# Simple Design Methodology for R.C. Slabs by Hybrid Reinforcing of Steel Rebars and Uniaxial or Triaxial Geogrids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Focused Literature Review on the Hybrid Reinforcing System of Geogrids and Steel Rebar

- The use of steel rebars and geogrids as a hybrid reinforcing material for concrete slabs provides a greater first-crack load and a greater ultimate load comparing to the conventional reinforcing material of steel rebars or using geogrids as the main reinforcing material. Meanwhile, it increased the deflection values.
- For the hybrid reinforcing of steel rebars and geogrids, the reduction of steel rebars’ reinforcing ratio to 0.13% led to a reduction in the geogrids’ contribution and its effectiveness as concrete slab-reinforcing material. Accordingly, it cannot be dependent on geogrids as the main reinforcing material for concrete slabs.
- The impact energy capacity and impact resistance of concrete slabs have improved by using the hybrid reinforcing steel rebars and geogrids with a positive relation to the count of the geogrids’ layers.
- The hybrid reinforcing of steel rebars and uniaxial geogrids provided efficient utilization and better performance, particularly for uniaxial geogrids with the grade of 120 kN/m and greater, as it provided greater benefits in terms of values, including and not limited to capacity of loads, capacity of energy absorption, and displacement ductility index. It also provided more effective utilization, including and not limited to better flexural performance and greater benefits in terms of cost compared to the case of using the conventional reinforcing of steel rebars using the hybrid reinforcing of steel rebars and triaxial geogrids. However, the hybrid reinforcing of steel rebars and triaxial geogrids provided smaller deflection values and greater values of first-crack load values. As a recommendation, the uniaxial geogrids is should be tension-stressed before the concrete pouring.

## 3. Characteristics of Used Materials

^{2}. The reinforcing materials used for concrete slabs in this investigation were steel rebars and uniaxial or triaxial geogrids. It should be indicated that because the tested concrete slabs are one-way concrete slabs, biaxial geogrids were overlooked, and the geogrid-reinforcing materials in this investigation were uniaxial and triaxial geogrids. The experimental characteristics of the steel rebars are illustrated in Table 2. As specified by the experimental tests and the manufacturer, the mechanical and physical characteristics of the used uniaxial and triaxial geogrids are illustrated in Figure 2, Table 3, and Table 4. Figure 3 illustrates the stress–strain curve per each grade of uniaxial and triaxial geogrids.

## 4. Summary of the Experimental Program and Results

## 5. Geogrids Tensile Force at the Post-Peak Load

_{G}= T W N

_{G}is the tensile strength of the geogrids (kN), T is the peak tensile strength of the geogrids (kN/m), W is the strip width of the geogrids (m), and N is the count of the geogrids’ layers.

## 6. Experimental Moment, Characteristics Resistance Moment, and Design Resistance Moment

## 7. Simple Design Relation to Estimate the Desired Grade of Geogrids and Its Count of Layers

_{EP}= 0.88 M

_{CR}= 0.97 M

_{DR}, (Where F

_{G}= T

_{G}= T W N and the case of using the hybrid reinforcing of steel rebars and uniaxial geogrids)

_{EP}is the experimental moment at the post-peak load; M

_{CR}is the characteristic resistance moment; M

_{DR}is the design resistance moment; F

_{G}is the tensile force of the uniaxial geogrids (kN); T

_{G}is the tensile strength of the geogrids (kN). T is the peak tensile strength of the uniaxial geogrids (kN/m); W is the strip width of the uniaxial geogrids (m), and N is the count of the uniaxial geogrids’ layers.

_{EP}= 1.26 M

_{CR}= 1.41 M

_{DR}, (Where F

_{G}= T

_{G}= T W N and the case of using the hybrid reinforcing of steel rebars and triaxial geogrids)

_{EP}is the experimental moment at the post-peak load, M

_{CR}is the characteristic resistance moment, M

_{DR}is the design resistance moment, F

_{G}is the tensile force of the triaxial geogrids (kN), T

_{G}is the tensile strength of the geogrids (kN), T is the peak tensile strength of the triaxial geogrids (kN/m), W is the strip width of the triaxial geogrids (m), and N is the count of the triaxial geogrids’ layers.

_{EP}) from Equation (2) or Equation (3) and their ratios to the experimental ones are illustrated in Table 8, which resulted in a variance that frequently has a range of ± 10% when compared with the actual experimental data.

