# Numerical Analysis of Cracked Double-Beam Systems

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Cracks: Modelling and Method of Solution

#### 3.1. Modelling of Cracks: An Overview of the Discrete Spring Model

#### 3.2. Method of Solution: Cell Discretisation Method (CDM)

**V**${\mathbf{v}}_{1}$ and ${\mathit{\varphi}}_{\mathbf{2}}=$

**V**${\mathbf{v}}_{2}$, where $\mathbf{V}$ is a rectangular transfer matrix with n− 1 rows and n columns.

**K**, with 2n rows and 2n columns, assumes the following form:

**Q**with 2n rows and 2n columns assumes the following form:

#### Boundary Conditions in Presence of a Crack

**K**must be modified as follows:

#### 3.3. Method of Solution: Finite Element Method (FEM)

## 4. Numerical Examples: Results and Discussion

#### 4.1. Case 1: Simply Supported-Simply Supported Double-Beam System in the Presence of a Crack on the Lower Beam and a Uniformly Distributed Load Applied to the Upper Beam

#### 4.2. Case 2: Simply Supported-Simply Supported Double-Beam System in the Presence of a Uniformly Distributed Load on the Upper Beam and a Crack on the Upper and Lower Beams

#### 4.3. Case 3: Effect of Elastic Constraints and Crack on the Static Behaviour of a Double-Beam System

#### 4.4. Case 4: Effect of Taper Ratio Coefficient and Crack on the Static Behaviour of a Double-Beam System

#### 4.5. Case 5: Effect of a Concentrated Force and Crack Located at the Mid-Span of the Upper and Lower Beams, Respectively

#### 4.6. Case 6: Effect of the Slenderness $\lambda $ on the Deflections for Both Upper and Lower Beams in the Presence of a Crack

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Double-beam system constrained at the ends by elastically flexible springs and in the presence of a Winkler-type elastic medium.

**Figure 7.**Displacements diagram for a clamped-clamped double-beam system with a crack on the lower beam and a uniformly distributed load applied to the upper beam.

**Figure 8.**Displacements diagram for a tapered double-beam system with different boundary conditions and for $\beta $ = 0.5.

**Figure 9.**Displacements diagram for a uniform double-beam system with different boundary conditions and for $\beta $ = 1.

**Figure 10.**Displacements diagram for a uniform double-beam system with different boundary conditions and for $\beta $ = 1.5.

**Figure 11.**Displacements diagram for a uniform double-beam system with different boundary conditions and for $\beta $ = 2.

**Table 1.**Maximum deflections values of two beams varying ${k}_{m}$ in the range $\left[{10}^{2},{10}^{3},{10}^{4},{10}^{5}\right]$ and with a crack acting on the lower beam, where C-FS = closed-form solution, C = CDM, F = FEM.

${\mathit{k}}_{\mathit{m}}$ | v${}_{1max}$ (C-FS) | v${}_{2max}$(C-FS) | v${}_{1max}$ (C) | v${}_{2max}$ (C) | v${}_{1max}$ (F) | v${}_{2max}$ (F) |
---|---|---|---|---|---|---|

10${}^{2}$ | 0.0348842 | 1.489 × 10${}^{-6}$ | 0.0348842 | 1.489 × 10${}^{-6}$ | 0.0348842 | 9.631 × 10${}^{-7}$ |

10${}^{3}$ | 0.0348756 | 0.0000149 | 0.0348754 | 0.0000149 | 0.0348756 | 0.0000149 |

10${}^{4}$ | 0.0347895 | 0.0001479 | 0.0347894 | 0.0001479 | 0.0347895 | 0.0001479 |

10${}^{5}$ | 0.0339827 | 0.0013952 | 0.0339826 | 0.0013952 | 0.0340380 | 0.0013964 |

**Table 2.**Maximum deflections values of two beams varying ${k}_{m}$ in the range $\left[{10}^{2},{10}^{3},{10}^{4},{10}^{5}\right]$ and in the absence of a crack, where C-FS = closed-form solution, C = CDM, F = FEM.

${\mathit{k}}_{\mathit{m}}$ | v${}_{1max}$ (C-FS) | v${}_{2max}$(C-FS) | v${}_{1max}$ (C) | v${}_{2max}$ (C) | v${}_{1max}$ (F) | v${}_{2max}$ (F) |
---|---|---|---|---|---|---|

10${}^{2}$ | 0.0348842 | 9.631 × 10${}^{-7}$ | 0.0348842 | 9.631 × 10${}^{-7}$ | 0.0348842 | 9.631 × 10${}^{-7}$ |

10${}^{3}$ | 0.0348756 | 9.626 × 10${}^{-6}$ | 0.0348754 | 9.626 × 10${}^{-6}$ | 0.0348756 | 9.626 × 10${}^{-6}$ |

10${}^{4}$ | 0.0347894 | 0.0000958 | 0.0347893 | 0.0000958 | 0.0347895 | 0.0000958 |

10${}^{5}$ | 0.0339723 | 0.0000913 | 0.0339721 | 0.0000913 | 0.0339271 | 0.0000913 |

**Table 3.**Maximum deflections values of two beams varying the non-dimensional rotational stiffness ${K}_{1\mathrm{RL}}$ = ${k}_{1\mathrm{RR}}$ = ${k}_{2\mathrm{RL}}$ = ${k}_{2\mathrm{RR}}$ = ${K}_{\mathrm{R}}$ in the range $\left[0,1,10,{10}^{2},{10}^{3},{10}^{4},{10}^{10}\right]$.

