# Influence of the Cross-Sectional Shape of a Reinforced Bimodular Beam on the Stress-Strain State in a Transverse Impact

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{t}< E

_{c}) and for heavy concrete (E

_{t}> E

_{c}) under the action of shock load with and without regard to the mass of the beam. The numerical study shows that taking into account the mass of the beam upon impact significantly decreases the magnitude of the normal stresses in both the tensioned and compressed zones. Beams of rectangular cross-section have the highest load-bearing capacity when the cross-section height is equal for both light and heavy concrete. An increase in the size of the flange of the I-beam in the stretched zone leads to a sharp decrease in normal tensile stresses and a slight increase in normal compressive stresses. The proposed engineering method makes it possible to numerically study the effect on the stress-strain state of a beam under the action of a concentrated impact of various geometric characteristics of the cross-section, bimodularity of the material, size, number and location of reinforcement.

## 1. Introduction

_{c}= 1.75 · 10

^{4}MPa, E

_{t}= 0.75 · 10

^{4}MPa [20], fiber-reinforced concrete: E

_{c}= 2.25 · 10

^{3}MPa; E

_{t}= 5 · 10

^{3}MPa [20].

## 2. Materials and Methods

- (a)
- I-beam

- (b)
- Rectangle

- (c)
- T-beam

_{t}—the diameter of the reinforcement in the tension zone; c

_{t}—the distance from the reinforcement in the tension zone to the neutral axis; n

_{t}—the number of rods in the tension zone; d

_{c}—the diameter of the reinforcement in the compression zone; c

_{c}—distance (coordinate) from the reinforcement in the compression zone to the neutral axis; n

_{c}—number of rods in the compression zone.

^{3}, and AFB-1 (heavy concrete), density ρ = 2000 kg/cm

^{3}, with cross-sections in the form of a rectangle, T-beam and I-beam. The corresponding reduced masses of beams M

_{B}are equal for beams made of fiber-reinforced concrete of rectangular cross-section M

_{B}

_{1}= 367.2 kg, T-section M

_{B}

_{2}= 163.2 kg and I-beam cross-section M

_{B}

_{3}= 184.96 kg, from AFB-1 M

_{B}

_{4}= 1049 kg, M

_{B}

_{5}= 466.3 kg and M

_{B}

_{6}= 528.5 kg.

_{B}L/M

_{A}= 7.71, a T-cross section m

_{B}L/M

_{A}= 3.423, an I-cross section m

_{B}L/M

_{A}= 3.9. For fiber-reinforced concrete beams of rectangular cross-section m

_{B}L/M

_{A}= 22, T-section m

_{B}L/M

_{A}= 9.77 and I-section cross-section m

_{B}L/M

_{A}= 11.1.

- (a)
- without considering the mass of the beam, Equation (1):$${k}_{d}=1+\sqrt{1+\frac{2h}{{f}_{s}}}$$
- (b)
- considering the mass of the beam, Equation (2):$${k}_{d}=1+\sqrt{1+\frac{2h}{{f}_{s}}{\left(1+\frac{{M}_{B}}{{M}_{A}}\right)}^{-3}}$$
_{s}is the static deflection of the beam under load without taking into account the mass of the beam; M_{A}is the mass of the falling weight, M_{B}is the reduced mass of the beam according to Cox H. [33]. For a simply supported beam, loaded in the middle of the span, M_{B}= $\frac{17}{35}$ m_{B}L where m_{B}is the distributed mass of the beam, L is the beam length.

_{d}(Equations (1) and (2)) depending on whether the mass of the beam is taken into account is shown in Figure 2.

_{d}on the relative height of the beam section 2h/f

_{s}at various ratios M

_{B}/M

_{A}. Curve 1 corresponds to M

_{B}/M

_{A}= 0, curve 2 corresponds to M

_{B}/M

_{A}= 0.1, curve 3 corresponds to M

_{B}/M

_{A}= 1, curve 4 corresponds to M

_{B}/M

_{A}= 100. The static deflection f

_{s}was determined without considering the bimodularity of the beam material. As can be seen from Figure 2, the dynamic coefficient k

_{d}reaches its highest value at M

_{B}/M

_{A}= 0 (Curve 1), which corresponds to the case if the beam mass is not taken into account in the calculations.