_{CG}= 0.76 T W N (case of using the hybrid reinforcing of steel rebars and uniaxial geogrids)

_{CG}= 2.06 T W N (case of using the hybrid reinforcing of steel rebars and triaxial geogrids)

_{CG}is the characteristic tensile force of the geogrids (kN), T is the peak tensile strength of the geogrids (kN/m), W is the strip width of the geogrids (m), and N is the count of the geogrids’ layers.

## 8. Conclusions and Recommendations

_{EP}= 0.88 M

_{CR}= 0.97 M

_{DR}, (based on the assumption of F

_{G}= T

_{G}= T W N)”, and for more accuracy, the uniaxial geogrids’ characteristic tensile force could be estimated by the equation of “F

_{CG}= 0.76 T W N”. For the design relations of the reinforced concrete slabs by the hybrid reinforcing of steel rebars and triaxial geogrids, the concrete slabs’ experimental moment at the post-peak load could be estimated by the equation of “M

_{EP}= 1.26 M

_{CR}= 1.41 M

_{DR}, (based on the assumption of F

_{G}= T

_{G}= T W N)”, and for more accuracy, the triaxial geogrids’ characteristic tensile force could be estimated by the equation of “F

_{CG}= 2.06 T W N”. Where M

_{EP}is the experimental moment at the post-peak load, M

_{CR}is the characteristic resistance moment, M

_{DR}is the design resistance moment, F

_{G}is the tensile force of the geogrids (kN), T

_{G}is the tensile strength of the geogrids (kN), T is the peak tensile strength of the geogrids (kN/m), W is the strip width of the geogrids (m), N is the count of the geogrids’ layers, and F

_{CG}is the characteristic tensile force of the geogrids (kN).

## 9. Future Studies

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**General arrangement and reinforcing material details per each group of the experimental program.

**Figure 6.**The design and the idealized characteristic stress–strain curves for the concrete and steel rebars material.

**Figure 7.**The design stain diagrams and the design stress diagrams per each case of using the hybrid reinforcing of steel rebars and uniaxial geogrids as a concrete slab-reinforcing material (the concrete slabs of group one) or using the hybrid reinforcing of steel rebars and triaxial geogrids as a concrete slab-reinforcing material (the concrete slabs of group two).

**Figure 11.**Probability distribution analysis for the ratios of the characteristic resistance moment or design resistance moment to the experimental moment at the post-peak load (case of using the hybrid reinforcing of steel rebars and uniaxial geogrids).

**Figure 12.**Probability distribution analysis for the ratios of the characteristic resistance moment or design resistance moment to the experimental moment at the post-peak load (case of using the hybrid reinforcing of steel rebars and triaxial geogrids).

**Figure 13.**Probability distribution analysis for the ratios of uniaxial or triaxial geogrids’ experimental tensile force at the post-peak load to their tensile strength.

Concrete–Mix, WC = 0.5, 1.5% Admixture | ||||||
---|---|---|---|---|---|---|

Cement Grade 42.5 | Sand | Crushed Limestone | Water | Water Reducing and Super-Plasticizer Concrete Admixture | Cone Slump | Compressive Strength after 28 days |

400 kg/m^{3} | 600 kg/m^{3} | 1200 kg/m^{3} | 200 kg/m^{3} | 6 kg/m^{3} | 6.5 cm | 40 N/mm^{2} |

Characteristics of the Steel Rebars | |||
---|---|---|---|

Diameter in mm (nominal) | 6 | Cross-Area in mm^{2} (nominal) | 28.29 |

Weight in kg/m (experimental) | 0.224 | Cross-Area in mm^{2} (experimental) | 28.52 |

Yield Load in kN (experimental) | 8.4 | Ultimate Load in kN (experimental) | 13.51 |

Yield Stress in N/mm^{2} (experimental) | 296.97 | Tensile Resistance in N/mm^{2} (experimental) | 477.51 |

Yield Strain in Micro-strain (experimental) | 1485 | Elongation After Break Down in % (experimental) | 32.7 |

