# Investigation of the Bearing Capacity of Transport Constructions Made of Corrugated Metal Structures Reinforced with Transversal Stiffening Ribs

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Analytical Method for Assessing the Stress-Strain State of Transport Facilities with CMS Reinforced with Stiffeners

_{1}and external S

_{2}surfaces of coating A are described correspondingly by the equations:

_{3}and external S

_{4}surfaces of coating B are described by the equations:

_{2}. The inner surface S

_{1}of coating A and the outer surface S

_{4}of coating B, except for lines (5), are free from loads. The inner surface S

_{3}of coating B and the side surfaces z = ±l of both coatings are also free from loads.

_{ij}(i, j = r, φ, z)—components of the stress tensor.

_{ij}(i, j = r, φ, z) and the displacement vector u

_{i}(i = r, φ, z) are related by the ratio:

_{ij}and deformations ε

_{ij}are related by Hooke’s law ratio, which has the form:

_{rr}+ ε

_{φφ}+ ε

_{zz}; $G=\frac{E}{2\left(1+\nu \right)}$—shear Kirchhof’s modulus; ν—Poisson’s ratio.

_{1}, S

_{3}, and S

_{4}, from which lines (5) are subtracted.

_{r}, n

_{z}, which correspond to the surfaces S

_{1}, S

_{2}, are determined by the formulas:

_{3}:

_{4}:

## 4. Initial Data for Calculation

_{II}and normal (axial) stiffness EA

_{II}, which are given in Table 2, are found based on calculations of the axial moment of inertia I

_{II}and the cross-sectional area A

_{II}, taking into account the transverse stiffening rib. The method for calculating the axial moment of inertia is given in the previous section of the article.

## 5. Method of Calculating Equivalent Loads from the Action of Mobile Transport Units

_{cal}= ${P}_{dyn}^{\mathrm{max}}$, and the influence of adjacent and further wheels is taken as an average dynamic pressure ${\overline{P}}_{dyn}^{}$. Given that the maximum dynamic pressure of the design wheel does not coincide with the maximum pressure of adjacent wheels (Figure 5), the action of two adjacent wheels located on both sides of the design wheel is taken into account in practical calculations, as further wheels have little effect on the load [31].

_{3}, which specifies the degree of influence on the calculated cross-section of every other wheel of the train (see Figure 4).

_{i}—ordinates of deflection lines influence of a rail in the cross sections of the track, located under the wheel loads from the axes of the carriage adjacent to the design axis.

_{i}depends on the value of kx

_{i}where x

_{i}corresponds to distances from the calculated cross section to each wheel, which are taken into account (Equation (15)):

_{s}—the distance between the axes of the sleepers; Q

_{p}

_{1}, Q

_{c}

_{2}, Q

_{c}

_{3}—respectively, the forces of rail pressure on the design and adjacent sleepers.

_{p}

_{1}, Q

_{c}

_{2}, Q

_{c}

_{3}, the ordinates η

_{i}of the influence lines of cross-section forces are situated depending on the distances to the sleepers under consideration (for which the pressure Q is determined) before taking into account the values of wheels (calculated or adjacent). Thus, the ordinates of influence lines η

_{2,3}take into account the influence of adjacent wheels on the calculated sleeper and correspond to the distances from the axis of the calculated sleeper to the adjacent wheels (left and right), i.e., for distances X

_{2}and X

_{3}(Figure 6).

_{p}

_{2}and η

_{p}

_{3}take into account loads influenced by the design wheel on the adjacent sleepers and are determined for the distance of the design load location from the considered adjacent and the design sleepers (left or right of it), i.e., for the distance l

_{s}= const, there is η

_{p}

_{2}= η

_{p}

_{3}.

_{c}

_{2}and η

_{c}

_{3}take into account the influence of the adjacent axes of the carriage on the load of the sleepers adjacent to the calculated one, i.e., correspond to the location distances respectively: for η

_{c}

_{2}—distances X

_{2}− l

_{s}and X

_{3}+ l

_{s}; for η

_{c}

_{3}—distances X

_{3}− l

_{s}and X

_{2}+ l

_{s}.

