# Calculation of Linear Buckling Load for Frames Modeled with One-Finite-Element Beams and Columns

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Linear Buckling Analysis of Canonical Cases Using a Single Element

## 3. Corrected Calculation of Critical Buckling Load in Canonical Cases Using a Single Element

Algorithm 1. Calculation of the corrected critical load factor of a bar. |

Calculate ${K}_{r}$ as a four-element refinement of the bar stiffness matrix; |

Calculate ${K}_{gr}$ as a four-element refinement of the bar geometric stiffness matrix; |

Calculate the projection matrix P in Equation (7); |

Solve the projected eigenvalue problem in Equation (8) for ${\lambda}_{c}$. |

## 4. Correction of Buckling Load Factor for Multiple-Bar-Element Frames

- The local correction of the buckling shape that we applied in the previous section to a single bar can be applied sequentially to all the bars in the frame, thereby obtaining local improvements of the buckling shape inside each bar.
- The reduced modeling of the refined bar of the previous section can account for the stiffness and geometric stiffness of the whole frame by expressing all the nodal displacements outside the bar being corrected as the product of the frame buckling shape $\varphi $ times an amplitude variable $\eta $.
- The overall corrected frame buckling factor can be obtained by dividing the frame stiffness quadratic form by the geometric stiffness quadratic form calculated for the corrected buckling shape.

Algorithm 2. Calculation of the corrected quadratic forms of a bar. |

Receive as inputs one-element frame magnitudes ${\varphi}^{T}K\varphi $, ${\varphi}^{T}{K}_{g}\varphi $; Receive as inputs one-element bar magnitudes ${\varphi}_{b}^{T}{K}_{b}{\varphi}_{b}$, ${\varphi}_{b}^{T}{K}_{gb}{\varphi}_{b}$, ${\varphi}_{b}$; |

Calculate ${K}_{r}$ as a four-element refinement of the bar stiffness matrix in bar reference frame; |

Perform the same step for ${K}_{gr}$; |

Transform ${\varphi}_{b}$ to bar reference frame as follows: ${\varphi}_{e}={R}_{b}^{T}{\varphi}_{b}$ (${R}_{b}$: bar rotation matrix); |

Calculate ${\varphi}_{i}$, ${\varphi}_{r}$ in Equations (5) and (19); |

Partition ${K}_{r}$ matrix according to external nodes (e) and internal ones (i) as follows: ${K}_{ee},{K}_{ei},{K}_{ie},{K}_{ii}$; |

Perform the same step with ${K}_{gr}$, ${K}_{gee},{K}_{gei},{K}_{gie},{K}_{gii}$; |

Calculate ${K}_{c}$, ${K}_{gc}$ in Equations (17) and (18); |

Solve projected eigenvalue problem in Equation (16) for ${\varphi}_{cb}$; |

Calculate corrected quadratic forms ${V}_{cb}$, ${V}_{gcb}$ by applying Equations (22)–(24). |

Algorithm 3. Calculation of improved corrected quadratic forms of a bar. |

Receive as inputs frame magnitudes ${V}_{c}$, ${V}_{cg}$; Receive as inputs bar magnitudes ${V}_{cb}$, ${V}_{cgb}$, ${\varphi}_{b}$; |

Calculate ${K}_{r}$ as a four-element refinement of the bar stiffness matrix in bar reference frame; |

Perform the same step for ${K}_{gr}$; |

Transform ${\varphi}_{b}$ to bar reference frame as follows: ${\varphi}_{e}={R}_{b}^{T}{\varphi}_{b}$ (${R}_{b}$: bar rotation matrix); |

Calculate ${\varphi}_{i}$, ${\varphi}_{r}$ in Equations (5) and (19); |

Partition ${K}_{r}$ matrix according to external nodes (e) and internal ones (i) as follows: ${K}_{ee},{K}_{ei},{K}_{ie},{K}_{ii}$ |

Perform the same step with ${K}_{gr}$, ${K}_{gee},{K}_{gei},{K}_{gie},{K}_{gii}$; |

Calculate ${K}_{c}$, ${K}_{gc}$ in Equations (28) and (29); |

Solve projected eigenvalue problem in Equation (16) for ${\varphi}_{cb}$; |

Update corrected quadratic forms ${V}_{cb}$, ${V}_{gcb}$ by applying Equations (22)–(24). |

Algorithm 4. Iterative calculation of corrected load critical factor of whole frame. |

Receive as inputs one-element frame magnitudes $K$, ${K}_{g}$, $\varphi $, $\lambda $; For all bars, receive as inputs one-element bar magnitudes ${K}_{b}$, ${K}_{gb}$, ${\varphi}_{b}$; |