Young’s modulus in N/mm^{2} (experimental) 200000 |

**Table 3.**Characteristics of the used uniaxial geogrids by the experimental tests and the manufacturer.

Characteristics | Uniaxial Geogrids (UG) Grades | |||
---|---|---|---|---|

UG45 | UG90 | UG120 | UG160 | |

RL in mm (physical) | 220 | 220 | 220 | 220 |

Rs in mm (physical) | 18 | 15 | 15 | 13 |

Bw in mm (physical) | 12.5 | 15 | 16.8 | 19.5 |

Rw in mm (physical) | 2.7 | 3.3 | 4 | 6.1 |

Bt in mm (physical) | 3.6 | 5.5 | 7 | 7.5 |

Rt in mm (physical) | 1.3 | 1.7 | 2.4 | 2.3 |

Mass per Unit Area in g/m^{2} | 300 | 600 | 800 | 1000 |

Tensile Strength at 2% Strain in kN/m (theoretical) | 11 | 26 | 36 | 45 |

Tensile Strength at 5% Strain in kN/m (theoretical) | 25 | 50 | 72 | 90 |

Tensile design Strength in kN/m (theoretical) | 21.2 | 42.4 | 56.5 | 75.4 |

Yield Point Elongation in % (theoretical) | 11.5 | 13 | 13 | 13 |

Peak Tensile Strength in kN/m (theoretical) | 45 | 90 | 120 | 160 |

Peak Tensile Strength in kN/m (experimental) | 45.56 | 79.36 | 103.91 | 143.46 |

Peak Strain in % (Experimental) | 30 | 30 | 30 | 30 |

Young’s modulus in N/mm^{2} (experimental) | 1500 | 1550 | 1200 | 1100 |

Material | High-Density Polyethylene (HDPE) |

**Table 4.**Characteristics of the used triaxial geogrids by the experimental tests, the manufacturer, and the numerical analysis.

Characteristics | Triaxial Geogrids (TG) Grades | |||
---|---|---|---|---|

TG150 | TG160 | |||

Transverse (1) | Diagonal (2) | Transverse (1) | Diagonal (2) | |

Mid-rib depth D in mm (physical) | 1.1 | 1.4 | 1.5 | 1.8 |

Mid-rib width W in mm (physical) | 1.2 | 1 | 1.3 | 1.1 |

Rib pitch P in mm (physical) | 40 | 40 | ||

Rib shape | Rectangular | Rectangular | ||

Aperture shape | Triangular | Triangular | ||

Radial Secant Stiffness at 0.5% Strain in kN/m (theoretical) | 360 (–75) | 390 (−75) | ||

Radial Secant Stiffness at 2% Strain in kN/m (theoretical) | 250 (−65) | 290 (−65) | ||

Hexagon Pitch in mm (theoretical) | 80 (±4) | 80 (±4) | ||

Radial Secant Stiffness Ratio (theoretical) | 0.8 | 0.8 | ||

Peak Tensile Strength in kN/m (experimental) | 17.21 | 19.45 | ||

Radial Secant Stiffness at 2% strain in kN/m (Experimental), according to BS EN 1SO 10319:1996 | 195 | 245 | ||

Peak Strain in % (experimental) | 14.5 | 14.5 | ||

Yield Strain in % (numerical) [22] | 6.4 | 6.4 | ||

Young’s modulus in N/mm^{2} (experimental) | 2800 | 2350 | ||

Material | Polypropylene with a Minimum of 2% Finely Divided Carbon Black Content |

Group No. | Concrete Slab Number | Loading Period Critical Points | Experimental Moment at the Post-Peak Load (kN·m) | ||||
---|---|---|---|---|---|---|---|

First-Crack Load Associated with Initial-Peak Load (kN) | Load Drop (kN) | Steel-Yield Load (kN) | Post-Peak Load (kN) | Failure Load (kN) | |||