_{B}» M

_{B}, curve 4), then the dynamic coefficient is the constant k

_{d}= 2.

_{dt}= k

_{d}·σ

_{t}, σ

_{dc}= k

_{d}·σ

_{c}, σ

_{da}= k

_{d}·σ

_{a}, where σ

_{dt}, σ

_{dc}, σ

_{da}, respectively, dynamic normal stresses arising in the tensile, compressed zone of the material beams and reinforcement; σ

_{t}, σ

_{c}, σ

_{a}, respectively, static normal stresses arising in the tensioned, compressed zone of the material of the beam and the reinforcement.

- (a)
- equilibrium equation for a heterogeneous beam, Equation (3):$${M}_{y}={M}_{yt}+{M}_{yc}+{M}_{ya}$$
- (b)
- deformation compatibility condition of a heterogeneous beam, Equation (4):$$\frac{1}{\rho}=\frac{1}{{\rho}_{t}}=\frac{1}{{\rho}_{c}}=\frac{1}{{\rho}_{a}}$$
_{y}, $\frac{1}{\rho}$ is the bending moment and beam curvature; M_{yt}, $\frac{1}{{\rho}_{t}}$ is the bending moment and curvature of the tensile zone beam; M_{yc}, $\frac{1}{{\rho}_{c}}$ is bending moment and curvature of the compressed zone beam; M_{ya}, $\frac{1}{{\rho}_{a}}$ is bending moment and curvature of rebars.

_{t}, A

_{t}are the normal stress and the cross-sectional area of the beam in the tensile zone, σ

_{c}, A

_{c}are the normal stress and the cross-sectional area of the beam in the compression, compression zone, σ

_{a}, A

_{a}are the normal stress and the cross-sectional area of the reinforcement.

_{t}is the tensile modulus of the material; E

_{c}is the elastic modulus of the material under compression; E

_{a}is the tensile modulus of reinforcement; ${J}_{y}^{t}$ is the moment of inertia about the neutral axis of the part of the cross-section that lies in the tension zone; ${J}_{y}^{c}$ is the moment of inertia about the neutral axis of the part of the cross-section that lies in the compression zone; ${J}_{y1}^{t}$ is the moment of inertia of the cross-section of the reinforcement, which lies in the tension zone, relative to its own central axis; ${J}_{y1}^{c}$ is the moment of inertia of the cross-section of the reinforcement, which lies in the compression zone, relative to its own central axis; n

_{t}is the number of reinforcement bars in the tension zone; n

_{c}is the number of reinforcement bars in the compression zone; ${A}_{a}^{t}$ is the cross-sectional area of the reinforcement in the tension zone; ${A}_{a}^{c}$ cross-sectional area of reinforcement in the compression zone; c

_{t}distance from the reinforcement in the tension zone to the neutral axis; c

_{c}is the distance from the reinforcement in the compression zone to the neutral axis.

_{yt}is the static moment of the cross-sectional area of the expanding zone relative to the neutral line, S

_{yc}is the static moment of the cross-sectional area of the contracting zone relative to the neutral line, S

_{ya}is the static moment of the area of reinforcement bars relative to the neutral line

_{yzt}(x) = 0, I

_{yzc}(x) = 0 since the cross-section is considered symmetric about the y-axis. Then Equation (13) is satisfied identically, and the position of the neutral line is determined from Equation (12).

_{c}is found from the condition h = h

_{c}+ h

_{t}, Equation (16):

_{c}is the height of the compressed area; h

_{t}is the height of the stretched area, h is the height of the cross-section, b is the width of the cross-section $\frac{{E}_{c}}{{E}_{t}}$ = k.

