# The Influence of Dome Geometry on the Results of Modal and Buckling Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Static-Strength Analysis

**K**—linear stiffness matrix of the structure,

_{L}**q**—vector of nodal displacements,

**P**—vector of nodal load.

**K**—geometrical stiffness matrix, $\mathsf{\mu}$—load multiplier.

_{G}**K**—secant stiffness matrix of the structure.

_{s}**K**(

_{T}**q**)—tangent stiffness matrix of the structure,

**R**=

**P**−

**F**—vector of residual forces,

**F**—vector of internal forces.

**R**= 0, while in the iterative process, the

**R**standard defines the distance from the equilibrium state. The iterative process converges if

**R**-> 0.

**K**of the structure is obtained by aggregating the stiffness matrix of the elements ${\mathbf{K}}_{\mathbf{T}}^{\mathbf{e}}$

_{T}**:**

_{x}, u

_{y}, u

_{z}, R

_{x}, R

_{y}, R

_{z}, where u denotes displacements in the X

_{G}, Y

_{G}, or Z

_{G}direction in a global coordinate system, and R denotes rotations around the axes). The local coordinate system of an element is a dextrorotary Cartesian system with the following orientation of axes. The local x axis is always the longitudinal element axis, whose direction is defined from the beginning point to the end point. The local axes y and z lie in the plane of the bar section and are arranged according to dextrorotary rotation. By standard, the axes mentioned represent: y axis—the axis of the greater moment of inertia of a bar, the z axis—the axis of the lesser moment of inertia of a bar (depending on the element section).

#### 2.2. Standard Modal Analysis

**K**—linear stiffness matrix of the structure,

_{L}**M**—mass matrix of the structure, ω

_{i}—natural pulsation (natural circular frequency) of i-th mode [rad/sec], ${\mathrm{f}}_{\mathrm{i}}=\frac{{\mathsf{\omega}}_{\mathrm{i}}}{2\mathsf{\pi}}$—frequency [Hz],

**q**—eigenmode vector of i-th mode.

_{i}_{i}= ${\mathsf{\omega}}_{\mathrm{i}}^{2}$ and the eigenvector

**q**

_{i}obtained during iteration for the i-th mode satisfy the eigenproblem equation. The participation factor is defined as:

**D**—vector of direction cosines is defined as follows:

**V**—i-th normalized eigenvector with:

_{i}#### 2.3. Modal Analysis Taking into Account Normal Forces

#### 2.3.1. Step 1—Linear or Nonlinear Static Analysis

#### 2.3.2. Step 2—Modal Analysis Taking into Account Normal Forces

_{i}(ω

_{i}> 0), while unstable equilibrium states are described by negative values (ω

_{i}< 0). In turn, the zero value (ω

_{i}= 0) indicates a stability (buckling) problem. If the matrices (

**K**+ ${\mathbf{K}}_{\mathbf{GN}}$) or (

_{L}**K**+ ${\mathbf{K}}_{\mathbf{G}}$) are not positively determined, the static load approaches the critical value (buckling).

_{L}## 3. Results

#### 3.1. Schwedler Dome

#### 3.2. Geodesic Dome

## 4. Discussion

## 5. Conclusions

_{cr}< 10.0, it is recommended to perform an approximate linear order II-elastic analysis (PDNA) or finer second-order analysis (GNA). First-order analysis may be used if the criterion that the value of the critical load multiplier μ

_{cr}in relation to design loads is not less than 10 is met (μ

_{cr}≥ 10). The low Schwedler dome presented in the article is characterized by a strongly nonlinear static and dynamic response. Ignoring nonlinear effects leads to large computational errors.

- N 1. EN 1990: 2002. Eurocode—Basis for structural design.
- N 2. EN 1991-1-3. Eurocode 1: Actions on structures—Part 1–3: General actions—Snow loads.
- N 3. EN 1991-1-4. Eurocode 1: Actions on structures—Part 1–4: General actions—Wind actions.
- N 4. EN 1993-1-1. Eurocode 3: Design of steel structures—Part 1–1: General rules and rules for buildings.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Graphic illustration of eigenvectors for low-rise Schwedler dome (standard modal analysis).

**Figure 5.**Graphic illustration of eigenvectors for low-rise Schwedler dome (modal analysis taking into account normal forces).