Control Slab | S1-ST-CONT | 22.16 | 17.9 | 18.37 | 21.18 | 21.15 | 3.71 |

Group One | S2-ST+1UG45 | 24.35 | 19.5 | 20.59 | 25.69 | 25.46 | 4.49 |

S3-ST+1UG90 | 24.18 | 21.4 | 21.39 | 26.59 | 26.59 | 4.65 | |

S4-ST+2UG45 | 25.92 | 18.1 | 21.84 | 27.68 | 27.39 | 4.84 | |

S5-ST+1UG120 | 25.22 | 21.4 | 22.14 | 29.2 | 29.12 | 5.11 | |

S6-ST+1UG160 | 27.17 | 20.3 | 22.92 | 33.82 | 33.74 | 5.91 | |

S7-ST+2UG90 | 25.33 | 23.4 | 22.88 | 32.75 | 32.64 | 5.73 | |

S8-ST+2UG120 | 26.33 | 22.6 | 24.22 | 36.7 | 34.59 | 6.42 | |

S9-ST+2UG160 | 28.16 | 24.1 | 25.17 | 39.67 | 39.61 | 6.94 | |

Group Two | S10-ST+1TG150 | 25.59 | 17.9 | 19.24 | 22.96 | 19.49 | 4.02 |

S11-ST+1TG160 | 27.84 | 18.9 | 20.12 | 26.15 | 21.66 | 4.57 | |

S12-ST+2TG150 | 26.59 | 19.7 | 19.91 | 25.78 | 18.05 | 4.51 | |

S13-ST+2TG160 | 28.94 | 20 | 21.39 | 27.27 | 22.12 | 4.77 | |

The Load-Deflection Behaviour and the Load-Loading Period Critical Points |

**Table 6.**The tensile force of the geogrids per each concrete slab and its ratio to geogrids’ peak tensile strength.

Group Number | Concrete Slab No. | The Tensile Force of Geogrids (F_{G}) at the Experimental Post-Peak Load (kN) | The Peak Tensile Strength of Geogrids (T_{G}), Estimated by Equation-(1) (kN) | F_{G}/T_{G} (%) |
---|---|---|---|---|

Group one | S2-ST+1UG45 | 26.52 | 20.51 | 129.37% |

S3-ST+1UG90 | 29.34 | 35.712 | 82.15% | |

S4-ST+2UG45 | 32.77 | 41.004 | 79.91% | |

S5-ST+1UG120 | 36.93 | 46.759.5 | 78.98% | |

S6-ST+1UG160 | 50.52 | 64.557 | 78.25% | |

S7-ST+2UG90 | 49.08 | 71.424 | 68.70% | |

S8-ST+2UG120 | 63.97 | 93.519 | 68.40% | |

S9-ST+2UG160 | 77.61 | 129.114 | 60.10% | |

Group two | S10-ST+1TG150 | 19.37 | 7.75 | 250.12% |

S11-ST+1TG160 | 28.15 | 8.76 | 321.72% | |

S12-ST+2TG150 | 27.59 | 15.49 | 178.13% | |

S13-ST+2TG160 | 32.14 | 17.51 | 183.65% | |

First note: The tensile force of the geogrids was estimated based on: ● Considering the characteristic state of the concrete slabs’ material (the strength reduction factors of the concrete slabs’ materials are not considered), the top concrete’s compressive strain is equal to 0.3%, and the bottom steel rebars tensile force equal to its yield force. ● The bending moment of the concrete slabs is equal to its experimental moment at the post-peak load. Second note: The geogrids’ rib-cuttings occurred only for group two’s concrete slabs, as numerous rib-cuttings of triaxial geogrids occurred. |

**Table 7.**The estimated values of the characteristic resistance moment (M

_{CR}), the design resistance moment (M

_{DR}), and their ratios to the experimental moment at the post-peak load (M

_{EP}).

Group No. | Concrete Slabs | M_{EP} (kN·m) | M_{CR} (kN·m) | M_{DR} (kN·m) | M_{CR}/M_{EP} (%) | M_{DR}/M_{EP} (%) |
---|---|---|---|---|---|---|

Group one | S2-ST+1UG45 | 4.50 | 4.09 | 3.70 | 90.97% | 82.30% |

S3-ST+1UG90 | 4.65 | 5.06 | 4.65 | 108.74% | 99.93% | |

S4-ST+2UG45 | 4.84 | 5.36 | 4.94 | 110.65% | 101.98% | |

S5-ST+1UG120 | 5.11 | 5.73 | 5.29 | 112.13% | 103.52% | |

S6-ST+1UG160 | 5.92 | 6.77 | 6.28 | 114.39% | 106.11% | |

S7-ST+2UG90 | 5.73 | 7.05 | 6.53 | 123.01% | 113.94% | |

S8-ST+2UG120 | 6.42 | 8.03 | 7.43 | 125.03% | 115.69% | |

S9-ST+2UG160 | 6.94 | 9.50 | 8.73 | 136.84% | 125.75% | |

Group two | S10-ST+1TG150 | 4.02 | 3.23 | 2.86 | 80.39% | 71.18% |

S11-ST+1TG160 | 4.58 | 3.29 | 2.92 | 71.89% | 63.81% | |

S12-ST+2TG150 | 4.51 | 3.72 | 3.34 | 82.46% | 74.03% | |

S13-ST+2TG160 | 4.77 | 3.84 | 3.45 | 80.47% | 72.29% |

**Table 8.**The estimated values of the experimental moment at the post-peak load (M

_{EP}) from Equation (2) or Equation (3) and their ratios to the experimental ones.