_{t}, t

_{t}are the width and thickness of the cross-section flange in the tension zone, b

_{c}, t

_{c}are the width and thickness of the cross-section flange in the compression zone, n

_{c}, c

_{c}

_{t}, c

_{t}are the the number and distance to the neutral line of reinforcing bars in the tensioned zone, A

_{c}, A

_{t}are the cross-sectional areas of the reinforcing bars in the compressed and tensile zones, respectively.

## 3. Results

_{A}= 100 kg from a height h = 4.00 cm to the middle of a reinforced beam made of a bimodular material were calculated.

#### 3.1. A Fiber-Reinforced Concrete Beam

_{c}= 2.25·10

^{3}MPa; E

_{t}= 5.0·10

^{3}MPa. Drop weight M

_{A}= 100 kg; drop height: h = 4 cm, the number of reinforcement bars n

_{t}= 2, n

_{c}= 2. Maximum dynamic stresses arising in beams of rectangular, T- and I-shape, considering bimodularity are presented in Table 1.

_{dt}increased by 1.5%, σ

_{dc}increased by 53%; for a T-beam σ

_{dt}increased by 8%, σ

_{dc}increased by 58%; for an I-beam, σ

_{dt}increased by 7%, σ

_{dc}increased by 51%.

#### 3.2. AFB-1 Concrete Beam

_{c}= 2.25·10

^{3}MPa; E

_{t}= 1.75·10

^{4}MPa. Drop weight M

_{A}= 100 kg; drop height: h = 4 cm, the number of reinforcement bars n

_{t}= 2, n

_{c}= 2. Maximum dynamic stresses arising in beams of rectangular, T- and I-shape, with bimodularity are presented in Table 3.

_{dc}was greater than σ

_{dt}, in contrast to beams made of fiber-reinforced concrete (Table 1). In concrete beams AFB-1, the largest maximum normal stresses occurred in the T-bar, the smallest in the rectangular cross-section. For fiber-reinforced concrete beams, σ

_{dt}was greater than σ

_{dc}, while for a lightweight concrete beam, σ

_{dt}was less than σ

_{dc}. Maximum dynamic stresses arising in beams of rectangular, T- and I-shape, without bimodularity are presented in Table 4.

_{dt}decreases within 5%, σ

_{dc}decreases within 55%, both with regard to the mass of the beam and without taking into account the mass of the beam.

#### 3.3. A Fiber Concrete Beam Reinforced in the Tensile Zone

_{c}= 2.25·10

^{3}MPa; E

_{t}= 5.0·10

^{3}MPa. Drop weight M

_{A}= 100 kg; drop height: h = 4 cm, the number of reinforcement bars n

_{t}= 2, n

_{c}= 0. Consider the influence of the location of the reinforcement on the value of the maximum dynamic normal stresses. Maximum dynamic stresses arising in a beam of rectangular, T-shaped and I-beams, considering bimodularity are presented in Table 5.

#### 3.4. Influence of the Width b_{t} and the Thickness of the Flanget_{t} of the I-Beam on the Values of the Maximum Normal Stresses

_{c}= 2.25·10

^{3}MPa; E

_{t}= 5.0·10

^{3}MPa. Drop weight M

_{A}= 100 kg, drop height: h=4 cm, the number of reinforcement bars n

_{t}= 2, n

_{c}= 2. Section dimensions: h = 90 cm, b

_{t}= 30 cm, b

_{c}= 30 cm, t

_{t}= 8 cm, t

_{c}= 8 cm, d

_{c}= 10 cm. Dependences of dynamic stresses on the width of the I-beam flange bp are shown in Figure 4.

_{t}practically did not affect the maximum compressive stress. The maximum tensile stress decreased in direct proportion to the increase in the flange length of the stretched zone.

_{t}are shown in Figure 5.

#### 3.5. Influence of the Dimensions of the Bottom Flange, the Number of Reinforcing Bars in the Tensioned Zone and the Elastic Moduli on the Value of the Maximum Tensile and Compressive Normal Stresses Upon Impact in the Example of an I-Beam

_{dt}(Figure 6). Therefore, it is important to know how the cumulative consideration of various geometrical and mechanical factors influences the values of the maximum tensile and compressive normal stresses upon impact arising in the cross-sections of the beam.

