stirrups (Figure 12b,d) or steel fibers (Figure 12e–g). The strains predicted numerically at SG1, which was close to the support, were lower than those recorded at other locations in all models. In agree-ment with the experimental results, the steel strains in other locations within the shear span were insignificantly different, verifying the arch action effect in all models. The ten-sile steel reinforcement did not reach the yielding strain in any of the models, which was verified experimentally.

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Figure 11. Numerical and experimental stirrup strain response of SR100-SF0-S: (a) west shear span;(b) east shear span.

4.1.4. Tensile Steel Strains

The numerical and experimental tensile steel strain responses of the solid RC deepbeam models are presented in Figure 12. The monitoring point of SG1 was located at adistance 125 mm away from the face of the support plate, whereas that of SG4 was underthe load point. Other monitoring points, SG2 and SG3, were located in between SG1 andSG4 at a spacing of 200 mm (refer to Figure 5). The numerical and experimental steel strainresponses showed a similar trend. The steel strain responses of the deep beam models SR0-SF0 and SR100-SF0 exhibited load decays due to development of shear cracks (Figure 12a,c).No load decays occurred in the deep beam models with steel stirrups (Figure 12b,d) orsteel fibers (Figure 12e–g). The strains predicted numerically at SG1, which was close tothe support, were lower than those recorded at other locations in all models. In agreementwith the experimental results, the steel strains in other locations within the shear spanwere insignificantly different, verifying the arch action effect in all models. The tensilesteel reinforcement did not reach the yielding strain in any of the models, which wasverified experimentally.

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(a) (b)

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Figure 12. Numerical and experimental steel strain responses of group S: (a) SR0-SF0; (b) SR0-SF0-S; (c) SR100-SF0; (d) SR100-SF0-S; (e) SR100-SF1; (f) SR100-SF2; and (g) SR100-SF3.

4.2. Deep Beam Models with Openings

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Figure 12. Cont.

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Buildings 2022, 12, 529 14 of 28

(a) (b)

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Figure 12. Numerical and experimental steel strain responses of group S: (a) SR0-SF0; (b) SR0-SF0-S; (c) SR100-SF0; (d) SR100-SF0-S; (e) SR100-SF1; (f) SR100-SF2; and (g) SR100-SF3.

4.2. Deep Beam Models with Openings

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Figure 12. Numerical and experimental steel strain responses of group S: (a) SR0-SF0; (b) SR0-SF0-S;(c) SR100-SF0; (d) SR100-SF0-S; (e) SR100-SF1; (f) SR100-SF2; and (g) SR100-SF3.

4.2. Deep Beam Models with Openings4.2.1. Shear Load–deflection Response

The predicted shear load–deflection curves of the deep beam models containingopenings are illustrated in Figure 13. The use of RCAs rather than NAs reduced thestiffness and shear capacity of the models, as shown in Figure 13a. The addition of steelfibers improved the stiffness and remarkably increased the shear capacity of the deepbeam models, as shown in Figure 13b. The beam models having steel fibers exhibited an

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improved stiffness even better than that of NR100-SF0-S with NAs and conventional steelstirrups. It is worth noting that when the steel fiber-reinforced concrete models reachedapproximately 90% of the shear capacity, they started to exhibit an increased midspandeflection without a significant increase in load until failure. This plastic response canbe attributed to the tension stiffening effect caused by steel fibers, which improved thedeformability of the models.

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4.2.1. Shear Load–deflection Response The predicted shear load–deflection curves of the deep beam models containing

openings are illustrated in Figure 13. The use of RCAs rather than NAs reduced the stiff-ness and shear capacity of the models, as shown in Figure 13a. The addition of steel fibers improved the stiffness and remarkably increased the shear capacity of the deep beam models, as shown in Figure 13b. The beam models having steel fibers exhibited an im-proved stiffness even better than that of NR100-SF0-S with NAs and conventional steel stirrups. It is worth noting that when the steel fiber-reinforced concrete models reached approximately 90% of the shear capacity, they started to exhibit an increased midspan deflection without a significant increase in load until failure. This plastic response can be attributed to the tension stiffening effect caused by steel fibers, which improved the de-formability of the models.

The shear load–deflection responses of the models without steel fibers predicted nu-merically are compared with those obtained experimentally in Figure 14a–d. The pre-dicted response of the deep beam models without steel fibers tended to be stiffer than that obtained from the experiment. This behavior may be attributed to the existence of mi-crocracks in the tested beams prior to testing caused by shrinkage or occurred during han-dling. Such microcracks would reduce the actual stiffness of the tested beams. The numer-ical shear load–deflection responses of the models with steel fibers are compared with those obtained from the tests in Figure 14e–g. The difference between the numerical and experimental stiffnesses of the deep beam models with steel fibers was less significant relative to that of the models without steel fibers.

The experimental and numerical shear capacities, VEXP and VFE, respectively, along with the deflections at failure, ΔEXP and ΔFE, are compared in Table 4. The predicted shear capacities were within the 11% error band, except for model NR100-SF3 with VFE/VEXP of 0.82. The ratio of VFE/VEXP was on average 0.92 with a corresponding standard deviation and coefficient of variation of 0.05% and 6%, respectively. The predicted deflections at failure for the models with steel fibers were generally in good agreement with those meas-ured experimentally with ΔFE/ΔEXP in the range of 0.81 to 0.98. The models tended, how-ever, to underestimate the deflection at the failure of the beams without steel fibers. It is worth noting that the measured deflections at the failure of the models without steel fibers were very small which made the ratio of ΔFE/ΔEXP very sensitive to any small difference (within 2 mm) between predicted and measured deflections. The ratio of ΔFE/ΔEXP of all models was on average 0.72 with a standard deviation of 0.18 and a coefficient of variation of 25%.

(a) (b)

Figure 13. Numerical shear load–deflection responses of group N: (a) beam models without steel fibers; (b) beam models with RCAs/steel fibers.

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Figure 13. Numerical shear load–deflection responses of group N: (a) beam models without steelfibers; (b) beam models with RCAs/steel fibers.

