# Minimum Shear Reinforcement for Reactive Powder Concrete Beams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{fb}to projected V

_{u}

_{,AFGC}is roughly 58.3%, whereas the mean proportion of vs. and V

_{c}is just 41.7%. The deformation response and ultimate shear strength of the examined RPC beams were well predicted by the designed model using finite elements when metal fibers were taken into account.

## 1. Introduction

## 2. Design Guidelines for the Shear of RC Beams

#### 2.1. Shear Design of RC Beams under ACI318-2014 Code

_{ud}, is calculated as the sum of the shear sustained by the shear reinforcement, V

_{s}, and the concrete, V

_{c}, i.e.,

_{c}:

_{w}and d are the beam’s width and depth, respectively, and f′c is the concrete’s cylinder compressive strength in MPa. It is possible to calculate the shear strength offered by shear reinforcement V

_{s}by using the following formula:

_{v}and f

_{yv}represent the area and required yield strength, respectively, of vertical reinforcement on the web located within a distance between stirrups (s

_{v}). The specified vertical web strengthening ratio (ρ

_{v}= A

_{v}/b

_{w}s

_{v}) should be greater than the minimal value (ρ

_{v}

_{,min}) demanded by this code, which is equal to the higher of the two values 0.35/f

_{yv}or (0.062 f′c/f

_{yv}).

#### 2.2. The ACI318-2014 Code for RC Beams Specifies the Minimal Amount of Shear Reinforcement

_{h}

_{,min}) and (A

_{v}

_{,min}), respectively:

#### 2.3. Limits of Spacing for Shear Reinforcement

_{v}

_{,max}) is equal to a minimum of 600 or 0.5 d mm; however, if the shear reinforcement’s contribution to the shear strength is above (0.33 $\sqrt{{f}^{\prime}c}$ b

_{w}d), s

_{v}

_{,max}should be decreased by half. The spacing limitations recommened by EC2 [33] are 0.75 d or 600 mm.

#### 2.4. Design Recommendations for RPC Beams in Regard to Shear

_{ud}of RPC is determined:

_{s}, V

_{c}, and V

_{fb}are the shear strengths contributed by the shear reinforcement, cement matrix, and steel fiber, respectively.

#### 2.5. The France Association of Civil Engineers (AFGC-2013)

_{c}is specified as follows:

_{cf}is the assumed value of 1.30 for the partial safety factor regarding fibers, k is the factor pertaining to the situation in which pre-stressing has been applied, and γ

_{E}is a safety factor, with γ

_{cf}γ

_{E}being equal to 1.5. The following formula can be used to compute the contribution of steel fibers, V

_{fb}:

_{fv}is the area of the a fiber effects, which can be taken as bw z for rectangular sections, where z = 0.9 d; and σ

_{Rd}

_{,f}is the remaining tensile strength, which can be calculated as follows:

_{f}(w) is a function of tensile stress and fracture breadth, where w

_{lim}= max (w

_{u}, w

_{max}); K is the orientation of the fiber factor, which can be considered to be equal to 1.25; and w

_{max}is the widest fracture possible.

#### 2.6. Korea Concrete Institute (KCI-2012)

_{w}is the beam’s breath, d is the beam’s effective depth, f′c is the cylinder’s compressive strength, and ϕ

_{b}is a member reduction factor of 0.77. According to the equation below, the steel fibers’ shear strength (V

_{fb}) is

_{vd}is the mean splitting tensile strength design in a location perpendicular to the diagonal tensile break, and z is the span from the location affected by compressive stresses to the centroid of tension steel, which is typically equal to d/1.15. This formula is used to determine f

_{vd}’s value:

_{v}is the ultimate fracture width at the area at which outer-fiber peak stress is applied; ϕ

_{c}is the material’s reducing factor, considered to be 0.8; σ

_{k}(w) is the tensile softening curve; w

_{v}= max (w

_{v}, 0.3 mm); and σ

_{d}(w) is equivalent to ϕ

_{c}σ

_{k}(w).

_{s}is the angle between a beam’s longitudinal direction and the shearing reinforcement, A

_{v}is the cross-sectional area of the reinforcement, f

_{yv}is its design yield strength, and s

_{v}is its spacing.