_{t}and the number of reinforcement bars in the tension zone of a fiber-reinforced concrete beam, taking into account the beam mass and bimodularity, are shown in Figure 7 and Table 6.

_{t}by 53% reduced σ

_{dt}by 18% with a constant number of reinforced bars. An increase in the number of reinforced bars by a factor of 3 reduced σ

_{dt}by 27% at a constant flange size. A simultaneous increase in the size of the shelf by 1.8 times and an increase in the number of reinforced flanges by 3 times reduced σ

_{dt}by 45%, while σ

_{dc}increased by only 1.6% (Figure 6, Table 6)

_{t}and the number of reinforcement bars in the tension zone of a fiber-reinforced concrete beam, taking into account the mass of the beam and excluding bimodularity, are shown in Figure 8 and Table 7.

_{t}of the I-beam and the number of reinforcement rods in the tension zone of the fiber-reinforced concrete beam, taking into account the beam mass and bimodularity are presented in Figure 9 and Table 8.

_{t}in the stretched zone, σ

_{dt}decreased, and σ

_{dc}increased. A simultaneous increase in t

_{t}by 1.5 times and in the number of reinforcement rods by 3 times decreased σ

_{dt}by 45%, while σ

_{dc}increased by only 3%.

_{t}of the I-beam and the number of reinforcement rods in the tension zone of a fiber-reinforced concrete beam, taking into account the mass of the beam and without taking into account the bimodularity, are shown in Figure 10 and Table 9.

_{dc}and a decrease in σ

_{dt}, but $\left[{\sigma}_{dc}\right]>\left[{\sigma}_{dt}\right]$ (Figure 10, Table 9).

## 4. Discussion

_{t}> E

_{c}in a T-section beam, regardless of the number of rods, σ

_{dt}differs from σ

_{dc}by only 0.7% (Table 1 and Table 4). At E

_{t}= E

_{c}σ

_{dc}> σ

_{dt}by 57% (Table 2). In the case of E

_{t}< E

_{c}σ

_{dc}> σ

_{dt}by 13% (Table 3). The data presented confirm the need to take into account bimodularity when calculating the strength of beams under shock loads.

_{t}> E

_{c}and the same number of reinforcing bars located in the compressed and tensile zones σ

_{dc}< σ

_{dt}by 45% for a rectangular cross-section and 15% for an I-beam cross-section (Table 1). In the absence of reinforcing bars in the compressed zone, the difference between the maximum tensile and compressive stresses increases: σ

_{dc}< σ

_{dt}by 57% for a rectangular cross-section and by 30% for an I-section (Table 5), the value of the maximum tensile normal stress in comparison with the arrangement of reinforcing bars in both the compressed and stretched zones, the increase for a rectangular cross-section is 9% and for I-beams by 13% (Table 1 and Table 5). As can be seen from the graphs shown in Figure 4 and Figure 5, by varying the dimensions of the cross-section, it is possible to significantly reduce the values of the maximum tensile normal stresses of the cross-section that arise in the beam under the action of shock loads, taking into account the bimodularity of the material.

_{dt}decreases within 5%, σ

_{dc}decreases within 55%, both taking into account the mass of the beam and excluding the mass of the beam. Chen et al. in [40] showed a similar effect when bimodularity not only displaced the neutral axis, which is obvious, but also changed the natural frequency of the beam vibrations.

## 5. Conclusions

_{t}< E

_{c}, the smallest tensile and compressive dynamic normal stresses arise in a beam of rectangular cross-section with and without regard to the mass of the beam (Table 1 and Table 2). When designing a reinforced concrete structure, however, it is necessary to take into account the weight of the beam, since although the bearing capacity of a rectangular cross-section beam is higher than that of beams of T- and I-cross-sections, the reduced mass M is twice as large as the reduced mass of beams of I- and T-cross-sections, and this is an additional load on the columns, walls and foundations.