The shear load–deflection responses of the models without steel fibers predicted nu-merically are compared with those obtained experimentally in Figure 14a–d. The predictedresponse of the deep beam models without steel fibers tended to be stiffer than that ob-tained from the experiment. This behavior may be attributed to the existence of microcracksin the tested beams prior to testing caused by shrinkage or occurred during handling. Suchmicrocracks would reduce the actual stiffness of the tested beams. The numerical shearload–deflection responses of the models with steel fibers are compared with those obtainedfrom the tests in Figure 14e–g. The difference between the numerical and experimentalstiffnesses of the deep beam models with steel fibers was less significant relative to that ofthe models without steel fibers.

The experimental and numerical shear capacities, VEXP and VFE, respectively, alongwith the deflections at failure, ∆EXP and ∆FE, are compared in Table 4. The predicted shearcapacities were within the 11% error band, except for model NR100-SF3 with VFE/VEXP of0.82. The ratio of VFE/VEXP was on average 0.92 with a corresponding standard deviationand coefficient of variation of 0.05% and 6%, respectively. The predicted deflections atfailure for the models with steel fibers were generally in good agreement with thosemeasured experimentally with ∆FE/∆EXP in the range of 0.81 to 0.98. The models tended,however, to underestimate the deflection at the failure of the beams without steel fibers. It isworth noting that the measured deflections at the failure of the models without steel fiberswere very small which made the ratio of ∆FE/∆EXP very sensitive to any small difference(within 2 mm) between predicted and measured deflections. The ratio of ∆FE/∆EXP of allmodels was on average 0.72 with a standard deviation of 0.18 and a coefficient of variationof 25%.


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Figure 14. Numerical and experimental responses of group N: (a) NR0-SF0; (b) NR0-SF0-S; (c) NR100-SF0; (d) NR100-SF0-S; (e) NR100-SF1; (f) NR100-SF2; and (g) NR100-SF3.

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Figure 14. Numerical and experimental responses of group N: (a) NR0-SF0; (b) NR0-SF0-S; (c) NR100-SF0; (d) NR100-SF0-S; (e) NR100-SF1; (f) NR100-SF2; and (g) NR100-SF3.

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Table 4. Comparison between numerical and experimental results of group N.

ModelShear Capacity Deflection Capacity

VFE/VEXP∆FE/∆EXP

VEXP (kN) VFE (kN) ∆EXP (mm) ∆FE (mm)

NR0-SF0 273 273 4.7 2.9 1.00 0.62

NR0-SF0-S 341 327 4.8 3.0 0.96 0.63

NR100-SF0 237 211 4.9 2.6 0.89 0.53

NR100-SF0-S 279 255 5.3 2.7 0.91 0.51

NR100-SF1 329 306 5.6 5.4 0.93 0.96

NR100-SF2 362 326 6.0 5.9 0.90 0.98

NR100-SF3 435 358 7.4 6.0 0.82 0.81

Average 0.92 0.72

Std dev 0.05 0.18

COV (%) 5.74 25.26

4.2.2. Crack Patterns

The crack patterns of the deep beam models with openings predicted numericallyare shown in Figure 15a. All models initially exhibited a diagonal shear crack crossingthe center of the opening. This crack did not propagate further as the load progressed.At higher loads, inclined shear cracks developed in the upper and lower chords in thedirection of the corresponding load paths. The models without steel stirrups nor steel fibersfailed due to the formation of independent diagonal splitting cracks in the direction ofupper and lower load paths. The models with stirrups exhibited multiple parallel inclinedcracks along in the direction of the upper and lower natural load paths prior to failure.

The minimum principal concrete strain of the deep beam models prior to failure areshown in Figure 15b. The profile of the minimum principal concrete strains of the modelswith steel stirrups or steel fibers verified the formation of diagonal struts in the direction ofthe upper and lower load paths. The deep beam models without steel fibers exhibited anaverage minimum principal concrete strain value of approximately −0.008 at peak load.The models with steel stirrups failed, however, a higher shear capacity than that of theircounterparts without stirrups, indicating a lower rate of increase in the minimum principalconcrete strains relative to that of their counterparts without stirrups. The deep beammodels with steel fibers at vf of 1%, 2%, and 3% reached their shear capacity at minimumprincipal strain values of approximately −0.014, −0.018, and −0.020, respectively. Theincreased minimum principal strain capacities exhibited by the models with steel fibersenabled the models to sustain higher load and deflection capacities prior to failure thanthose of their counterparts without steel fibers.

Figure 15c shows photographs of the tested beams with openings at failure. The crackpatterns obtained from the tests (Figure 15c) are in good agreement with those predictednumerically (Figure 15a). Experimental results verified the frame-type mode of failure inthe deep beams without shear reinforcement and the shear compression mode of failurealong the upper and lower load paths in the deep beams with stirrups or steel fibers.

Buildings 2022, 12, 529 17 of 28

Table 4. Comparison between numerical and experimental results of group N.

Model Shear Capacity Deflection Capacity

VFE/VEXP ΔFE/ΔEXP

VEXP (kN) VFE (kN) ΔEXP (mm) ΔFE (mm) NR0-SF0 273 273 4.7 2.9 1.00 0.62

NR0-SF0-S 341 327 4.8 3.0 0.96 0.63 NR100-SF0 237 211 4.9 2.6 0.89 0.53

NR100-SF0-S 279 255 5.3 2.7 0.91 0.51 NR100-SF1 329 306 5.6 5.4 0.93 0.96 NR100-SF2 362 326 6.0 5.9 0.90 0.98 NR100-SF3 435 358 7.4 6.0 0.82 0.81

Average 0.92 0.72 Std dev 0.05 0.18

COV (%) 5.74 25.26

4.2.2. Crack Patterns The crack patterns of the deep beam models with openings predicted numerically

are shown in Figure 15a. All models initially exhibited a diagonal shear crack crossing the center of the opening. This crack did not propagate further as the load progressed. At higher loads, inclined shear cracks developed in the upper and lower chords in the direc-tion of the corresponding load paths. The models without steel stirrups nor steel fibers failed due to the formation of independent diagonal splitting cracks in the direction of upper and lower load paths. The models with stirrups exhibited multiple parallel inclined cracks along in the direction of the upper and lower natural load paths prior to failure.