## 3. Materials and Methods

#### 3.1. Materials Used

^{2}/g, and a medium diameter of particles of 15 μm was utilized. It was made from natural sand that was 0.5 mm in size and had a specific density of 2.65. In addition to silica fume powder, which has a specific gravity of 2.20, a specific surface area of 170,000 cm

^{2}/g, and an average diameter of around 0.15 mm, crushed quartz powder with an average diameter of 1 um to 100 um, a Blaine fineness of 3100 cm

^{2}/gm, and a specific gravity of 2.85 was also utilized. All mixes contained a superplasticizer of a new generation of polycarboxlic ether. The steel fibers employed in this investigation were a form of locally accessible hook-ended straight fiber. The steel fibers had an equivalent diameter and length of 13.0 mm and 2.0 mm, respectively. By performing a direct tension test, the fibers’ yield strength and tensile strength were calculated, and the results were 550 MPa and 820 MPa, respectively. The volume fraction of steel fibers was maintained at a consistent level of 2.0% for each tested beam. The regular tap water that was used complied with the standards for concrete mixing water.

#### 3.2. Concrete Mix Proportions and Specimen Casting

^{3}of RPC. A high-speed mixer was used to combine the mixture’s contents for 10 min. Then, 75% of the water was added to the mixer after all powders and natural sand had been dry-mixed for 2 min at low speed. The mixer was stopped for 1 min after 2 min of mixing at low speed (140 ± 5 rpm). The mixture was then blended for three minutes while the remaining water and superplasticizer were added. The final step involved mixing the mixture for two minutes at high speed (285 ± 10 rpm). For the first 24 h, concrete specimens were allowed to cure at ambient temperature (21 ± 2 °C). The samples were damp-cured up until the testing day after demolding. This RPC mix’s cube concrete compressive strength (f

_{cu}) is 157 MPa depending on a typical of three cube specimens, and its cylinder concrete compressive strength (f′c) and splitting cylinder tensile strength (f

_{sp}) are 144.3 MPa and 11.9 MPa, respectively. Its flexural strength (f

_{r}) is 39.7 MPa based on 100 mm × 100 mm × 500 mm prisms.

#### 3.3. Experimental Program for the RPC Beams

_{yl}) of 410 MPa served as the compression reinforcement for the beams. Each bar was 10 mm in diameter. The yield strengths (f

_{yv}) of stirrup bars with diameters of 6 mm and 8 mm are 330 MPa and 310 MPa, respectively.

_{v}= 200 and 100 mm) and different bar stirrup diameters (d

_{v}= 6 and 8 mm) so as to test the suitability of the minimal vertical reinforcement of the web required according to ACI 318-14 when applied to RPC beams. In accordance with ACI 318-14, s

_{v}

_{,max}is equal to the lesser of 600 mm or 0.5d, and ρ

_{v}

_{,min}is equivalent to the larger of 0.35/f

_{y}or (0.062$\sqrt{f{c}^{\prime}}$/f

_{yv}), while s

_{v}

_{,max}must be decreased by half in cases where the shear strength given by shear reinforcement (V

_{s}) is above (0.33 $\sqrt{f{c}^{\prime}}$ bd). Table 2 compares the provided vertical web reinforcement ratio (ρ

_{v}= A

_{v}/(b

_{w}·s

_{v})) of the examined RPC beams with the minimum requirements (ρ

_{v}

_{,min}) of the code, as well as the distance between stirrups (s

_{v}) of the examined RPC beams with the highest permitted distance between stirrups (s

_{v}

_{,max}) required by the ACI 318-2014 code. Clearly, it can be observed that the provided area of vertical web reinforcement (A

_{v}) for all the tested beams satisfies the minimum requirements (A

_{v}

_{,min}) of the ACI 318-2014 code. The specified stirrup spacing (s

_{v}) of each of the examined beams, however, is significantly greater than the highest stirrup distance (s

_{v}

_{,max}) allowed by the aforementioned code. It should be noted that the stated stirrup spacing (s

_{v}) of all examined beams is more than double the highest stirrup spacing (s

_{v}

_{,max}) permitted by ACI 318-2014.