_{t}> E

_{c}, the smallest tensile and compressive normal stresses also arise in a beam of rectangular cross-section with and without regard to the mass of the beam (Table 3 and Table 4). As for lightweight concrete, the reduced mass M

_{B}of a rectangular beam is twice as large as the reduced mass of I-beams and T-beams. If the normal tensile stresses σ

_{dt}differ little in magnitude, then the normal compressive stresses σ

_{dc}in a rectangular cross-section beam are 65% less than in a T-cross-section beam, and by 22% less than in an I-beam cross-section.

_{dt}with a slight increase in σ

_{dc}. The number and location of reinforcement bars also affect the stress state of beams. The arrangement of the rods only in the tensile zone has practically no effect on σ

_{dc}, while σ

_{dt}increases by 8% (Table 5).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Calculation scheme: (

**b**) beam width in tension zone; (

_{t}**b**) beam width in the compressed zone; (

_{c}**h**), (

**h**), (

_{t}**h**) beam height, beam height in tension and compressed zones; (

_{c}**t**), (

_{t}**t**)the dimensions of the flanges in the stretched and compressed zones; (

_{c}**d**) web width.

**Figure 2.**Dependence of the dynamic coefficient k

_{d}on the relative height of the beam section 2h/f

_{s}at various ratios of the beam mass to the mass of the falling load M

_{B}/M

_{A}: (

**1**) M

_{B}/M

_{A}= 0; (

**2**) M

_{B}/M

_{A}= 0.1; (

**3**) M

_{B}/M

_{A}= 1.0; (

**4**) M

_{B}/M

_{A}= 100.0.

**Figure 3.**Examples of simply supported reinforced concrete beams of T, I, and rectangular cross-sections used in building structures.

**Figure 4.**Dependences of dynamic stresses on the width of the I-beam flange b

_{t}in the tension zone without taking into account the mass of the fiber-reinforced concrete beam: (

**1**) dynamic tensile stress; (

**2**) dynamic compressive stress.

**Figure 5.**Dependences of dynamic stresses on the thickness of the I-beam flange t

_{t}in the tension zone without taking into account the mass of the fiber-reinforced concrete beam, (

**1**) dynamic tensile stress; (

**2**) dynamic compressive stress.

**Figure 6.**Destruction of a simply supported reinforced concrete beam under the action of concentrated forces.

**Figure 7.**Dependences of dynamic normal stresses on the width of the I-beam flange b

_{t}and the number of reinforcement rods in the tension zone of a fiber-reinforced concrete beam taking into account the beam mass and bimodularity at n

_{c}= 2, E

_{c}= 2250 MPa, E

_{t}= 5000 MPa: (

**1**) dynamic tensile stresses at n

_{t}= 2; (

**2**) dynamic compressive stress at n

_{t}= 2; (

**3**) dynamic tensile stresses at n

_{t}= 4; (

**4**) dynamic compressive stress at n

_{t}= 4; (

**5**) dynamic tensile stresses at n

_{t}= 6; (

**6**) dynamic compressive stresses at n

_{t}= 6.

**Figure 8.**Dependence of dynamic normal stresses on the width of the I-beam flange b

_{t}and the number of reinforcement rods in the tension zone of a fiber-reinforced concrete beam, taking into account the mass of the beam and without taking into account the bimodularity at n

_{c}= 2, E

_{c}= 5000 MPa, E

_{t}= 5000 MPa: (

**1**) dynamic tensile stresses at n

_{t}= 2; (

**2**) dynamic compressive stress at n

_{t}= 2; (

**3**) dynamic tensile stresses at n

_{t}= 4; (

**4**) dynamic compressive stress at n

_{t}= 4; (

**5**) dynamic tensile stresses at n

_{t}= 6; (

**6**) dynamic compressive stresses at n

_{t}= 6.