The minimum principal concrete strain of the deep beam models prior to failure are shown in Figure 15b. The profile of the minimum principal concrete strains of the models with steel stirrups or steel fibers verified the formation of diagonal struts in the direction of the upper and lower load paths. The deep beam models without steel fibers exhibited an average minimum principal concrete strain value of approximately -0.008 at peak load. The models with steel stirrups failed, however, a higher shear capacity than that of their counterparts without stirrups, indicating a lower rate of increase in the minimum princi-pal concrete strains relative to that of their counterparts without stirrups. The deep beam models with steel fibers at vf of 1%, 2%, and 3% reached their shear capacity at minimum principal strain values of approximately -0.014, -0.018, and -0.020, respectively. The in-creased minimum principal strain capacities exhibited by the models with steel fibers en-abled the models to sustain higher load and deflection capacities prior to failure than those of their counterparts without steel fibers.

Figure 15c shows photographs of the tested beams with openings at failure. The crack patterns obtained from the tests (Figure 15c) are in good agreement with those predicted numerically (Figure 15a). Experimental results verified the frame-type mode of failure in the deep beams without shear reinforcement and the shear compression mode of failure along the upper and lower load paths in the deep beams with stirrups or steel fibers.

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Figure 15. Crack patterns and principal strains: (a) numerical crack patterns; (b) minimum principal strains; and (c) experimental crack patterns.

4.2.3. Stirrup Strains The stirrup strain responses predicated numerically of the deep beam models NR0-

SF0-S and NR100-SF0-S are presented in Figure 16a,b respectively. The locations of the monitoring points are shown in Figure 6. All predicted stirrup strains were below the yielding value indicating that failure was controlled by concrete due to the presence of the openings. Although experimental stirrup strain data at locations of most of the monitoring points were not available, the few stirrup strain data obtained from the experimental tests showed no yielding of stirrups [28]. The full-depth vertical stirrups located at the left and right sides of the opening in NR0-SF0-S and NR100-SF0-S exhibited minimal strains as shown in Figures 16a,b, respectively. These reduced strains occurred because the major shear cracks were developed in the upper and lower chords rather than at the sides of the opening. Upper and lower chord stirrup strains indicate that shear cracks occurred in NR0-SF0-S at a load value of approximately 170 kN (Figure 16a). Following cracking, there was no significant difference in the rate of increase in the horizontal and vertical stirrup strains in the upper and lower chords. The strains in the vertical and horizontal stirrups at peak load were insignificantly different. The stirrups in the lower chord tended, how-ever, to exhibit slightly higher strains at peak load than those exhibited by the stirrups in the upper chord. Upper and lower chord stirrup strains of model NR100-SF0-S, shown in Figure 16b, indicate the initiation of shear cracks at a load value of approximately 130 kN. Figure 16c shows the horizontal and vertical stirrup strain responses in the upper chord for models NR0-SF0-S and NR100-SF0-S. It can be seen that the deep beam model NR100-SF0-S made with RCAs exhibited shear cracks at a lower load than that of NR0-SF0-S made with NAs. Following shear cracking, the stirrup strains of NR100-SF0-S increased at a

Figure 15. Crack patterns and principal strains: (a) numerical crack patterns; (b) minimum principalstrains; and (c) experimental crack patterns.

4.2.3. Stirrup Strains

The stirrup strain responses predicated numerically of the deep beam models NR0-SF0-S and NR100-SF0-S are presented in Figure 16a,b respectively. The locations of themonitoring points are shown in Figure 6. All predicted stirrup strains were below theyielding value indicating that failure was controlled by concrete due to the presence of theopenings. Although experimental stirrup strain data at locations of most of the monitoringpoints were not available, the few stirrup strain data obtained from the experimental testsshowed no yielding of stirrups [28]. The full-depth vertical stirrups located at the left andright sides of the opening in NR0-SF0-S and NR100-SF0-S exhibited minimal strains asshown in Figure 16a,b, respectively. These reduced strains occurred because the majorshear cracks were developed in the upper and lower chords rather than at the sides ofthe opening. Upper and lower chord stirrup strains indicate that shear cracks occurred inNR0-SF0-S at a load value of approximately 170 kN (Figure 16a). Following cracking, therewas no significant difference in the rate of increase in the horizontal and vertical stirrupstrains in the upper and lower chords. The strains in the vertical and horizontal stirrups atpeak load were insignificantly different. The stirrups in the lower chord tended, however, toexhibit slightly higher strains at peak load than those exhibited by the stirrups in the upperchord. Upper and lower chord stirrup strains of model NR100-SF0-S, shown in Figure 16b,indicate the initiation of shear cracks at a load value of approximately 130 kN. Figure 16cshows the horizontal and vertical stirrup strain responses in the upper chord for modelsNR0-SF0-S and NR100-SF0-S. It can be seen that the deep beam model NR100-SF0-S madewith RCAs exhibited shear cracks at a lower load than that of NR0-SF0-S made with NAs.Following shear cracking, the stirrup strains of NR100-SF0-S increased at a slightly higherrate than that of model NR0-SF0-S. The reduced shear cracking load and the increasedstirrup strains of model NR100-SF0-S compared with those of model NR0-SF0-S indicatedless contribution of RCAs to the shear resistance than NAs.

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Buildings 2022, 12, 529 19 of 28

slightly higher rate than that of model NR0-SF0-S. The reduced shear cracking load and the increased stirrup strains of model NR100-SF0-S compared with those of model NR0-SF0-S indicated less contribution of RCAs to the shear resistance than NAs.

(a) (b) (c)

Figure 16. Numerical stirrup strain response of group N: (a) NR0-SF0-S; (b) NR100-SF0-S; and (c) horizontal versus vertical stirrup strain responses in the upper chord.