_{z}

_{0}under the center of the loaded rectangle is determined. According to work [14], the actual load from the rolling stock is converted by the formulas of stress distribution in the Boussinesq half-space into an equivalent linear load, which gives the same vertical stress at the upper level of CMS. Vertical stresses ${\sigma}_{v}$ at the depth z of an elastic body (vertically under the load), caused by linear load p in half-space, are determined by Equation (21) [32,33]:

## 6. Estimation of Bearing Capacity of a Corrugated Metal Structure Reinforced with Stiffening Ribs

#### 6.1. Calculation of the Axial Moment of Inertia of the Reinforced Structure

_{d}= e

_{g}= 0.5e.

_{x}.

#### 6.2. Research Results

## 7. Conclusions

- Based on CMS’s bearing capacity studies, the following conclusions can be drawn. It is established that the magnitude of loads from the rolling stock of railways on structures made of corrugated metal increases with increasing speed. At a locomotive speed of 2 × M62 40 km/h, the equivalent load is 99.42 kPa; at 80 km/h–101.54 kPa; at 100 km/h–102.63 kPa and at a speed of 120 km/h, 103.77 kPa.
- The stress that occurs in the corrugated metal sheets of the structure during double corrugation and the speed of locomotive 2 × M62 equal to 120 km/h is 38.92 MPa, and deformation—5.56 mm. In the absence of a stiffening rib at the thickness of a corrugated sheet of the structure of 6 mm, the stress is 47.03 MPa, and the deformation is 11.04 mm. Thus, corrugation increases the bearing capacity of corrugated metal structures.
- It is found that additional corrugation of metal corrugated structures leads to a reduction of stresses in the structure by up to 20% and deflections by 50% from the action of railway vehicles.
- Installing axial stiffeners in the most loaded places of a metal corrugated design is recommended.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Scheme of MCS reinforcement with stiffening ribs (

**a**) and the scheme of bending moments distribution (

**b**) (on the basis of [20]).

**Figure 4.**Cross section of the reinforced CMS: 1—crushed stone layer of the railway track; 2—soil crushed stone-sand compacting backfill; 3—loam backfill; 4—the basis of a corrugated metal structure.

**Figure 5.**Lines of deflection influence from the action of a single wheel load P

_{1}= 1, applied in the calculated cross section and during influence line loading with a three-axle bogie (on the basis of [31]).

**Figure 6.**Calculation scheme for determining pressure forces on sleepers from train load (on the basis of [31].)

**Figure 8.**Variant calculation of the values of equivalent forces from locomotive 2 × M62 locomotives with an elasticity modulus of the subrail base equal to 73.6 MPa.

**Figure 9.**Distribution of vertical pressure on the top of CMS from locomotive 2 × M62 at speeds: (

**a**)—40 km/h; (

**b**)—120 km/h.

**Figure 10.**The scheme of CMS reinforcement with a stiffening rib (on the basis of [34]).

**Figure 12.**Stress-strain state of a transport construction with CMS with corrugated sheet thickness of 6 mm: (

**a**)—distribution of stresses; (

**b**)—distribution of deformations.

**Figure 13.**Stress-strain state of a transport construction with CMS at double corrugation and at a thickness of 6 mm: (

**a**)—distribution of stresses; (

**b**)—distribution of deformations.

**Table 1.**Physical and mechanical features of a corrugated metal structure design (381 × 140 × 6 mm) and stiffening ribs.