**Table 1.**Results of linear buckling analysis and nonlinear buckling analysis for low-rise Schwedler dome.

Form | Eigenvalues [-] | |
---|---|---|

Linear Buckling Analysis | Nonlinear Buckling Analysis | |

1 | 3.78 | 2.42 |

2 | 3.80 | 2.43 |

3 | 5.47 | 3.53 |

4 | 5.47 | 3.54 |

5 | 5.80 | 3.92 |

Form | Frequency [Hz] | |
---|---|---|

Standard Modal Analysis | Modal Analysis Taking into Account Normal Forces | |

1 | 3.57 | 2.85 |

2 | 3.57 | 2.85 |

3 | 5.75 | 4.87 |

4 | 5.86 | 4.87 |

5 | 5.86 | 5.24 |

6 | 6.44 | 5.33 |

7 | 6.44 | 5.33 |

8 | 6.97 | 6.18 |

9 | 7.36 | 6.18 |

10 | 7.36 | 6.45 |

Form | Standard Modal Analysis | ||
---|---|---|---|

Current Mass UX % | Current Mass UY % | Current Mass UZ % | |

1 | 2.13 | 0.21 | 0.00 |

2 | 0.21 | 2.13 | 0.00 |

3 | 5 × 10^{−07} | 0.00 | 80.33 |

4 | 3 × 10^{−08} | 2 × 10^{−08} | 8 × 10^{−06} |

5 | 3 × 10^{−08} | 2 × 10^{−08} | 2 × 10^{−07} |

6 | 2 × 10^{−07} | 0.00 | 7 × 10^{−07} |

7 | 0.00 | 1 × 10^{−07} | 1 × 10^{−07} |

8 | 4 × 10^{−07} | 3 × 10^{−09} | 8.25 |

9 | 3 × 10^{−08} | 4 × 10^{−09} | 6 × 10^{−06} |

10 | 4 × 10^{−09} | 2 × 10^{−09} | 3 x 10^{−07} |

Form | Standard Modal Analysis | ||
---|---|---|---|

Effective Mass UX % | Effective Mass UY % | Effective Mass UZ % | |

1 | 2.13 | 0.20 | 0.00 |

2 | 2.33 | 2.33 | 0.00 |

3 | 2.33 | 2.33 | 80.33 |

4 | 2.33 | 2.33 | 80.33 |

5 | 2.33 | 2.33 | 80.33 |

6 | 2.33 | 2.33 | 80.33 |

7 | 2.33 | 2.33 | 80.33 |

8 | 2.33 | 2.33 | 88.58 |

9 | 2.33 | 2.33 | 88.58 |

10 | 2.33 | 2.33 | 88.58 |

**Table 5.**Current masses for low-rise Schwedler dome (modal analysis taking into account normal forces).

Form | Modal Analysis Taking into Account Normal Forces | ||
---|---|---|---|

Current Mass UX % | Current Mass UY % | Current Mass UZ % | |

1 | 0.23 | 1.76 | 0.00 |

2 | 1.76 | 0.23 | 0.00 |

3 | 0.00 | 0.00 | 2 × 10^{−09} |

4 | 5 × 10^{−08} | 5 × 10^{−08} | 8 × 10^{−09} |

5 | 3 × 10^{−08} | 2 × 10^{−08} | 88.75 |

6 | 0.00 | 7 × 10^{−09} | 1 × 10^{−08} |

7 | 1 × 10^{−07} | 1 × 10^{−09} | 4 × 10^{−06} |

8 | 1 × 10^{−08} | 3 × 10^{−09} | 1 × 10^{−06} |

9 | 2 × 10^{−08} | 8 × 10^{−09} | 8 × 10^{−09} |

10 | 5 × 10^{−09} | 2 × 10^{−09} | 0.07 |

**Table 6.**Effective masses for low-rise Schwedler dome (modal analysis taking into account normal forces).

Form | Modal Analysis Taking into Account Normal Forces | ||
---|---|---|---|

Effective Mass UX % | Effective Mass UY % | Effective Mass UZ % | |

1 | 0.23 | 1.76 | 0.00 |

2 | 1.99 | 1.99 | 0.00 |

3 | 1.99 | 1.99 | 0.00 |

4 | 1.99 | 1.99 | 0.00 |

5 | 1.99 | 1.99 | 88.75 |

6 | 1.99 | 1.99 | 88.75 |

7 | 1.99 | 1.99 | 88.75 |

8 | 1.99 | 1.99 | 88.75 |

9 | 1.99 | 1.99 | 88.75 |

10 | 1.99 | 1.99 | 88.81 |

Form | Eigenvalues [-] | |
---|---|---|

Linear Buckling Analysis | Nonlinear Buckling Analysis | |

1 | 12.42 | 11.45 |

2 | 12.42 | 11.45 |