4.2.4. Tensile Steel Strains The numerical and experimental tensile steel strain responses of the deep beam mod-

els containing openings without steel fibers are presented in Figure 17a–d, whereas those with steel fibers are shown in Figure 17e–g. The monitoring point of SG1 was located at a distance 85 mm away from the face of the support plate, whereas that of SG4 was under the load point. Other monitoring points, SG2 and SG3, were located in between SG1 and SG4 at a spacing of 80 mm (refer to Figure 6). The numerical and experimental steel strain responses showed a similar trend. The steel strains predicted numerically in all models at different locations within the shear span were insignificantly different, verifying the deep beam action. This behavior was verified experimentally despite being less evident in the steel strain data measured experimentally in beam NR100-SF0 (Figure 17c). The strains recorded numerically at SG1, which was close to the support, were slightly lower than those recorded at other locations. The tensile steel reinforcement did not reach the yielding strain in any of the models, which has been verified experimentally, except at one location (SG2) in beam NR100-SF3 (Figure 17g).

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Figure 16. Numerical stirrup strain response of group N: (a) NR0-SF0-S; (b) NR100-SF0-S; and(c) horizontal versus vertical stirrup strain responses in the upper chord.

4.2.4. Tensile Steel Strains

The numerical and experimental tensile steel strain responses of the deep beam modelscontaining openings without steel fibers are presented in Figure 17a–d, whereas those withsteel fibers are shown in Figure 17e–g. The monitoring point of SG1 was located at adistance 85 mm away from the face of the support plate, whereas that of SG4 was underthe load point. Other monitoring points, SG2 and SG3, were located in between SG1 andSG4 at a spacing of 80 mm (refer to Figure 6). The numerical and experimental steel strainresponses showed a similar trend. The steel strains predicted numerically in all models atdifferent locations within the shear span were insignificantly different, verifying the deepbeam action. This behavior was verified experimentally despite being less evident in thesteel strain data measured experimentally in beam NR100-SF0 (Figure 17c). The strainsrecorded numerically at SG1, which was close to the support, were slightly lower thanthose recorded at other locations. The tensile steel reinforcement did not reach the yieldingstrain in any of the models, which has been verified experimentally, except at one location(SG2) in beam NR100-SF3 (Figure 17g).

Buildings 2022, 12, 529 20 of 28

(a) (b)

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Figure 17. Numerical and experimental tensile steel strain responses: (a) NR0-SF0; (b) NR0-SF0-S; (c) NR100-SF0; (d) NR100-SF0-S; (e) NR100-SF1; (f) NR100-SF2; and (g) NR100-SF3.

5. Parametric Study and Discussion It was not possible to test solid RC deep beams with a/h of 0.8 due to the limited

capacity of the laboratory facility. The models developed and verified in the current study

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Figure 17. Cont.

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Buildings 2022, 12, 529 20 of 28

(a) (b)

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Figure 17. Numerical and experimental tensile steel strain responses: (a) NR0-SF0; (b) NR0-SF0-S; (c) NR100-SF0; (d) NR100-SF0-S; (e) NR100-SF1; (f) NR100-SF2; and (g) NR100-SF3.

5. Parametric Study and Discussion It was not possible to test solid RC deep beams with a/h of 0.8 due to the limited

capacity of the laboratory facility. The models developed and verified in the current study

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Figure 17. Numerical and experimental tensile steel strain responses: (a) NR0-SF0; (b) NR0-SF0-S;(c) NR100-SF0; (d) NR100-SF0-S; (e) NR100-SF1; (f) NR100-SF2; and (g) NR100-SF3.

5. Parametric Study and Discussion

It was not possible to test solid RC deep beams with a/h of 0.8 due to the limitedcapacity of the laboratory facility. The models developed and verified in the current studyare considered a valid alternative to laboratory testing. As such, an additional sevensolid deep models with a/h of 0.8 were developed. The performance of these models waspredicted numerically and compared with that of other models presented earlier with andwithout openings.

5.1. Effect of Shear Span-to-Depth Ratio

Figure 18 presents the shear load–deflection responses of RCA-based solid deep beammodels with different a/h ratios. The models with a/h of 0.8 (Figure 18a) were significantlystiffer than their counterparts with a/h of 1.6 (Figure 18b). Figure 18a shows that at a/h of 0.8,the use of steel fibers with vf of 1% (model R100-SF1) was sufficient to restore the stiffness ofthe model with RCAs and minimum steel stirrups (model R100-SF0-S). However, at a/h of1.6, the use of steel fibers at vf of 2% (model SR100-SF2) was necessary to restore the stiffnessof the counterpart model with RCAs and minimum steel stirrups (model SR100-SF0-S), asshown in Figure 18b. At both a/h values, the stiffness of the models with vf of 3% coincidedwith that of the model with NAs and conventional steel stirrups. Figure 19 shows the effectof a/h on the shear capacity of the RCA-based deep beam models. The shear capacity of thedeep beam models with steel fibers increased almost linearly with an increase in the steelfiber volume fraction. Obviously, the shear capacity of the deep beam models with a/h of0.8 was higher than that of their counterparts with a/h of 1.6. For the deep beam modelswithout steel fibers, the increase in the shear capacity due to the decrease in the a/h from 1.6to 0.8 ranged from 31 to 66% with an average of 51%, whereas for those with steel fibers,the increase in shear capacity ranged from 42 to 88% with an average of 67%.

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Buildings 2022, 12, 529 21 of 28

to restore the stiffness of the counterpart model with RCAs and minimum steel stirrups (model SR100-SF0-S), as shown in Figure 18b. At both a/h values, the stiffness of the mod-els with vf of 3% coincided with that of the model with NAs and conventional steel stir-rups. Figure 19 shows the effect of a/h on the shear capacity of the RCA-based deep beam models. The shear capacity of the deep beam models with steel fibers increased almost linearly with an increase in the steel fiber volume fraction. Obviously, the shear capacity of the deep beam models with a/h of 0.8 was higher than that of their counterparts with a/h of 1.6. For the deep beam models without steel fibers, the increase in the shear capacity due to the decrease in the a/h from 1.6 to 0.8 ranged from 31 to 66% with an average of 51%, whereas for those with steel fibers, the increase in shear capacity ranged from 42 to 88% with an average of 67%.