**Figure 9.**Dependences of the dynamic normal stresses on the thickness of the flange t

_{t}of the I-beam and the number of reinforcement rods in the tension zone of the fiber-reinforced concrete beam, taking into account the beam mass and bimodularity at n

_{c}= 2, E

_{c}= 2250 MPa, E

_{t}= 5000 MPa: (

**1**) dynamic tensile stresses at n

_{t}= 2; (

**2**) dynamic compressive stress at n

_{t}= 2; (

**3**) dynamic tensile stresses at n

_{t}= 4; (

**4**) dynamic compressive stress at n

_{t}= 4; (

**5**) dynamic tensile stresses at n

_{t}= 6; (

**6**) dynamic compressive stresses at n

_{t}= 6.

**Figure 10.**Dependence of the dynamic normal stresses on the thickness of the flange t

_{t}of the I-beam and the number of reinforcement bars in the tension zone of the fiber-reinforced concrete beam, taking into account the mass of the beam and without the bimodularity at n

_{c}= 2, E

_{c}= 5000 MPa, E

_{t}= 5000 MPa: (

**1**) dynamic tensile stresses at n

_{t}= 2; (

**2**) dynamic compressive stress at n

_{t}= 2; (

**3**) dynamic tensile stresses at n

_{t}= 4; (

**4**) dynamic compressive stress at n

_{t}= 4; (

**5**) dynamic tensile stresses at n

_{t}= 6; (

**6**) dynamic compressive stresses at n

_{t}=6.

**Table 1.**Maximum dynamic stresses in beams of rectangular, T- and I-shape, considering bimodularity. n

_{t}= 2, n

_{c}= 2, E

_{t}= 5000 MPa; E

_{C}= 2250 MPa.

Beam Material | Sectional Shape | |||||
---|---|---|---|---|---|---|

Rectangular | T-Beam | I-Beam | ||||

Fiber-foam concrete material | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa |

Without considering the mass of the beam | 1.685 | 1.157 | 1.708 | 1.824 | 1.711 | 1.489 |

Considering the mass of the beam | 0.192 | 0.132 | 0.428 | 0.457 | 0.381 | 0.332 |

**Table 2.**Maximum dynamic stresses arising in beams of rectangular, T- and I-shape, without bimodularity. E

_{t}= 5000 MPa; E

_{c}= 5000 MPa.

Beam Material | Sectional Shape | |||||
---|---|---|---|---|---|---|

Rectangular | T-Beam | I-Beam | ||||

Fiber-foam concrete material | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa |

Without considering the mass of the beam | 1.710 | 1.770 | 1.825 | 2.876 | 1.828 | 2.243 |

Considering the mass of the beam | 0.190 | 0.197 | 0.452 | 0.712 | 0.402 | 0.493 |

**Table 3.**Maximum dynamic stresses in beams of rectangular, T- and I-shape, considering bimodularity n

_{t}= 2, n

_{c}= 2, E

_{t}= 17500 MPa; E

_{c}= 7500 MPa.

Beam Material | Sectional Shape | |||||
---|---|---|---|---|---|---|

Rectangular | T-Beam | I-Beam | ||||

Fiber-foam concrete material | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa |

Without considering the mass of the beam | 2.157 | 3.379 | 2.439 | 5.585 | 2.438 | 4.106 |

Considering the mass of the beam | 0.077 | 0.120 | 0.210 | 0.480 | 0.180 | 0.303 |

**Table 4.**Maximum dynamic stresses arising in beams of rectangular, T- and I-shape, without bimodularity. E

_{t}= 7500 MPa; E

_{c}= 7500 MPa.

Beam Material | Sectional Shape | |||||
---|---|---|---|---|---|---|

Rectangular | T-Beam | I-Beam | ||||

Fiber-foam concrete material | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa |

Without considering the mass of the beam | 2.143 | 2.193 | 2.315 | 3.611 | 2.310 | 2.790 |

Considering the mass of the beam | 0.081 | 0.083 | 0.205 | 0.320 | 0.176 | 0.212 |

**Table 5.**Maximum dynamic stresses arising in beams of rectangular, T- and I-shape, considering bimodularity. n

_{t}= 2, n

_{c}= 0, E

_{t}= 5000 MPa; E

_{c}= 2250 MPa.