## 4. Analysis Software and Model Calibration

_{max}) and the corresponding displacement (δ

_{max}) but also the ultimate load (P

_{u}) and the corresponding displacement (δ

_{u}). The final parameter was the energy absorption capacity of the beam, which is equal to the area below the force–deflection response curve [44,45].

#### 4.1. Concrete and Reinforcement Constitutive Models

#### 4.2. Finite Element Modelling and Analysis Procedure

## 5. Results and Discussion

#### 5.1. Damage and Crack Patterns

_{u}

_{,exp}) and diagonal cracking strength (P

_{cr}) of the examined beams in this study are shown in Table 4. Following the development of flexural fractures in the middle of the span, diagonal cracks often developed in the beam’s two shear spans. The breadth of the flexural crack narrowed significantly after the formation of diagonal shear cracks. Typically, a diagonal shear crack began abruptly in the center of the span of a shear crack and spread toward the supports and load points as a result of an increase in the load being applied. The already-present diagonal shear cracks only slightly spread further as the load applied was increased, but a few new inclined cracks were also created. Finally, the concrete fractured abruptly across the inclined crack due to shear failure. The images demonstrate that the shear spans of the examined beams B-1, B-2, B-4, and B-5 failed due to significant concrete degradation, whereas beam B-3 fell due to crushing of the compression zone in the middle of the span with an a/d of 3.0. According to the test results, flexural failure may occur before shear failure when minimal reinforcement for shear is provided within the distance of 0.5d recommended in ACI code [34]. The yielding of shear reinforcement, however, may be seen before the yielding of flexural reinforcement and the compression failure of beams with a distance that is higher than the minimal levels required by code.

#### 5.2. Load–Displacement Relationships

_{v}) are displayed.

_{v}% values showed only slight variations after being subjected to the ultimate load. Figure 3 shows that beam B-2 with an a/d ratio of 3.0 and a supplied ρv of 0.10, the latter of which is lower than the minimum specified by the code, exhibits a stiffness like that of beam B-1, with the same a/d ratio but providing a ρ

_{v}of 0.20. The rigidity of the examined beams significantly decreases as the a/d ratio rises. In comparison to beam B-4, which has an a/d ratio of 3.0, beam B-5, which has an a/d ratio of 3.2, is less stiff.

#### 5.3. Strain Response

#### 5.4. Effect of Web Reinforcement Ratio

_{v}%) on the examined beams’ ultimate shear strength and the strength of diagonal cracks in terms of the distance between stirrups and the diameter of the stirrups’ bars. It is evident that while diagonal cracks formed more slowly with a smaller distance between stirrups (s

_{v}), they formed more quickly with a smaller diameter of the stirrup bars (d

_{v}). The growth of diagonal cracks was significantly slower for beam B-1, with a 100 mm stirrup spacing (s

_{v}), than it was for the comparable B-2 beam, with a 200 mm stirrup spacing (s

_{v}). The emergence of diagonal cracking occurred more slowly in beam B-4, with an 8 mm stirrup bar diameter and a 200 mm stirrup spacing (s

_{v}), than in beam B-2, with a 6 mm stirrup bar diameter and a 200 mm stirrup spacing (s

_{v}). As can be observed, decreasing the distance between stirrups had a greater inhibitory effect on the growth of diagonal cracks than increasing stirrup diameter. The given stirrup spacings (s

_{v}) for all the examined RPC beams are noticeably greater than the maximum stirrup spacing (s

_{v}

_{,max}) stipulated in the ACI code [34]. The RPC beams strengthened using the widest possible stirrup spacing, however, exhibited good overall performance.

_{v}

_{,max}requirements of ACI 318-2014 are not practically appropriate and can be safely adjusted to 0.75d instead of 0.50d. Table 3 shows that for the beams with an identical a/d ratio, the ultimate shear strength improves marginally as the given vertical reinforcement of the web ratio (ρ

_{v}%) rises. It should be noticed that, in accordance with ACI 318-2014, the given reinforcement of the web ratio (ρ

_{v}%) for beams B-1, B-2, B-4, and B-5 is lower than the minimal web reinforcement ratio (ρ

_{v}

_{,min}). Nevertheless, excellent general performance was seen for all of the examined RPC beams that were strengthened via reinforcement of the web ratios (ρ

_{v}%) smaller than the minimal web reinforcement ratios (ρ

_{v}

_{,min}) stipulated by the ACI code. This shows that when used with RPC beams with 2.0% steel fibers, the minimal vertical reinforcement of the web ratio specified by the ACI code can be properly lowered.