(a) (b)

Figure 18. Shear load–deflection responses of solid deep beam models: (a) a/h = 0.8; (b) a/h = 1.6.

Figure 19. Effect of a/h on the shear capacity of RCA-based models.

Figure 20 illustrates the effect of the inclusion of minimum steel stirrups on the shear strength gain of the beam models with different types of aggregates at different a/h ratios. The shear strength gains exhibited by the models with NAs and those of their counterparts with RCAs were insignificantly different, regardless of a/h. Figure 21 shows the effect of steel fibers on the shear strength gain of the deep beam models with RCAs at different a/h ratios. The strength gains caused by the steel stirrups are also included in Figure 21 to serve as a benchmark. The shear strength gain was calculated relative to the shear capacity of the corresponding control model made with RCAs without steel stirrups nor steel fi-bers. At a/h of 1.6, the addition of steel fibers at vf of 1%, 2%, and 3% resulted in respective shear strength gains of 39%, 64%, and 114%. For the models with a/h of 0.8, the addition of steel fibers at vf of 1%, 2%, and 3% resulted in shear strength gains of 78%, 91%, and 108%, respectively. At vf of 1 and 2%, the shear strength gain was higher for the models

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Buildings 2022, 12, 529 21 of 28

to restore the stiffness of the counterpart model with RCAs and minimum steel stirrups (model SR100-SF0-S), as shown in Figure 18b. At both a/h values, the stiffness of the mod-els with vf of 3% coincided with that of the model with NAs and conventional steel stir-rups. Figure 19 shows the effect of a/h on the shear capacity of the RCA-based deep beam models. The shear capacity of the deep beam models with steel fibers increased almost linearly with an increase in the steel fiber volume fraction. Obviously, the shear capacity of the deep beam models with a/h of 0.8 was higher than that of their counterparts with a/h of 1.6. For the deep beam models without steel fibers, the increase in the shear capacity due to the decrease in the a/h from 1.6 to 0.8 ranged from 31 to 66% with an average of 51%, whereas for those with steel fibers, the increase in shear capacity ranged from 42 to 88% with an average of 67%.

(a) (b)



Figure 20 illustrates the effect of the inclusion of minimum steel stirrups on the shear strength gain of the beam models with different types of aggregates at different a/h ratios. The shear strength gains exhibited by the models with NAs and those of their counterparts with RCAs were insignificantly different, regardless of a/h. Figure 21 shows the effect of steel fibers on the shear strength gain of the deep beam models with RCAs at different a/h ratios. The strength gains caused by the steel stirrups are also included in Figure 21 to serve as a benchmark. The shear strength gain was calculated relative to the shear capacity of the corresponding control model made with RCAs without steel stirrups nor steel fi-bers. At a/h of 1.6, the addition of steel fibers at vf of 1%, 2%, and 3% resulted in respective shear strength gains of 39%, 64%, and 114%. For the models with a/h of 0.8, the addition of steel fibers at vf of 1%, 2%, and 3% resulted in shear strength gains of 78%, 91%, and 108%, respectively. At vf of 1 and 2%, the shear strength gain was higher for the models

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vf = 2% vf = 3%vf = 1%


Figure 20 illustrates the effect of the inclusion of minimum steel stirrups on the shearstrength gain of the beam models with different types of aggregates at different a/h ratios.The shear strength gains exhibited by the models with NAs and those of their counterpartswith RCAs were insignificantly different, regardless of a/h. Figure 21 shows the effect ofsteel fibers on the shear strength gain of the deep beam models with RCAs at different a/hratios. The strength gains caused by the steel stirrups are also included in Figure 21 to serveas a benchmark. The shear strength gain was calculated relative to the shear capacity ofthe corresponding control model made with RCAs without steel stirrups nor steel fibers.At a/h of 1.6, the addition of steel fibers at vf of 1%, 2%, and 3% resulted in respectiveshear strength gains of 39%, 64%, and 114%. For the models with a/h of 0.8, the additionof steel fibers at vf of 1%, 2%, and 3% resulted in shear strength gains of 78%, 91%, and108%, respectively. At vf of 1 and 2%, the shear strength gain was higher for the modelswith the lower a/h of 0.8. The improved effectiveness of steel fibers in increasing the shearcapacity at the lower a/h has also been reported in previous studies for concrete beamsmade with NAs. Ashour et al. [23] reported shear strength gains of 97% and 32% at shearspan-to-effective depth (a/d) ratios of 1 and 6, respectively, due to the addition of steel fibersat vf of 1.5%. Kwak et al. [24] indicated that the shear strength gain caused by the additionof steel fibers at vf of 0.75% was in the range of 69 to 80% at a/d =2, whereas for larger a/d of3 and 4, a reduced shear strength gain of 22 to 38% was reported.

Buildings 2022, 12, 529 22 of 27

Buildings 2022, 12, 529 22 of 28

with the lower a/h of 0.8. The improved effectiveness of steel fibers in increasing the shear capacity at the lower a/h has also been reported in previous studies for concrete beams made with NAs. Ashour et al. [23] reported shear strength gains of 97% and 32% at shear span-to-effective depth (a/d) ratios of 1 and 6, respectively, due to the addition of steel fibers at vf of 1.5%. Kwak et al. [24] indicated that the shear strength gain caused by the addition of steel fibers at vf of 0.75% was in the range of 69 to 80% at a/d =2, whereas for larger a/d of 3 and 4, a reduced shear strength gain of 22 to 38% was reported.

