# Buckling of Tapered Heavy Columns with Constant Volume

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Solution Methods

## 4. Results and Discussion

^{3}, $E=20$ GPa, and $\lambda =0$ (i.e., without self-weight, with varying end conditions, a side number $k$, and taper ratio $n$) are compared. The results of this study and those presented by Riley [22] are in good agreement (0.3% error). Second, the buckling self-weight parameters ${c}_{1}^{2}{c}_{3}^{2}\Gamma /{c}_{2}\text{}(=\gamma A{L}^{3}/EI)$ for $n=1$ (i.e., uniform column) in this study and previous studies [2,6,8] with various end conditions are compared. Note that the parameters of ${c}_{1}^{2}{c}_{3}^{2}\Gamma /{c}_{2}$ for the buckling self-weight parameter $\Gamma $ have also been adopted in the literature [2,6]. If $n=1$, then the parameters are the same, regardless of $k$. The results of this study and the references [6] agree well, and the results of this study and the references [2,8] are the same to within five significant figures. Thus, the analytical theories and numerical methods developed in this study are validated when considering all of the column parameters, including the end conditions, $k$, and $n$.

^{3}, $E=20$ GPa, and $\gamma =23$ kN/m

^{3}for a concrete material. In the case of (a) self-weight, $\sigma $ decreases along the column axis and $\sigma $ is maximized as ${\sigma}_{max}={\sigma}_{t}$ at $\xi =0$, which is the expected behavior. For the buckling column length $L$, the C-C column is the longest and the C-F column is the shortest, which is the expected behavior. Considering the ultimate stress of ${\sigma}_{u}=40$ MPa for the concrete material, ${\sigma}_{t}$ values between 0.825 and 1.249 MPa are relatively small compared to ${\sigma}_{u}$, meaning heavy column ruptures are caused by buckling, rather than fracturing. In the case of (b), the external load of $B=5$ MN, $\sigma $ increases along the column axis, where $\sigma $ is minimized as ${\sigma}_{min}={\sigma}_{t}$ and maximized as ${\sigma}_{max}={\sigma}_{h}$ because the column is subjected to an external load and the column area decreases (i.e., $n=0.5$). Additionally, the buckling length $l$ of the C-C column is the largest and that of the C-F column is the smallest. Even when an external load is applied to the column, the column ruptures as a result of buckling, rather than fracturing, just as in the case of self-weight buckling.

^{3}, with the other parameters kept constant. The buckling behavior of steel columns is similar to that of concrete columns. It is noteworthy that the self-eight buckling length $L$ (see Equation (36)) and buckling length $l$ (see Equation (34)) of the steel column do not increase significantly beyond those of the concrete column, despite the Young’s moduli of $E=20$ GPa for the concrete column and $E=210$ GPa for the steel column. Note that under the same column parameters given above, if the length of a particular column is shorter than the tallest length $L$ or $l$ shown in Table 5 and Table 6, the column is safe from column buckling. For example, the H-H column with a specific column length of 10 m $(<L=62.64\text{}\mathrm{m})$ will not be buckling. The corresponding circumradii of the column are ${r}_{t}=0.752$ m and ${r}_{h}=0.376$ m ($n=0.5,V=10{\text{}\mathrm{m}}^{3}$ and column length $=$ 10 m), which are practical in real engineering systems. The column stress ${\sigma}_{t}\left(=\gamma V/{A}_{t}\right)$ at the toe end is computed as ${\sigma}_{t}=0.129$ MPa $(<{\sigma}_{all}=2\text{}\mathrm{MPa},\text{}\mathrm{approximately})$, and therefore this column is safe from self-weight buckling.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of a buckled heavy column with a $k$-sided regular polygon cross-section and (

**b**) forces imposed on a buckled element.

**Figure 2.**Convergence analysis for a suitable ${n}_{e}\text{}\left(=1/\Delta \xi \right)$ in the Runge–Kutta scheme.

**Figure 7.**Examples of buckling stress $\sigma $ for a concrete column with a circular cross-section, $n=0.5$, $V=10$ m

^{3}, $E=20$ GPa, and $\gamma =23$ kN/m

^{3}: (

**a**) self-weight buckling ($P=0$) and (

**b**) external load buckling ($B=5$ MN).

$\mathbf{End}\text{}\mathbf{Condition},\text{}\mathit{k}$$\text{}\mathbf{and}\text{}\mathit{n}$ | $\mathbf{Buckling}\text{}\mathbf{Load}\text{}\mathit{B}$ in MN | ||
---|---|---|---|

This Study | Riley [22] | % Error | |

H-H, $k=3$, $n=0.4$ | 49.95 | 49.95 | 0. |

H-C, $k=4$, $n=0.5$ | 109.88 | 109.72 | 0.15 |

C-F, $k=5$, $n=0.6$ | 22.07 | 22.01 | 0.27 |

C-H, $k=6$, $n=0.7$ | 132.39 | 132.33 | 0.05 |

C-C, $k=\infty $, $n=0.8$ | 270.17 | 270.17 | 0. |

^{3}, and $E=20$ GPa for the concrete column.