Initialize ${V}_{c}={\varphi}^{T}K\varphi $, ${V}_{gc}={\varphi}^{T}{K}_{g}\varphi $, ${\lambda}_{c}=\lambda $; |

Perform For b = 1:nbars; If b is not in compression, go to End; If first iteration, set ${V}_{cb}={\varphi}_{b}^{T}{K}_{b}{\varphi}_{b}$, ${V}_{gcb}={\varphi}_{b}^{T}{K}_{gb}{\varphi}_{b}$; Update ${V}_{cb}$, ${V}_{gcb}$ with Algorithm 3; End; Update ${V}_{c}$, ${V}_{gc}$ with Equations (26) and (27); |

Calculate ${\lambda}_{c}={V}_{c}/{V}_{gc}$; While relative change in ${\lambda}_{c}$ does not fall below 0.01. |

## 5. Results

#### 5.1. 2D Building Portal Frame

^{2}, a moment of inertia of 1000 cm

^{4}and a length of 4 m. The “exact” buckling load factor was calculated via Abaqus using 10 elements per member. The Abaqus calculation was also used to validate our program in Matlab, in which we coded the proposed novel algorithm. The buckling shape is shown in the same figure with red dashed lines.

#### 5.2. 3D Stand Structure

^{2}and an isotropic moment of inertia of 1000 cm

^{4}. The columns are 4 m high; the highest point lies at a height of 7 m, and the column bases are on the vertices of a 4 m sided square.

#### 5.3. 3D Building Structure

^{2}, an isotropic moment of inertia of 1000 cm

^{4}and a length of 4 m. The “exact” buckling load factor was calculated with our program using 10 elements per member. The buckling shape is shown in the same figure with red dashed lines.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Table 1.**Relative error of the buckling load calculation in percentage as a function of the number of elements in the discretization for each of the five canonical cases.

Nel | CC | CP | PP | CM | CF |
---|---|---|---|---|---|

1 | - | 48.94% | 21.59% | 1.32% | 0.75% |

2 | 1.32% | 2.81% | 0.75% | 0.75% | 0.05% |

3 | 2.19% | 0.86% | 0.16% | 0.16% | 0.01% |

4 | 0.75% | 0.45% | 0.05% | 0.05% | 0.00% |

5 | 0.32% | 0.33% | 0.02% | 0.02% | 0.00% |

6 | 0.16% | 0.28% | 0.01% | 0.01% | 0.00% |

**Table 2.**Relative error in critical load factor calculation with one element per member after applying the proposed correction.

Nel | CC | CP | PP | CM | CF |
---|---|---|---|---|---|

1 | 0.75% | 0.45% | 0.05% | 0.05% | 0.00% |

Exact λ | 1-Elem λ | Relative Error | Corrected λ | Relative Error |
---|---|---|---|---|

75.331 | 75.851 | 0.69% | 75.351 | 0.027% |

Exact λ | 1-Elem λ | Relative Error | Corrected λ | Relative Error |
---|---|---|---|---|

217.33 | 373.10 | 71.67% | 219.44 | 0.97% |

Exact λ | 1-Elem λ | Relative Error | Corrected λ | Relative Error |
---|---|---|---|---|

227.11 | 408.79 | 80.00% | 229.97 | 1.08% |

Exact λ | 1-Elem λ | Relative Error | Corrected λ | Relative Error |
---|---|---|---|---|

4655.4 | 4800.8 | 3.12% | 4667.42 | 0.26% |

Exact λ | 1-Elem λ | Relative Error | Corrected λ | Relative Error |
---|---|---|---|---|

74.336 | 74.889 | 0.74% | 74.317 | 0.02% |

Exact λ | 1-Elem λ | Relative Error | Corrected λ | Relative Error |
---|---|---|---|---|

206.30 | 338.64 | 64.15% | 207.08 | 0.38% |

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**MDPI and ACS Style**

Urruzola, J.; Garmendia, I.
Calculation of Linear Buckling Load for Frames Modeled with One-Finite-Element Beams and Columns. *Computation* **2023**, *11*, 109.
https://doi.org/10.3390/computation11060109

**AMA Style**

Urruzola J, Garmendia I.
Calculation of Linear Buckling Load for Frames Modeled with One-Finite-Element Beams and Columns. *Computation*. 2023; 11(6):109.
https://doi.org/10.3390/computation11060109

**Chicago/Turabian Style**

Urruzola, Javier, and Iñaki Garmendia.
2023. "Calculation of Linear Buckling Load for Frames Modeled with One-Finite-Element Beams and Columns" *Computation* 11, no. 6: 109.
https://doi.org/10.3390/computation11060109