#### 5.5. Analyzing the RPC Beam Test Results in Relation to the ACI 318-2014 Code’s Shear Requirements

_{v}) of the examined beams with the minimal demands of the ACI 318-14 code, while Table 4 gives a ratio between the estimated ultimate shear strength (V

_{u}

_{,cal}) utilizing the ACI 318-14 code and the estimated experimental ultimate shear strength (V

_{u}

_{,exp}). As can be observed, the average value for the evaluated beams’ V

_{u}

_{,cal}to V

_{u}

_{,exp}ratio according to the ACI 318-14 code is 0.403. This shows that despite every one of the examined beams having stirrups with a spacing (s

_{v}) significantly greater than the maximum stirrup spacing (s

_{v}

_{,max}) required by ACI 318-14, the measured readings of V

_{u}

_{,exp}for all of the examined beams were significantly higher than those specified by the code. This demonstrates that the shear strength estimation formulae established by ACI 318-14 are not suitable for RPC beams as they do not account for the significant role that steel fibers play in providing resistance to shear stresses. It should be noted that, in accordance with ACI 318-14, steel fibers may be utilized as the beam’s reinforcement for shear if their normalized shear strength is more than 0.29 (for f′c ≤ MPa, d ≤ 600 mm). For all of the examined RPC beams with a fiber percent of 2.0%, the normalized shear strength values in Table 3 are significantly higher than 0.29, having an average value of 0.60.

#### 5.6. Comparison of Test Results for RPC Beams with AFGC-2013 and KCI-2012 Design Recommendations

_{u}

_{,exp}and V

_{u}

_{,AFGC}ratios is 1.280, while the average value for their V

_{u}

_{,exp}and V

_{u}

_{,KCI}ratios is 1.220. This shows that, when used for RPC beams equipped with shear reinforcement below the minimum requirement allowed by ACI 318-14, the KCI-2012 and AFGC-2013 forecasts for the ultimate shear strength are secure and cautious. When AFGC-2013 and KCI-2012 recommendations were compared for all of the examined beams, it was found that there were only very slight variations among the projections of the ultimate shear strength. In actuality, the tiny variation in the safety parameters taken into account by each methodology is what caused the small variance in the forecasts of the two recommendations. In accordance with the KCI-2012 recommendations, Table 5 demonstrates that the mean percent of the forecast contributions of the fibers of steel (V

_{fb}) in comparison to the forecast ultimate shear strength of the examined beams (V

_{u}

_{,KCI}) is roughly 59.2%, while the mean percent of the forecast contributions of the shear reinforcement (V

_{s}) and concrete (V

_{c}) is only 40.8%. According to the AFGC-2013 criteria, the mean proportion of V

_{fb}to projected V

_{u}

_{,AFGC}is roughly 58.3%, whereas the mean proportion of V

_{s}to V

_{c}is just 41.7%.

#### 5.7. RPC Beam Analytical Modeling Utilizing Finite Element Software

_{s}) was assumed to be 200 GPa. It was believed that the concrete and reinforcement would always stick together perfectly.

_{c}) of RPC:

_{cu}) equal to 157 MPa, the normative value of E

_{c}is equal to 48,929.5 MPa. The Poisson’s ratio is assumed to be equal to 0.20, and the tensile strength of RPC is considered to be 11 MPa.

_{cr}

_{,exp}to V

_{cr}

_{,NUM}is equivalent to 1.034, whereas the average proportion of V

_{u}

_{,exp}to V

_{u}

_{,NUM}is equivalent to 1.007. Additionally, the proportion of ∆

_{cr}

_{,exp}to ∆

_{cr}

_{,NUM}has a mean value of 0.92, while the proportion of ∆

_{u}

_{,exp}to ∆

_{u}

_{,NUM}has a mean value of 0.979. This demonstrates that for the examined RPC beams, the FEM model can accurately predict the ultimate shear load and the diagonal cracking load. Figure 8 displays a comparison of the empirical and numerical load–displacement curves for the examined beams. It can be observed that the load–displacement response of the examined RPC beams can be well predicted by the suggested nonlinear FEM.