**Table 2.**Comparisons of buckling self-weight parameter $\Gamma $ in terms of ${c}_{1}^{2}{c}_{3}^{2}\Gamma /{c}_{2}$ for $n=1$ *.

End Condition | ${\mathit{c}}_{1}^{2}{\mathit{c}}_{3}^{2}\mathit{\Gamma}/{\mathit{c}}_{2}$ | ||||
---|---|---|---|---|---|

H-H | H-C | C-F | C-H | C-C | |

This study | 18.5687 | 30.0094 | 7.8373 | 52.5007 | 74.6286 |

Duan and Wang [2] | 18.5687 | - | 7.8373 | 52.5007 | 74.6286 |

Wang and Ang [6] | 18.58 | - | 7.84 | 53.91 | 78.96 |

Lee and Lee [8] | 18.5687 | 30.0094 | 7.8373 | 52.5007 | 74.6286 |

$\mathit{k}$ | $\mathbf{Buckling}\text{}\mathbf{Load}\text{}\mathbf{Parameter}\text{}\mathit{\beta}$ | Ratio of | ||||
---|---|---|---|---|---|---|

H-H | H-C | C-F | C-H | C-C | ${\mathit{I}}_{\mathit{k}}/{\mathit{I}}_{\mathit{k}=3}$ | |

3 (triangle) | 0.3934 | 1.0123 | 0.1578 | 1.2814 | 2.5595 | 1.0 |

4 (square) | 0.2970 | 1.8193 | 0.1170 | 1.0896 | 2.1850 | 0.8660 |

5 (pentagon) | 0.2789 | 0.7833 | 0.1092 | 1.0538 | 2.1152 | 0.8410 |

6 (hexagon) | 0.2734 | 0.7722 | 0.1069 | 1.0428 | 2.0936 | 0.8333 |

$\infty $(circular) | 0.2688 | 0.7630 | 0.1049 | 1.0337 | 2.0759 | 0.8270 |

$\mathit{k}$ | $\mathbf{Buckling}\text{}\mathbf{Self}$-$\mathbf{Weight}\text{}\mathbf{Parameter}\text{}\mathit{\Gamma}$ | ||||
---|---|---|---|---|---|

H-H | H-C | C-F | C-H | C-C | |

3 (triangle) | 2.1405 | 3.2497 | 1.9883 | 8.0144 | 10.453 |

4 (square) | 1.8537 | 2.8143 | 1.7219 | 6.9407 | 9.0523 |

5 (pentagon) | 1.8002 | 2.7331 | 1.6772 | 6.7403 | 8.7911 |

6 (hexagon) | 1.7837 | 2.7080 | 1.6569 | 6.6787 | 8.7106 |

$\infty $(circular) | 1.7701 | 2.6874 | 1.6443 | 6.6278 | 8.6443 |

**Table 5.**Tallest buckling length $L$ and buckling stresses ${\sigma}_{t}$ of heavy columns * without compressive load.

$\mathit{L}$$\mathbf{and}\text{}{\mathit{\sigma}}_{\mathit{t}}$ | H-H | H-C | C-F | C-H | C-C |
---|---|---|---|---|---|

(a) Circular ($k=\infty $) concrete column | |||||

$L\text{}$(m) | 62.64 | 69.52 | 61.49 | 87.13 | 93.11 |

${\sigma}_{t}\text{}$(MPa) | 0.840 | 0.933 | 0.825 | 1.169 | 1.249 |

(b) Square ($k=4$) steel column | |||||

$L\text{}$(m) | 84.32 | 93.60 | 82.78 | 117.3 | 125.3 |

${\sigma}_{t}\text{}$(MPa) | 3.787 | 4.204 | 3.718 | 5.269 | 5.630 |

**Table 6.**Tallest buckling length * $l$ and buckling stresses ${\sigma}_{h}$ of heavy columns * with compressive load.

$\mathit{l}$$\mathbf{and}\text{}{\mathit{\sigma}}_{\mathit{h}}$ | H-H | H-C | C-F | C-H | C-C |
---|---|---|---|---|---|

(a) Circular concrete column | |||||

$l\text{}$(m) | 21.53 | 25.87 | 17.26 | 25.94 | 30.77 |

${\sigma}_{h}\text{}$(MPa) | 25.11 | 30.18 | 20.14 | 30.26 | 35.90 |

(b) Square steel column | |||||

$l\text{}$(m) | 39.41 | 46.97 | 31.66 | 47.34 | 55.94 |

${\sigma}_{h}\text{}$(MPa) | 45.98 | 54.80 | 36.93 | 55.24 | 65.26 |

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Lee, B.K.; Lee, J.K.
Buckling of Tapered Heavy Columns with Constant Volume. *Mathematics* **2021**, *9*, 657.
https://doi.org/10.3390/math9060657

**AMA Style**

Lee BK, Lee JK.
Buckling of Tapered Heavy Columns with Constant Volume. *Mathematics*. 2021; 9(6):657.
https://doi.org/10.3390/math9060657

**Chicago/Turabian Style**

Lee, Byoung Koo, and Joon Kyu Lee.
2021. "Buckling of Tapered Heavy Columns with Constant Volume" *Mathematics* 9, no. 6: 657.
https://doi.org/10.3390/math9060657