Group No. | Concrete Slabs | M_{EP}(kN·m) | M_{EP} = 0.88 M_{CR}(where F_{G} = T_{G})(kN·m) | M_{EP} = 0.97 M_{DR}(where F_{G} = T_{G})(kN·m) | 0.88 M_{CR}/M_{EP}(%) | 0.97 M_{DR}/M_{EP}(%) |

Group one | S2-ST+1UG45 | 4.5 | 3.5992 | 3.589 | 79.98% | 79.76% |

S3-ST+1UG90 | 4.65 | 4.4528 | 4.5105 | 95.76% | 97.00% | |

S4-ST+2UG45 | 4.84 | 4.7168 | 4.7918 | 97.45% | 99.00% | |

S5-ST+1UG120 | 5.11 | 5.0424 | 5.1313 | 98.68% | 100.42% | |

S6-ST+1UG160 | 5.92 | 5.9576 | 6.0916 | 100.64% | 102.90% | |

S7-ST+2UG90 | 5.73 | 6.204 | 6.3341 | 108.27% | 110.54% | |

S8-ST+2UG120 | 6.42 | 7.0664 | 7.2071 | 110.07% | 112.26% | |

S9-ST+2UG160 | 6.94 | 8.36 | 8.4681 | 120.46% | 122.02% | |

Group No. | Concrete Slabs | M_{EP}(kN·m) | M_{EP} = 1.26 M_{CR}(where F_{G} = T_{G})(kN·m) | M_{EP} = 1.41 M_{DR}(where F_{G} = T_{G})(kN·m) | 1.26 M_{CR}/M_{EP}(%) | 1.41 M_{DR}/M_{EP}(%) |

Group two | S10-ST+1TG150 | 4.02 | 4.0698 | 4.0326 | 101.24% | 100.31% |

S11-ST+1TG160 | 4.58 | 4.1454 | 4.1172 | 90.51% | 89.90% | |

S12-ST+2TG150 | 4.51 | 4.6872 | 4.7094 | 103.93% | 104.42% | |

S13-ST+2TG160 | 4.77 | 4.8384 | 4.8645 | 101.43% | 101.98% |

**Table 9.**The estimated values of the characteristic resistance moment (M

_{CR}) considering Equation (4) for group one and Equation (5) for group two and their ratios to the experimental moment at the post-peak load (M

_{EP}).

Group number | Concrete Slabs | M_{EP} (kN·m) | M_{CR} (kN·m) | M_{CR}/M_{EP} (%) |
---|---|---|---|---|

Group one | S2-ST+1UG45 | 4.50 | 3.77 | 83.86% |

S3-ST+1UG90 | 4.65 | 4.51 | 96.92% | |

S4-ST+2UG45 | 4.84 | 4.74 | 97.85% | |

S5-ST+1UG120 | 5.11 | 5.02 | 98.24% | |

S6-ST+1UG160 | 5.92 | 5.83 | 98.50% | |

S7-ST+2UG90 | 5.73 | 6.04 | 105.39% | |

S8-ST+2UG120 | 6.42 | 6.81 | 106.03% | |

S9-ST+2UG160 | 6.94 | 7.98 | 114.95% | |

Group two | S10-ST+1TG150 | 4.02 | 3.79 | 94.33% |

S11-ST+1TG160 | 4.58 | 3.91 | 85.44% | |

S12-ST+2TG150 | 4.51 | 4.78 | 105.95% | |

S13-ST+2TG160 | 4.77 | 5.01 | 104.98% |

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**MDPI and ACS Style**

Mohamed, R.N.A.; El Sebai, A.M.; Gabr, A.S.A.-H.
Simple Design Methodology for R.C. Slabs by Hybrid Reinforcing of Steel Rebars and Uniaxial or Triaxial Geogrids. *Inventions* **2021**, *6*, 32.
https://doi.org/10.3390/inventions6020032

**AMA Style**

Mohamed RNA, El Sebai AM, Gabr ASA-H.
Simple Design Methodology for R.C. Slabs by Hybrid Reinforcing of Steel Rebars and Uniaxial or Triaxial Geogrids. *Inventions*. 2021; 6(2):32.
https://doi.org/10.3390/inventions6020032

**Chicago/Turabian Style**

Mohamed, Ramy Nasr Abdelmonem, Ahmed Mohamed El Sebai, and Ahmed Shaban Abdel-Hay Gabr.
2021. "Simple Design Methodology for R.C. Slabs by Hybrid Reinforcing of Steel Rebars and Uniaxial or Triaxial Geogrids" *Inventions* 6, no. 2: 32.
https://doi.org/10.3390/inventions6020032