The presence of steel fibers reduced the severity of cracks within the shear span, im-proved the tension stiffening effect, and hence, increased the strut capacity. The shear ca-pacity (V) of RC deep beams is sensitive to the strut capacity (Fs) and its angle of inclina-tion in the shear span (θ) through the relationship V = Fssinθ. The impact of increasing the strut capacity on the shear strength is more pronounced for struts with a greater angle of inclination θ. This relationship could explain why the shear strength gain caused by steel fibers was higher at the lower a/h of 0.8. The effect of a/h on the shear strength gain caused by steel fibers diminished at the higher vf of 3%. It seems that, at vf of 3%, the increase in the shear capacity was too high to show an effect for the strut angle of inclination. From Figure 21, it can be seen that at a/h of 0.8, the use of vf of 1% was sufficient to provide a shear strength gain higher than that provided by the minimum steel stirrups. At a/h of 1.6, it was necessary to use vf of 2% to substitute the minimum steel stirrups.

Figure 20. Shear strength gain of the models with minimum stirrups at different a/h ratios.

Figure 21. Shear strength gain of the models with RCAs and steel fibers at different a/h ratios.

5.2. Effect of Web Openings

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Buildings 2022, 12, 529 22 of 28

with the lower a/h of 0.8. The improved effectiveness of steel fibers in increasing the shear capacity at the lower a/h has also been reported in previous studies for concrete beams made with NAs. Ashour et al. [23] reported shear strength gains of 97% and 32% at shear span-to-effective depth (a/d) ratios of 1 and 6, respectively, due to the addition of steel fibers at vf of 1.5%. Kwak et al. [24] indicated that the shear strength gain caused by the addition of steel fibers at vf of 0.75% was in the range of 69 to 80% at a/d =2, whereas for larger a/d of 3 and 4, a reduced shear strength gain of 22 to 38% was reported.

The presence of steel fibers reduced the severity of cracks within the shear span, im-proved the tension stiffening effect, and hence, increased the strut capacity. The shear ca-pacity (V) of RC deep beams is sensitive to the strut capacity (Fs) and its angle of inclina-tion in the shear span (θ) through the relationship V = Fssinθ. The impact of increasing the strut capacity on the shear strength is more pronounced for struts with a greater angle of inclination θ. This relationship could explain why the shear strength gain caused by steel fibers was higher at the lower a/h of 0.8. The effect of a/h on the shear strength gain caused by steel fibers diminished at the higher vf of 3%. It seems that, at vf of 3%, the increase in the shear capacity was too high to show an effect for the strut angle of inclination. From Figure 21, it can be seen that at a/h of 0.8, the use of vf of 1% was sufficient to provide a shear strength gain higher than that provided by the minimum steel stirrups. At a/h of 1.6, it was necessary to use vf of 2% to substitute the minimum steel stirrups.




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The presence of steel fibers reduced the severity of cracks within the shear span,improved the tension stiffening effect, and hence, increased the strut capacity. The shearcapacity (V) of RC deep beams is sensitive to the strut capacity (Fs) and its angle of inclina-tion in the shear span (θ) through the relationship V = Fssinθ. The impact of increasing thestrut capacity on the shear strength is more pronounced for struts with a greater angle ofinclination θ. This relationship could explain why the shear strength gain caused by steelfibers was higher at the lower a/h of 0.8. The effect of a/h on the shear strength gain causedby steel fibers diminished at the higher vf of 3%. It seems that, at vf of 3%, the increase inthe shear capacity was too high to show an effect for the strut angle of inclination. FromFigure 21, it can be seen that at a/h of 0.8, the use of vf of 1% was sufficient to provide ashear strength gain higher than that provided by the minimum steel stirrups. At a/h of 1.6,it was necessary to use vf of 2% to substitute the minimum steel stirrups.


Figure 22a,b present the shear load–deflection responses of the deep beam modelshaving a/h of 0.8 without and with web openings, respectively. The deep beam models withthe web opening (Figure 22b) exhibited a reduced stiffness and significant reduction in theshear capacity relative to those of their counterparts without opening (Figure 22a). At botha/h values, the inclusion of steel fibers at vf of 1% was sufficient to restore the stiffness of thecorresponding counterpart model with RCAs and minimum steel stirrups. This behavior ismanifested in the responses of counterpart models R100-SF1 and R100-SF0-S in Figure 22aand those of models SR100-SF1 and SR100-SF0-S in Figure 22b.

Buildings 2022, 12, 529 23 of 27

Buildings 2022, 12, 529 23 of 28

Figure 22a,b present the shear load–deflection responses of the deep beam models having a/h of 0.8 without and with web openings, respectively. The deep beam models with the web opening (Figure 22b) exhibited a reduced stiffness and significant reduction in the shear capacity relative to those of their counterparts without opening (Figure 22a). At both a/h values, the inclusion of steel fibers at vf of 1% was sufficient to restore the stiffness of the corresponding counterpart model with RCAs and minimum steel stirrups. This behavior is manifested in the responses of counterpart models R100-SF1 and R100-SF0-S in Figure 22a and those of models SR100-SF1 and SR100-SF0-S in Figure 22b.

Figure 23 shows the effect of the web opening on the shear capacity of the RCA-based deep models at different steel fiber volume fractions. The shear capacity of the models with and without openings increased almost linearly with an increase in vf. For the models without steel fibers, the reduction in shear capacity due to the presence of the web opening ranged from 17 to 28%, with an average of 23%. For the models with steel fibers, the re-duction in shear capacity caused by the web opening ranged from 35 to 36%, with an average of 35%. It is worth mentioning that the shear capacity of the deep beam model having vf of 3% and the web opening was equal to that of the solid beam model with steel stirrups. Numerical results of the deep beam models with a web opening are consistent with those published in the literature for RC deep beams with openings made with NAs [25,26]. Al Sarraf et al. [25] reported a shear strength reduction of 33% due to the presence of a square opening with ho/h of 0.2 in RC deep beams (a/h = 0.7) without steel fibers, whereas their counterparts with vf of 1% exhibited a shear strength reduction of 40% due to the web opening. Zewair et al. [26] reported that the existence of a circular web opening (ho/h of 0.31) reduced the shear capacity of RC deep beams (a/h = 1) without steel fibers by 27%, whereas it caused a shear strength reduction of 43 to 58% in their counterparts with vf of 1%.