3 | 12.43 | 11.46 |

4 | 12.44 | 11.48 |

5 | 12.45 | 11.48 |

6 | 14.48 | 13.51 |

Form | Frequency [Hz] | |
---|---|---|

Standard Modal Analysis | Modal Analysis Taking into Account Normal Forces | |

1 | 17.41 | 17.35 |

2 | 17.69 | 17.66 |

3 | 17.69 | 17.66 |

4 | 19.27 | 19.13 |

5 | 19.40 | 19.29 |

6 | 19.40 | 19.29 |

7 | 19.52 | 19.41 |

8 | 19.53 | 19.41 |

9 | 19.99 | 19.86 |

10 | 20.00 | 19.87 |

Form | Standard Modal Analysis | Modal Analysis Taking into Account Normal Forces | ||||
---|---|---|---|---|---|---|

Current Mass UX % | Current Mass UY % | Current Mass UZ % | Current Mass UX % | Current Mass UY % | Current Mass UZ % | |

1 | 3 × 10^{−03} | 1 × 10^{−03} | 5.31 | 3 × 10^{−03} | 1 × 10^{−03} | 5.15 |

2 | 23.25 | 33.08 | 5 × 10^{−05} | 23.31 | 32.95 | 7 × 10^{−05} |

3 | 32.87 | 23.37 | 3 × 10^{−04} | 32.72 | 23.44 | 2 × 10^{−04} |

4 | 1 × 10^{−04} | 8 × 10^{−04} | 1.27 | 0.00 | 1 × 10^{−03} | 1.31 |

5 | 1.95 | 0.81 | 2 × 10^{−05} | 2.04 | 0.36 | 0.00 |

6 | 0.86 | 1.86 | 5 × 10^{−04} | 0.40 | 1.92 | 1 × 10^{−03} |

7 | 3 × 10^{−04} | 7 × 10^{−04} | 4 × 10^{−04} | 5 × 10^{−04} | 1 × 10^{−03} | 2 × 10^{−04} |

8 | 1 × 10^{−04} | 2 × 10^{−03} | 9 × 10^{−06} | 9 × 10^{−05} | 2 × 10^{−03} | 1 × 10^{−05} |

9 | 2 × 10^{−03} | 2 × 10^{−04} | 2 × 10^{−04} | 1 × 10^{−03} | 1 × 10^{−03} | 2 × 10^{−04} |

10 | 4 × 10^{−03} | 0.03 | 1 × 10^{−04} | 3 × 10^{−03} | 0.04 | 4 × 10^{−04} |

Form | Standard Modal Analysis | Modal Analysis Taking into Account Normal Forces | ||||
---|---|---|---|---|---|---|

Effective Mass UX % | Effective Mass UY % | Effective Mass UZ % | Effective Mass UX % | Effective Mass UY % | Effective Mass UZ % | |

1 | 0.00 | 0.00 | 5.31 | 0.00 | 0.00 | 5.15 |

2 | 23.25 | 33.08 | 5.31 | 23.32 | 32.95 | 5.15 |

3 | 56.12 | 56.45 | 5.31 | 56.03 | 56.39 | 5.15 |

4 | 56.12 | 56.46 | 6.58 | 56.03 | 56.39 | 6.46 |

5 | 58.07 | 57.26 | 6.58 | 58.08 | 56.75 | 6.46 |

6 | 58.94 | 59.13 | 6.58 | 58.48 | 58.67 | 6.46 |

7 | 58.94 | 59.13 | 6.58 | 58.48 | 58.68 | 6.46 |

8 | 58.94 | 59.13 | 6.58 | 58.48 | 58.68 | 6.46 |

9 | 58.94 | 59.13 | 6.58 | 58.48 | 58.68 | 6.46 |

10 | 58.94 | 59.16 | 6.58 | 58.48 | 58.71 | 6.46 |

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**MDPI and ACS Style**

Radoń, U.; Zabojszcza, P.; Sokol, M.
The Influence of Dome Geometry on the Results of Modal and Buckling Analysis. *Appl. Sci.* **2023**, *13*, 2729.
https://doi.org/10.3390/app13042729

**AMA Style**

Radoń U, Zabojszcza P, Sokol M.
The Influence of Dome Geometry on the Results of Modal and Buckling Analysis. *Applied Sciences*. 2023; 13(4):2729.
https://doi.org/10.3390/app13042729

**Chicago/Turabian Style**

Radoń, Urszula, Paweł Zabojszcza, and Milan Sokol.
2023. "The Influence of Dome Geometry on the Results of Modal and Buckling Analysis" *Applied Sciences* 13, no. 4: 2729.
https://doi.org/10.3390/app13042729