## 6. Conclusions

- i
- The offered shear reinforcement barely affects the maximum shear strength of the tested RPC beams with a volume content of 2.0% steel fibers. These steel fibers are crucial in helping RPC beams endure shear loads. Despite the fact that the examined RPC shallow beams’ vertical web reinforcement ratio was far below the smallest proportion specified by ACI 318-14, all of the examined RPC shallow beams displayed excellent performance in general.
- ii
- ACI 318-14’s shear strength calculation formulas significantly understate the shear strength of the examined RPC beams that have a minimal vertical web reinforcement ratio. As a result, these formulas are not suitable for RPC beams as they do not account for the significant role that steel fibers play in resisting shear stresses. For RPC beams with relatively low heights, the highest distance between stirrups (s
_{v}_{,max}) specified according to the ACI 318-14 guidelines can safely be extended from 0.50 d to 0.75 d. - iii
- In light of the predicted ultimate shear strengths of the examined beams, the design suggestions for RPC specified by KCI-2012 and AFGC-2013 are secure and restrained. The ultimate shearing strength forecasts made by KCI-2012 and AFGC-2013 are roughly equivalent (the mean proportion of the experimental ultimate shearing strength and the predicted ultimate shearing strength utilizing KCI-2012 and AFGC-2013 are approximately 1.462 and 1.446, respectively).
- iv
- According to the AFGC-2013 criteria, the mean proportion of V
_{fb}to projected V_{u}_{,AFGC}is roughly 58.3%, whereas the mean proportion of V_{s}to V_{c}is just 41.7%. - v
- The deformation response and the ultimate shear strength of the examined RPC beams with vertical reinforcement of the web ratio much below the lowest value permitted by the code were reliably predicted by the suggested FEM when steel fibers were taken into account.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Details of the reinforcement employed in the RPC beam samples (depending on a/d ratio 3 or 3.2 the a: 465 or 496 mm; c: 758 or 820 mm; d: 155 mm).

**Figure 4.**Relationship between the examined beams’ total load and strain in the stirrups’ vertical legs.

**Figure 6.**(

**a**) A wire-frame model reinforcement cage profile showing longitudinal and transverse reinforcements and (

**b**) a shaded model showing the loading and boundary condition profile with the load applied taken to be P/2.

**Figure 7.**RPC beam B-1 meshing and stress distribution determined using a finite element model (

**a**) designed been (

**b**) loaded beam.

**Figure 8.**Comparison of the load–displacement curves for the examined beams using experimental and numerical data for (

**a**) B-1, (

**b**) B-2, (

**c**) B-3, (

**d**) B-4, and (

**e**) B-5.

C ^{a}, kg | FS ^{b}, kg | QP ^{c}, kg | SF ^{d}, kg | W ^{e}, kg | SP ^{f}, kg |
---|---|---|---|---|---|

760 | 1026 | 228 | 190 | 144.4 | 30.4 |

Beam | f_{cu}, MPa | b × h (mm) | a/d Ratio | Main Longitudinal Bars | Provided Stirrups (Vertical Web Reinforcement) | Requirements of ACI 318-14 for s _{v}_{,max} and ρ_{v}_{,min} | ||||
---|---|---|---|---|---|---|---|---|---|---|

Lower | Upper | s_{v}, (mm) | d_{v}, (mm) | ρ_{v}, (%) | s_{v}_{,max}, (mm) | ρ_{v}_{,min} (%) | ||||