(a) (b)

Figure 22. Shear load–deflection responses: (a) solid models; (b) models with web openings.

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Figure 23 shows the effect of the web opening on the shear capacity of the RCA-baseddeep models at different steel fiber volume fractions. The shear capacity of the modelswith and without openings increased almost linearly with an increase in vf. For the modelswithout steel fibers, the reduction in shear capacity due to the presence of the web openingranged from 17 to 28%, with an average of 23%. For the models with steel fibers, thereduction in shear capacity caused by the web opening ranged from 35 to 36%, with anaverage of 35%. It is worth mentioning that the shear capacity of the deep beam modelhaving vf of 3% and the web opening was equal to that of the solid beam model with steelstirrups. Numerical results of the deep beam models with a web opening are consistent withthose published in the literature for RC deep beams with openings made with NAs [25,26].Al Sarraf et al. [25] reported a shear strength reduction of 33% due to the presence of asquare opening with ho/h of 0.2 in RC deep beams (a/h = 0.7) without steel fibers, whereastheir counterparts with vf of 1% exhibited a shear strength reduction of 40% due to the webopening. Zewair et al. [26] reported that the existence of a circular web opening (ho/h of 0.31)reduced the shear capacity of RC deep beams (a/h = 1) without steel fibers by 27%, whereasit caused a shear strength reduction of 43 to 58% in their counterparts with vf of 1%.

Buildings 2022, 12, 529 23 of 28

Figure 22a,b present the shear load–deflection responses of the deep beam models having a/h of 0.8 without and with web openings, respectively. The deep beam models with the web opening (Figure 22b) exhibited a reduced stiffness and significant reduction in the shear capacity relative to those of their counterparts without opening (Figure 22a). At both a/h values, the inclusion of steel fibers at vf of 1% was sufficient to restore the stiffness of the corresponding counterpart model with RCAs and minimum steel stirrups. This behavior is manifested in the responses of counterpart models R100-SF1 and R100-SF0-S in Figure 22a and those of models SR100-SF1 and SR100-SF0-S in Figure 22b.

Figure 23 shows the effect of the web opening on the shear capacity of the RCA-based deep models at different steel fiber volume fractions. The shear capacity of the models with and without openings increased almost linearly with an increase in vf. For the models without steel fibers, the reduction in shear capacity due to the presence of the web opening ranged from 17 to 28%, with an average of 23%. For the models with steel fibers, the re-duction in shear capacity caused by the web opening ranged from 35 to 36%, with an average of 35%. It is worth mentioning that the shear capacity of the deep beam model having vf of 3% and the web opening was equal to that of the solid beam model with steel stirrups. Numerical results of the deep beam models with a web opening are consistent with those published in the literature for RC deep beams with openings made with NAs [25,26]. Al Sarraf et al. [25] reported a shear strength reduction of 33% due to the presence of a square opening with ho/h of 0.2 in RC deep beams (a/h = 0.7) without steel fibers, whereas their counterparts with vf of 1% exhibited a shear strength reduction of 40% due to the web opening. Zewair et al. [26] reported that the existence of a circular web opening (ho/h of 0.31) reduced the shear capacity of RC deep beams (a/h = 1) without steel fibers by 27%, whereas it caused a shear strength reduction of 43 to 58% in their counterparts with vf of 1%.

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Figure 23. Effect of web opening on the shear capacity of RCA-based models.

Figure 24 illustrates the effect of the minimum steel stirrups on the shear strength gainof the deep beam models having a/h of 0.8 with and without openings. The inclusion ofminimum steel stirrups was less effective in improving the shear capacity of the modelswith web openings, irrespective of the type of aggregates. This occurred because of thefailure mode of these models that was controlled by properties of concrete since no yieldingof stirrups took place. Figure 25 shows the shear strength gain caused by the addition ofsteel fibers to RCA deep beams having a/h of 0.8 with and without web openings. Thestrength gains caused by the steel stirrups are also provided in Figure 25 as a benchmark.The strength gain was calculated relative to the shear capacity of the corresponding control

Buildings 2022, 12, 529 24 of 27

models made with RCAs without steel stirrups nor steel fibers. The shear strength gaincaused by steel fibers for the deep beam models with the web opening was in the rangeof 45 to 70%, whereas their solid counterparts exhibited a shear strength gain of 78 to108%. Nevertheless, for the models with and without openings, the addition of steel fibersat vf of 1% was sufficient to achieve a shear strength gain higher than that provided bythe minimum conventional shear reinforcement. The reduced shear strength gain in thepresence of a web opening has also been reported in the literature for deep beams madewith NAs. Zewair et al. [26] reported a shear strength gain due to the addition of steel fibersin the range of 60 to 110% for solid RC deep beams with a/d of 1, whereas their counterpartswith a web opening showed a reduced shear strength gain in the range of 20 to 29%.

Buildings 2022, 12, 529 24 of 28


Figure 24 illustrates the effect of the minimum steel stirrups on the shear strength gain of the deep beam models having a/h of 0.8 with and without openings. The inclusion of minimum steel stirrups was less effective in improving the shear capacity of the models with web openings, irrespective of the type of aggregates. This occurred because of the failure mode of these models that was controlled by properties of concrete since no yield-ing of stirrups took place. Figure 25 shows the shear strength gain caused by the addition of steel fibers to RCA deep beams having a/h of 0.8 with and without web openings. The strength gains caused by the steel stirrups are also provided in Figure 25 as a benchmark. The strength gain was calculated relative to the shear capacity of the corresponding con-trol models made with RCAs without steel stirrups nor steel fibers. The shear strength gain caused by steel fibers for the deep beam models with the web opening was in the range of 45 to 70%, whereas their solid counterparts exhibited a shear strength gain of 78 to 108%. Nevertheless, for the models with and without openings, the addition of steel fibers at vf of 1% was sufficient to achieve a shear strength gain higher than that provided by the minimum conventional shear reinforcement. The reduced shear strength gain in the presence of a web opening has also been reported in the literature for deep beams made with NAs. Zewair et al. [26] reported a shear strength gain due to the addition of steel fibers in the range of 60 to 110% for solid RC deep beams with a/d of 1, whereas their counterparts with a web opening showed a reduced shear strength gain in the range of 20 to 29%.

