# Buckling of Cracked Euler–Bernoulli Columns Embedded in a Winkler Elastic Medium

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## Abstract

**:**

## 1. Introduction

## 2. Euler–Benoulli Column Model in an Elastic Medium

#### 2.1. Theoretical Formulation of an Intact Column

#### 2.2. Problem Formulation in Cracked Columns

- Continuity in deflection:

- Continuity in bending moment:

- Continuity in shear force:

- Jump in the slope deflection:

## 3. Direct Solution

## 4. Numerical Results

#### 4.1. Influence of the Crack on the Buckling Load

#### 4.2. Influence of the Crack and Elastic Medium on the Buckling Load

^{4}, and a Young’s modulus $E=200$ GPa [29] is considered. The values of the critical buckling load, ${P}_{c}$, obtained for different values of the dimensionless parameter representative of the Winkler medium, ${K}_{w}$ [0, 5, 10, 50, 100], along with those calculated by Jančo in 2013 [29] using the analytical solution proposed for the simply supported column (Equation (22)), are presented in Table 2:

#### 4.3. Combined Influence of the Crack and the Elastic Medium on the Buckling Load

#### 4.3.1. Simply Supported Column

#### 4.3.2. Clamped–Supported Column

#### 4.3.3. Clamped–Clamped Column

#### 4.3.4. Cantilever Column

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**Left**): Cracked Euler–Bernoulli column in an elastic medium of Winkler type. (

**Right**): Equivalent model with elastic spring for the cracked section.

**Figure 3.**Variation of the first normalised critical buckling load with $(a/W)$ for different boundary conditions and $\beta =0.25$.

**Figure 4.**Variation of the first normalised critical buckling load with $(a/W)$ for different boundary conditions and $\beta =0.50$.

**Figure 5.**Variation of the first normalised critical buckling load with ${K}_{w}$ and different boundary conditions for a non-cracked beam.

**Figure 6.**Simply supported column, $\beta =0.25$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

**Figure 7.**Simply supported column, $\beta =0.5$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

**Figure 8.**Clamped–supported column, $\beta =0.25$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

**Figure 9.**Clamped–supported column, $\beta =0.5$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

**Figure 10.**Clamped–clamped column, $\beta =0.25$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

**Figure 11.**Position $\xi $ at which the bending moment becomes null in a clamped–clamped column, for different crack lengths $(a/W)$ and ${K}_{w}$.

**Figure 12.**Clamped–clamped column, $\beta =0.5$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

**Figure 13.**Cantilever column, $\beta =0.25$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

**Figure 14.**Cantilever column, $\beta =0.5$. Variation of the first normalised critical buckling load with $(a/W)$ and for different ${K}_{w}$.

Type of Support | Restrictions |
---|---|

Simple support | $V=\overline{M}=0$ |

Fixed support | $V=\theta =0$ |

Free end | $\overline{M}=\overline{Q}=0$ |

${\mathit{P}}_{\mathit{c}}$ [N] | ${\mathit{K}}_{\mathit{w}}$ | ||||
---|---|---|---|---|---|

0 | 5 | 10 | 50 | 100 | |

Theoretical [29] | 1644.94 | 1729.36 | 1814.82 | 2489.28 | 3334.65 |

Proposed | 1644.94 | 1729.44 | 1813.94 | 2489.95 | 3334.97 |

Error [%] | 0 | 0 | 0.01 | 0.03 | 0.04 |

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**MDPI and ACS Style**

Loya, J.A.; Santiuste, C.; Aranda-Ruiz, J.; Zaera, R.
Buckling of Cracked Euler–Bernoulli Columns Embedded in a Winkler Elastic Medium. *Math. Comput. Appl.* **2023**, *28*, 87.
https://doi.org/10.3390/mca28040087

**AMA Style**

Loya JA, Santiuste C, Aranda-Ruiz J, Zaera R.
Buckling of Cracked Euler–Bernoulli Columns Embedded in a Winkler Elastic Medium. *Mathematical and Computational Applications*. 2023; 28(4):87.
https://doi.org/10.3390/mca28040087

**Chicago/Turabian Style**

Loya, José Antonio, Carlos Santiuste, Josué Aranda-Ruiz, and Ramón Zaera.
2023. "Buckling of Cracked Euler–Bernoulli Columns Embedded in a Winkler Elastic Medium" *Mathematical and Computational Applications* 28, no. 4: 87.
https://doi.org/10.3390/mca28040087