B-1 | 157 | 140 × 220 | 3.0 | 6Φ18 | 2Φ10 | 100 | 6 | 0.20 | 90 | 0.235 |

B-2 | 157 | 140 × 220 | 3.0 | 6Φ18 | 2Φ10 | 200 | 6 | 0.10 | 90 | 0.235 |

B-3 | 157 | 140 × 220 | 3.0 | 6Φ18 | 2Φ10 | 100 | 8 | 0.36 | 90 | 0.25 |

B-4 | 157 | 140 × 220 | 3.0 | 6Φ18 | 2Φ10 | 200 | 8 | 0.18 | 90 | 0.25 |

B-5 | 157 | 140 × 220 | 3.2 | 6Φ18 | 2Φ10 | 200 | 8 | 0.18 | 90 | 0.25 |

**Table 3.**Summary of model calibrations [47].

Details | ||
---|---|---|

Concrete | Finite element type | 8-node isoparametric solid elements |

Failure mode | Fracture under tension, plasticity under compression | |

Crack formulation | Smeared | |

Numerical modification | Shear factor reduction | |

Reinforcement | Finite element type | 2-node truss elements |

Constitutive model | Uniaxial multilinear law | |

Modeling type | Embedded reinforcement | |

Bond type | Full reinforcement–concrete bond |

Beam | a/d | 2V_{cr} (kN) | 2V_{u}_{,exp} (kN) | V_{cr} (kN) | V_{u}_{,exp} (kN) | $\frac{\mathit{V}\mathit{c}\mathit{r}}{\mathit{V}\mathit{u},\mathit{e}\mathit{x}\mathit{p}}$ | $\frac{\mathit{V}\mathit{u},\mathit{e}\mathit{x}\mathit{p}}{\mathit{b}\mathit{d}\sqrt{{\mathit{f}}^{\prime}\mathit{c}}}$ |
---|---|---|---|---|---|---|---|

B-1 | 3.0 | 130 | 403 | 65 | 201.5 | 0.323 | 0.638 |

B-2 | 3.0 | 110 | 369 | 55 | 184.5 | 0.298 | 0.584 |

B-3 | 3.0 | 170 | 450 | 85 | 225 | 0.378 | 0.713 |

B-4 | 3.0 | 120 | 369 | 60 | 184.5 | 0.325 | 0.584 |

B-5 | 3.2 | 110 | 310 | 55 | 155 | 0.355 | 0.491 |

**Table 5.**Comparison of the experimental findings with the maximum shear strength specified by the ACI 318-14 code.

Beam | V_{u}_{,exp} (kN) | ACI 318-14 Code | |||
---|---|---|---|---|---|

V_{c} (kN) | V_{s} (kN) | V_{u}_{,cal} (kN) | $\frac{\mathit{V}\mathit{u},\mathit{c}\mathit{a}\mathit{l}}{\mathit{V}\mathit{u},\mathit{e}\mathit{x}\mathit{p}}$ | ||

B-1 | 201.5 | 45.6 | 32.3 | 77.9 | 0.387 |

B-2 | 184.5 | 45.6 | 16.2 | 61.8 | 0.335 |

B-3 | 225 | 45.6 | 52.7 | 98.3 | 0.437 |

B-4 | 184.5 | 45.6 | 26.4 | 72 | 0.390 |

B-5 | 155 | 45.6 | 26.4 | 72 | 0.465 |

**Table 6.**Comparison of testing results and maximum shear strength computed using the RPC design guidelines.

Beam | V_{u,exp} (kN) | KCI-2012 | AFGC-2013 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

V_{c} (kN) | V_{fb} (kN) | V_{s} (kN) | V_{u,KCI} (kN) | $\frac{\mathit{V}\mathit{u},\mathit{e}\mathit{x}\mathit{p}}{\mathit{V}\mathit{u},\mathit{K}\mathit{C}\mathit{I}}$ | V_{c} (kN) | V_{fb} (kN) | V_{s} (kN) | V_{u,AFGC} (kN) | $\frac{\mathit{V}\mathit{u},\mathit{e}\mathit{x}\mathit{p}}{\mathit{V}\mathit{u},\mathit{A}\mathit{F}\mathit{G}\mathit{C}}$ | ||