${\mathit{K}}_{\mathit{R}}$ | v${}_{1max}$ | v${}_{2max}$ |
---|---|---|

0 | 0.0291244 | 0.00890045 |

1 | 0.0220762 | 0.00509119 |

10 | 0.0107519 | 0.00119981 |

10${}^{2}$ | 0.00713561 | 0.000536748 |

10${}^{3}$ | 0.00669172 | 0.000474023 |

10${}^{4}$ | 0.00666452 | 0.000467836 |

10${}^{10}$ | 0.00664013 | 0.000467150 |

$\mathit{\beta}$ | v${}_{1max}$ | v${}_{2max}$ | Figure |
---|---|---|---|

0.5 | 0.051597 | 0.026785 | 7 |

1 | 0.013401 | 0.003783 | 8 |

1.5 | 0.005433 | 0.0010456 | 9 |

2 | 0.002697 | 0.0004161 | 10 |

**Table 5.**The maximum deflections value for different values of dimensional concentrated force F and for height of crack “a” = 0.09 m.

$\mathit{F}\left(\mathbf{N}\right)$ | v${}_{1max}$ | v${}_{2max}$ |
---|---|---|

10 | 1.87065 × 10${}^{-6}$ | 5.59848 × 10${}^{-7}$ |

10${}^{2}$ | 1.87065 × 10${}^{-5}$ | 5.59848 × 10${}^{-6}$ |

10${}^{3}$ | 1.87065 × 10${}^{-4}$ | 5.59848 × 10${}^{-5}$ |

10${}^{4}$ | 1.87065 × 10${}^{-3}$ | 5.59848 × 10${}^{-4}$ |

10${}^{5}$ | 1.87065 × 10${}^{-2}$ | 5.59848 × 10${}^{-3}$ |

**Table 6.**The maximum deflections for different values of crack depth and for a non-dimensional concentrated force ${F}_{t}=\left(0.107167\right)$.

$\mathit{\xi}$ | v${}_{1max}$ | v${}_{2max}$ |
---|---|---|

0.1 | 0.00184331 | 0.000397534 |

0.2 | 0.00184666 | 0.000417421 |

0.3 | 0.00185267 | 0.000453055 |

0.4 | 0.00186293 | 0.000513990 |

0.5 | 0.00187065 | 0.000559848 |

**Table 7.**Numerical comparison among the closed-form solutions (C-FS) and numerical results based on the CDM for different values of length $\lambda $ and ($\zeta $ = 0.5).

$\mathit{\lambda}$ | v_{1}(C-FS) | v_{2}(C-FS) | v${}_{1}\left(\mathbf{CDM}\right)$ | v${}_{2}\left(\mathbf{CDM}\right)$ |
---|---|---|---|---|

2 | 0.00001500 | 8.5277 × 10${}^{-9}$ | 0.000014500 | 8.5276 × 10${}^{-9}$ |

3 | 0.00007589 | 1.6061 × 10${}^{-7}$ | 0.00007589 | 1.6061 × 10${}^{-7}$ |

5 | 0.00058326 | 6.7375 × 10${}^{-6}$ | 0.00058326 | 6.7374 × 10${}^{-6}$ |

7 | 0.00221279 | 0.00007947 | 0.00221279 | 0.00007947 |

9 | 0.00588599 | 0.00048809 | 0.00588599 | 0.00048808 |

13.8889 | 0.0291243 | 0.00890046 | 0.02912429 | 0.00890037 |

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**MDPI and ACS Style**

De Rosa, M.A.; Lippiello, M.
Numerical Analysis of Cracked Double-Beam Systems. *Appl. Mech.* **2023**, *4*, 1015-1037.
https://doi.org/10.3390/applmech4040052

**AMA Style**

De Rosa MA, Lippiello M.
Numerical Analysis of Cracked Double-Beam Systems. *Applied Mechanics*. 2023; 4(4):1015-1037.
https://doi.org/10.3390/applmech4040052

**Chicago/Turabian Style**

De Rosa, Maria Anna, and Maria Lippiello.
2023. "Numerical Analysis of Cracked Double-Beam Systems" *Applied Mechanics* 4, no. 4: 1015-1037.
https://doi.org/10.3390/applmech4040052