Beam Material | Sectional Shape | |||||
---|---|---|---|---|---|---|

Rectangular | T-Beam | I-Beam | ||||

Fiber-foam concrete material | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa | ${\sigma}_{dt}$, MPa | ${\sigma}_{dc}$, MPa |

Without considering the mass of the beam | 1.834 | 1.168 | 1.942 | 1.826 | 1.934 | 1.485 |

Considering the mass of the beam | 0.210 | 0.134 | 0.488 | 0.459 | 0.432 | 0.332 |

**Table 6.**Dependences of dynamic normal stresses on the width of the I-beam flange b

_{t}and the number of reinforcement bars in the tension zone of a fiber-reinforced concrete beam, taking into account the beam mass and bimodularity at n

_{c}= 2.

Flange Width | ${\mathit{n}}_{\mathit{t}}=2$ | ${\mathit{n}}_{\mathit{t}}=4$ | ${\mathit{n}}_{\mathit{t}}=6$ | |||
---|---|---|---|---|---|---|

${\mathit{b}}_{\mathit{t}},\text{}\mathbf{cm}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}\text{}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ |

30 | 0.474 | 0.359 | 0.416 | 0.361 | 0.372 | 0.362 |

32 | 0.463 | 0.36 | 0.408 | 0.361 | 0.366 | 0.363 |

34 | 0.453 | 0.36 | 0.401 | 0.361 | 0.36 | 0.363 |

36 | 0.443 | 0.36 | 0.393 | 0.362 | 0.354 | 0.363 |

38 | 0.434 | 0.36 | 0.386 | 0.362 | 0.349 | 0.364 |

40 | 0.426 | 0.361 | 0.38 | 0.362 | 0.343 | 0.364 |

42 | 0.417 | 0.361 | 0.373 | 0.363 | 0.338 | 0.364 |

44 | 0.409 | 0.361 | 0.367 | 0.363 | 0.333 | 0.365 |

46 | 0.402 | 0.362 | 0.361 | 0.363 | 0.328 | 0.365 |

**Table 7.**Dependences of dynamic normal stresses on the width of the I-beam flange b

_{t}and the number of reinforcement bars in the tension zone of a fiber-reinforced concrete beam, taking into account the mass of the beam and without taking into account the bimodularity at n

_{c}= 2.

Flange Width | ${\mathit{n}}_{\mathit{t}}=2$ | ${\mathit{n}}_{\mathit{t}}=4$ | ${\mathit{n}}_{\mathit{t}}=6$ | |||
---|---|---|---|---|---|---|

${\mathit{b}}_{\mathit{t}}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ |

30 | 0,499 | 0,536 | 0,445 | 0,537 | 0,403 | 0,539 |

32 | 0,489 | 0,536 | 0,437 | 0,537 | 0,397 | 0,539 |

34 | 0,479 | 0,536 | 0,43 | 0,538 | 0,391 | 0,54 |

36 | 0,47 | 0,537 | 0,423 | 0,538 | 0,385 | 0,54 |

38 | 0,462 | 0,537 | 0,416 | 0,538 | 0,38 | 0,54 |

40 | 0,453 | 0,537 | 0,41 | 0,539 | 0,374 | 0,541 |

42 | 0,445 | 0,537 | 0,403 | 0,539 | 0,369 | 0,541 |

44 | 0,438 | 0,538 | 0,397 | 0,54 | 0,364 | 0,542 |

46 | 0,43 | 0,538 | 0,391 | 0,54 | 0,359 | 0,542 |

**Table 8.**Dependences of dynamic normal stresses on the thickness of the flange t

_{t}of the I-beam and the number of reinforcement bars in the tension zone of a fiber-reinforced concrete beam, taking into account the beam mass and bimodularity at n

_{c}= 2.