B-1 | 201.5 | 37.2 | 87.4 | 24.9 | 149.5 | 1.348 | 37.6 | 90.4 | 29.1 | 157.1 | 1.283 |

B-2 | 184.5 | 37.2 | 87.4 | 12.5 | 137.1 | 1.346 | 37.6 | 90.4 | 14.5 | 142.5 | 1.295 |

B-3 | 225 | 37.2 | 87.4 | 40.6 | 165.2 | 1.362 | 37.6 | 90.4 | 47.4 | 175.4 | 1.283 |

B-4 | 184.5 | 37.2 | 87.4 | 20.3 | 144.9 | 1.273 | 37.6 | 90.4 | 23.7 | 151.7 | 1.216 |

B-5 | 155 | 37.2 | 87.4 | 20.3 | 144.9 | 1.07 | 37.6 | 90.4 | 23.7 | 151.7 | 1.022 |

Beam | Cracking Load and Ultimate Load | Cracking Displacement and Maximum Displacement | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{V}}_{\mathit{c}\mathit{r},\mathit{N}\mathit{U}\mathit{M}}$ (kN) | ${\mathit{V}}_{\mathit{c}\mathit{r},\mathit{e}\mathit{x}\mathit{p}}$ (kN) | $\frac{{\mathit{V}}_{\mathit{c}\mathit{r},\mathit{e}\mathit{x}\mathit{p}}}{{\mathit{V}}_{\mathit{c}\mathit{r},\mathit{N}\mathit{U}\mathit{M}}}$ | ${\mathit{V}}_{\mathit{u},\mathit{e}\mathit{x}\mathit{p}}$ (kN) | ${\mathit{V}}_{\mathit{u},\mathit{N}\mathit{U}\mathit{M}}$ (kN) | $\frac{{\mathit{V}}_{\mathit{u},\mathit{e}\mathit{x}\mathit{p}}}{{\mathit{V}}_{\mathit{u},\mathit{N}\mathit{U}\mathit{M}}}$ | ${\u2206}_{\mathit{c}\mathit{r},\mathit{N}\mathit{U}\mathit{M}}$ (mm) | ${\u2206}_{\mathit{c}\mathit{r},\mathit{e}\mathit{x}\mathit{p}}$ (mm) | $\frac{{\u2206}_{\mathit{c}\mathit{r},\mathit{e}\mathit{x}\mathit{p}}}{{\u2206}_{\mathit{c}\mathit{r},\mathit{N}\mathit{U}\mathit{M}}}$ | ${\u2206}_{\mathit{u},\mathit{e}\mathit{x}\mathit{p}}$ (mm) | ${\u2206}_{\mathit{u},\mathit{N}\mathit{U}\mathit{M}}$ (mm) | $\frac{{\u2206}_{\mathit{u},\mathit{e}\mathit{x}\mathit{p}}}{{\u2206}_{\mathit{u},\mathit{N}\mathit{U}\mathit{M}}}$ | |

B-1 | 72 | 76.5 | 1.06 | 201.5 | 185 | 1.089 | 0.5 | 0.34 | 0.68 | 1.75 | 1.95 | 0.897 |

B-2 | 51 | 57 | 1.12 | 184.5 | 178 | 1.037 | 0.48 | 0.4 | 0.83 | 1.85 | 2.1 | 0.881 |

B-3 | 88 | 83.5 | 0.95 | 225 | 255 | 0.882 | 0.31 | 0.33 | 1.07 | 2.2 | 1.98 | 1.11 |

B-4 | 58 | 65 | 1.12 | 184.5 | 174 | 1.06 | 0.24 | 0.26 | 1.08 | 1.78 | 1.88 | 0.947 |

B-5 | 60 | 55 | 0.92 | 155 | 160.5 | 0.966 | 0.47 | 0.44 | 0.94 | 2.1 | 1.98 | 1.06 |

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## Share and Cite

**MDPI and ACS Style**

Sultan, H.K.; Huseien, G.F.
Minimum Shear Reinforcement for Reactive Powder Concrete Beams. *Eng* **2024**, *5*, 801-818.
https://doi.org/10.3390/eng5020043

**AMA Style**

Sultan HK, Huseien GF.
Minimum Shear Reinforcement for Reactive Powder Concrete Beams. *Eng*. 2024; 5(2):801-818.
https://doi.org/10.3390/eng5020043

**Chicago/Turabian Style**

Sultan, Hussein Kareem, and Ghasan Fahim Huseien.
2024. "Minimum Shear Reinforcement for Reactive Powder Concrete Beams" *Eng* 5, no. 2: 801-818.
https://doi.org/10.3390/eng5020043