Name of Characteristics | Physical and Mechanical Features |
---|---|

Corrugated metal structure without a stiffening rib | |

Young’s module E_{I}, MPa | 2.05 × 10^{5} |

Poisson’s ratio, ν_{I} | 0.3 |

Axial moment of inertia I_{I}, mm^{4}/mm | 18,141 |

Cross-sectional area A_{I}, mm^{2}/mm | 7.766 |

Corrugated metal structure reinforced with a stiffening rib | |

Young’s module E_{II}, MPa | 2.05 × 10^{5} |

Poisson’s ratio, ν_{II} | 0.3 |

Axial moment of inertia I_{II}, mm^{4}/mm | 37,432 |

Cross-sectional area A_{II}, mm^{2}/mm | 15.532 |

Name of a Mechanical Characteristic of a Soil Backfill | Area 1 (Crushed Ballast) | Area 2 (Crushed Stone and Sand Backfill) | Area 3 | Area 2 (Crushed Stone and Sand Backfill) |
---|---|---|---|---|

Specific weight, γ kN/m^{3} | 13.8 | 21.7 | 22.6 | 27.04 |

Poisson’s ratio, ν | 0.27 | 0.27 | 0.27 | 0.27 |

Coefficient of adhesion c, kPa | 0.1 | 5.0 | 13.0 | 30.0 |

Angle of internal friction, φ ° | 43.0 | 37.0 | 25.0 | 17.0 |

Dilatation angle, ψ ° | 0.0 | 1.0 | 1.0 | 2.0 |

E, MPa | 150.0 | 110.0 | 28.0 | 16.4 |

No. | Type of Initial Data | Dimensions | Meaning |
---|---|---|---|

1 | Rolling stock | - | 2 × M62 locomotives |

2 | Speed | km/h | 40, 80, 100 |

3 | Rail type | - | R65 |

4 | Sleeper type | - | reinforced concrete |

5 | Kind of a ballast bed | - | crushed stone and sand |

6 | Height of a soil backfill | m | 2.57 |

7 | Elasticity modulus of the subrail base | MPa | 73.6 |

8 | Distance between the axes of the sleepers | m | 0.5 |

9 | Coefficient α_{0} | - | 0.403 |

10 | Coefficient γ | - | 1.0 |

11 | Coefficient α_{1} | - | 0.931 |

12 | Coefficient ε | - | 0.332 |

13 | Coefficient β | - | 0.87 |

14 | Moment of rail inertia relative to the horizontal axis | cm^{4} | 3548 |

15 | Moment of rail resistance relative to the horizontal axis | cm^{3} | 436 |

16 | Base area of the substrate | cm^{2} | 262.5 |

17 | Sleeper’s length | cm | 270 |

18 | Width of the lower bed of a sleeper | cm | 27.5 |

No. | Speed, km/h | Stress, MPa | Deflection, mm |
---|---|---|---|

Corrugated metal structure without stiffening rib | |||

1 | 40 | 46.55 | 10.84 |

2 | 80 | 46.77 | 10.91 |

3 | 100 | 46.85 | 10.93 |

4 | 120 | 47.03 | 11.04 |

Corrugated metal structure with stiffening ribs | |||

1 | 40 | 38.53 | 5.01 |

2 | 80 | 38.70 | 5.32 |

3 | 100 | 38.81 | 5.41 |

4 | 120 | 38.92 | 5.56 |

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**MDPI and ACS Style**

Kovalchuk, V.; Sysyn, M.; Movahedi Rad, M.; Fischer, S.
Investigation of the Bearing Capacity of Transport Constructions Made of Corrugated Metal Structures Reinforced with Transversal Stiffening Ribs. *Infrastructures* **2023**, *8*, 131.
https://doi.org/10.3390/infrastructures8090131

**AMA Style**

Kovalchuk V, Sysyn M, Movahedi Rad M, Fischer S.
Investigation of the Bearing Capacity of Transport Constructions Made of Corrugated Metal Structures Reinforced with Transversal Stiffening Ribs. *Infrastructures*. 2023; 8(9):131.
https://doi.org/10.3390/infrastructures8090131

**Chicago/Turabian Style**

Kovalchuk, Vitalii, Mykola Sysyn, Majid Movahedi Rad, and Szabolcs Fischer.
2023. "Investigation of the Bearing Capacity of Transport Constructions Made of Corrugated Metal Structures Reinforced with Transversal Stiffening Ribs" *Infrastructures* 8, no. 9: 131.
https://doi.org/10.3390/infrastructures8090131