Flange Thickness | ${\mathit{n}}_{\mathit{t}}$ = 2 | ${\mathit{n}}_{\mathit{t}}$ = 4 | ${\mathit{n}}_{\mathit{t}}$ = 6 | |||
---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{t}}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}}$, MPa | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}}$, MPa | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}}$, MPa | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}}$, MPa | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}}$, MPa | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}}$, MPa |

8 | 0.474 | 0.359 | 0.416 | 0.361 | 0.372 | 0.362 |

10 | 0.454 | 0.361 | 0.402 | 0.362 | 0.361 | 0.364 |

12 | 0.438 | 0.363 | 0.39 | 0.364 | 0.352 | 0.365 |

14 | 0.425 | 0.364 | 0.38 | 0.365 | 0.344 | 0.366 |

16 | 0.415 | 0.366 | 0.373 | 0.367 | 0.339 | 0.367 |

18 | 0.407 | 0.367 | 0.367 | 0.368 | 0.335 | 0.369 |

20 | 0.401 | 0.369 | 0.363 | 0.369 | 0.332 | 0.369 |

22 | 0.396 | 0.37 | 0.36 | 0.37 | 0.33 | 0.37 |

24 | 0.393 | 0.371 | 0.358 | 0.37 | 0.329 | 0.37 |

**Table 9.**Dependences of dynamic normal stresses on the thickness of the flange t

_{t}of the I-beam and the number of reinforcement bars in the tension zone of the fiber-reinforced concrete beam, taking into account the mass of the beam and without the bimodularity at n

_{c}= 2.

Flange Thickness | ${\mathit{n}}_{\mathit{t}}=2$ | ${\mathit{n}}_{\mathit{t}}=4$ | ${\mathit{n}}_{\mathit{t}}=6$ | |||
---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{t}}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{t}},\text{}\mathbf{MPa}$ | ${\mathit{\sigma}}_{\mathit{d}\mathit{c}},\text{}\mathbf{MPa}$ |

8 | 0.499 | 0.536 | 0.445 | 0.537 | 0.403 | 0.539 |

10 | 0.479 | 0.538 | 0.43 | 0.539 | 0.391 | 0.541 |

12 | 0.463 | 0.54 | 0.417 | 0.541 | 0.381 | 0.543 |

14 | 0.449 | 0.542 | 0.406 | 0.543 | 0.372 | 0.545 |

16 | 0.437 | 0.545 | 0.397 | 0.546 | 0.365 | 0.547 |

18 | 0.428 | 0.547 | 0.39 | 0.548 | 0.359 | 0.549 |

20 | 0.42 | 0.55 | 0.384 | 0.55 | 0.354 | 0.551 |

22 | 0.413 | 0.552 | 0.379 | 0.552 | 0.35 | 0.553 |

24 | 0.408 | 0.554 | 0.375 | 0.554 | 0.347 | 0.555 |

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## Share and Cite

**MDPI and ACS Style**

Beskopylny, A.; Kadomtseva, E.; Meskhi, B.; Strelnikov, G.; Polushkin, O.
Influence of the Cross-Sectional Shape of a Reinforced Bimodular Beam on the Stress-Strain State in a Transverse Impact. *Buildings* **2020**, *10*, 248.
https://doi.org/10.3390/buildings10120248

**AMA Style**

Beskopylny A, Kadomtseva E, Meskhi B, Strelnikov G, Polushkin O.
Influence of the Cross-Sectional Shape of a Reinforced Bimodular Beam on the Stress-Strain State in a Transverse Impact. *Buildings*. 2020; 10(12):248.
https://doi.org/10.3390/buildings10120248

**Chicago/Turabian Style**

Beskopylny, Alexey, Elena Kadomtseva, Besarion Meskhi, Grigory Strelnikov, and Oleg Polushkin.
2020. "Influence of the Cross-Sectional Shape of a Reinforced Bimodular Beam on the Stress-Strain State in a Transverse Impact" *Buildings* 10, no. 12: 248.
https://doi.org/10.3390/buildings10120248