Figure 24. Shear strength gain of models with minimum stirrups with and without openings.

Figure 25. Shear strength gain of RCA steel fiber-reinforced models with and without openings.

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Buildings 2022, 12, 529 24 of 28


Figure 24 illustrates the effect of the minimum steel stirrups on the shear strength gain of the deep beam models having a/h of 0.8 with and without openings. The inclusion of minimum steel stirrups was less effective in improving the shear capacity of the models with web openings, irrespective of the type of aggregates. This occurred because of the failure mode of these models that was controlled by properties of concrete since no yield-ing of stirrups took place. Figure 25 shows the shear strength gain caused by the addition of steel fibers to RCA deep beams having a/h of 0.8 with and without web openings. The strength gains caused by the steel stirrups are also provided in Figure 25 as a benchmark. The strength gain was calculated relative to the shear capacity of the corresponding con-trol models made with RCAs without steel stirrups nor steel fibers. The shear strength gain caused by steel fibers for the deep beam models with the web opening was in the range of 45 to 70%, whereas their solid counterparts exhibited a shear strength gain of 78 to 108%. Nevertheless, for the models with and without openings, the addition of steel fibers at vf of 1% was sufficient to achieve a shear strength gain higher than that provided by the minimum conventional shear reinforcement. The reduced shear strength gain in the presence of a web opening has also been reported in the literature for deep beams made with NAs. Zewair et al. [26] reported a shear strength gain due to the addition of steel fibers in the range of 60 to 110% for solid RC deep beams with a/d of 1, whereas their counterparts with a web opening showed a reduced shear strength gain in the range of 20 to 29%.



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6. Conclusions

Tensile softening laws of concrete made with RCAs and steel fibers were establishedbased on an inverse analysis of flexural test data of concrete prisms. Numerical deep beammodels were developed and validated through a comparative analysis with publishedexperimental results. A parametric study was conducted to investigate the effect of a/h andpresence of a web opening on the shear response of RC deep beam models at different steelfiber volume fractions. Main conclusions of the work are summarized hereafter.

The tensile softening law of the concrete made with RCAs and steel fibers can beapproximated by a bilinear relationship. The modulus of rupture of the mixture with vfof 1% was 71% higher than that of a benchmark mixture without fibers. The addition ofsteel fibers at vf ≥ 2% resulted in more than a two-fold increase in the modulus of rupture.The tension function of the concrete with vf of 2 and 3% exhibited a slightly reduced rate ofdegradation in the tensile stress after cracking relative to that of the concrete with vf of 1%.

The numerical deep beam models developed in the current study predicted the shearbehavior of previously tested RC deep beams with good accuracy. The numerical results of

Buildings 2022, 12, 529 25 of 27

the solid RC deep beam models indicated that the shear capacity increased almost linearlywith an increase in the steel fiber volume fraction. At a/h of 1.6, the addition of steel fibers atvf of 1%, 2%, and 3% increased the shear capacity by 39%, 64%, and 114%, respectively. Ata/h of 0.8, the respective shear strength gains were 78%, 91%, and 108%. At a/h of 0.8, a steelfiber volume fraction of vf = 1% was sufficient to substitute the minimum steel stirrups,whereas at a/h of 1.6, it was necessary to use vf of 2% to substitute the steel stirrups.

The addition of steel fibers was more effective in improving the shear capacity of thesolid deep beam models rather than those with the web opening at the same a/h of 0.8. Theshear strength gain caused by the addition of steel fibers for the deep beam models withthe web opening was in the range of 45 to 70%, whereas their solid counterparts exhibiteda shear strength of 78 to 108%.

For the deep beam models without steel fibers, the reduction in shear capacity due tothe presence of the web opening ranged from 17 to 28%, with an average of 23%. For thedeep beam models with steel fibers, a more pronounced reduction in the shear capacity ofapproximately 35% was reported because of the web opening.

The addition of steel fibers to the deep beam models with the web opening at vf of 1%was sufficient to achieve a shear strength gain higher than that provided by the minimumsteel stirrups. The shear capacity of the deep beam model with the web opening and vf of3% was equal to that of a similar solid deep beam model with steel stirrups.

This research provided insight into the shear behavior of steel-fiber reinforced recycledaggregates concrete deep beams with and without web openings. Steel fibers are vulnerableto corrosion under harsh environmental conditions. In such cases, appropriate measuresshould be adopted to protect the steel fibers from corrosion. Future work should focus onstudying the behavior of conventional and geopolymer concrete beams reinforced withnonmetallic fibers.

Author Contributions: Conceptualization, T.E.-M., N.K. and H.E.-H.; methodology, T.E.-M., N.K.and H.E.-H.; software, T.E.-M. and N.K., validation, N.K. and T.E.-M.; formal analysis, N.K. andT.E.-M.; investigation, N.K., T.E.-M. and H.E.-H.; resources, T.E.-M.; data curation, N.K.; writing—original draft preparation, N.K.; writing—review and editing, T.E.-M., H.E.-H., B.E.-A. and N.K.;visualization, N.K.; supervision, T.E.-M., H.E.-H. and B.E.-A.; project administration, T.E.-M.; fundingacquisition, T.E.-M. All authors have read and agreed to the published version of the manuscript.

Funding: This project is supported by the United Arab Emirates University (UAEU) (grantnumber 12N004).

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The data presented in this study are available on request from thecorresponding author. The data are not publicly available due to privacy issues.

Conflicts of Interest: The authors declare no conflict of interest.

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https://www.bekaert.com/en/product-catalog/content/dop/~//-/media/Files/ImportFiles/DRAMIX-3D/Dramix-3D-6535BG-6535BG.pdf?sc_lang=en-US

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