# Design and Non-Linear Modeling of New Wind Girder Used for Bolted Tanks

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## Abstract

**:**

## 1. Introduction

#### 1.1. Procedures for the Evaluation of Tank Buckling Capacity

- Plastic limit state;
- Cyclic plasticity limit state;
- Buckling limit state;
- Fatigue limit state.

#### 1.2. Description of Wind Girders

#### 1.3. Possibilities of New Girders Design

## 2. Materials and Methods

#### 2.1. Design of New Wind Girder

#### 2.2. Design of Non-Linear Numerical Model

#### 2.3. Analytical Models According to European Standards

#### 2.4. Parametric Study

## 3. Results

#### 3.1. Application of New Wind Girder to Low-Height Tanks

#### 3.2. Application of New Wind Girders to Middle-Height Tanks

#### 3.3. Application of New Wind Girders to High Tanks

#### 3.4. Comparison of Results Obtained from LBA and GMNIA with Analytical Models

#### 3.5. Geometrical Imperfection Study

#### 3.6. Evaluation of NWG Results

## 4. Discussion

## 5. Conclusions

- The cross-section of NWG segments is very simple and can be easily produced by common manufacturers.
- Although the total mass of NWG is slightly higher than for L60, due to the obtained results, NWG could be compared with the larger cross-section profiles usually used for tanks with larger diameters and possibly replace them.
- Although the total mass of NWG is slightly higher than for L60, the total price should be less because of the simpler cross-section and therefore lower material prices.
- The geometry is designed so that NWG can be used for different tank curvatures and stiffness requirements. By modifying the basic dimensions of segments, the stiffness is increased and overall buckling capacity is also increased.
- Because NWG is composed of simple segments with various curvatures, it is convenient for storage and assembly.
- An important difference of NWG compared to conventional girders is the variable moment of inertia around the circumference of the tank, which requires a more detailed assessment using GMNIA.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

3D | Three-dimensional |

FEM | Finite element method |

FSI | Fluid-structure interaction |

GMNA | Geometrically and materially non-linear analysis |

GMNIA | Geometrically and materially non-linear analysis with imperfections |

GNA | Geometrically non-linear elastic analysis |

GNIA | Geometrically non-linear elastic analysis with imperfections |

K300T | KOSMALT E 300 T |

L60 | Round-edged equal leg steel angle bar |

EN 10056-1 - L60 × 60 × 6 - EN 10025-2 - S235JR | |

LBA | Linear elastic bifurcation analysis |

MA | Modal analysis |

MNA | Materially non-linear analysis |

NWG | New wind girder |

P-segment | Plate segment of new wind girder |

S235 | S235JR EN 10025-2 |

TK | TANK |

V-segment | V-shape segment of new wind grider |

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**Figure 1.**Scheme of pressure distribution of wind around the tank circumference: (

**a**) idealized; (

**b**) real.

**Figure 2.**Types of conventional stiffeners: (

**a**) vertical stiffeners; (

**b**) horizontal girders; (

**c**) detail A of horizontal girder.

**Figure 4.**Shape of designed segments forming NWG: Item 1—V-segment; Item 2—P-segment; Item 3—bolted connection; Item 4—tank wall.

**Figure 5.**Finite element meshes for variants with NWG and diameter D = 4.3 m: (

**a**) low-height tank; (

**b**) middle-height tank; (

**c**) high tank.

**Figure 7.**Applied boundary conditions illustrated on the variant with NWG and diameter D = 4.3 m: (

**a**) uniform pressure; and (

**b**) fixed support.

**Figure 8.**Tank TK1–shape under critical pressure from GMNIA: (

**a**) L60; (

**b**) NWG; first mode shape from MAl; (

**c**) L60; (

**d**) NWG.

**Figure 9.**Tank TK3–shape under critical pressure from GMNIA: (

**a**) L60; (

**b**) NWG; first mode shape from MA; (

**c**) L60; (

**d**) NWG.

**Figure 10.**Comparison of low-height tanks: on the left is critical pressure, and on the right is first natural frequency.

**Figure 11.**Tank TK5—shape under critical pressure from GMNIA: (

**a**) L60; (

**b**) NWG; first mode shape from MA; (

**c**) L60; (

**d**) NWG.

**Figure 12.**Comparison of middle-height tanks: on the left is critical pressure, and on the right is first natural frequency.

**Figure 13.**Comparison of high tanks: on the left is critical pressure, and on the right is first natural frequency.

**Figure 14.**Tank TK7—shape under critical pressure from GMNIA: (

**a**) L60; (

**b**) NWG; first mode shape from MA; (

**c**) L60; (

**d**) NWG.

Type of Tank Analysis | Shell Theory |
---|---|

Linear elastic shell analysis (LA) | Linear bending and stretching |

Linear elastic bifurcation analysis (LBA) | Linear bending and stretching |

Geometrically non-linear elastic analysis (GNA) | Non-linear |

Materially non-linear analysis (MNA) | Linear |

Geometrically and materially non-linear analysis (GMNA) | Non-linear |

Geometrically non-linear elastic analysis with imperfections (GNIA) | Non-linear |

Geometrically and materially non-linear analysis with imperfections (GMNIA) | Non-linear |

Thickness t | Height h | Width w |
---|---|---|

4 mm | 150 mm | 50 mm |

Tank Wall | Girders | |
---|---|---|

Steel grade | K300T | S235 |

Young’s modulus E | 210,000 MPa | 210,000 MPa |

Poisson’s ratio $\mu $ | 0.30 | 0.30 |

Yield stress ${\sigma}_{y}$ | 300 MPa | 235 MPa |

Elongation $\delta $ | 28% | 26% |

Ultimate stress ${\sigma}_{ult}$ | 360 MPa | 235 MPa |

Tangent modulus ${E}_{T}$ | 531 MPa | 844 MPa |

Variant Label | D (m) | H (m) | ${\mathit{t}}_{\mathit{w}}$ (m) | Volume (m^{3}) | Number of Nodes (-) | Number of Elements (-) | Number of Segments (-) |
---|---|---|---|---|---|---|---|

TK1 | 4.3 | 2.9 | 0.003 | 42 | 28,900 | 26,800 | 70 |

TK2 | 21.4 | 2.9 | 0.003 | 1043 | 143,900 | 133,400 | 350 |

TK3 | 42.0 | 2.9 | 0.003 | 4018 | 208,900 | 189,100 | 686 |

TK4 | 4.3 | 8.6 | 0.003 | 125 | 78,700 | 76,600 | 70 |

TK5 | 21.4 | 8.6 | 0.003 | 3093 | 392,600 | 382,500 | 350 |

TK6 | 42.0 | 8.6 | 0.003 | 11,915 | 549,900 | 530,100 | 686 |

TK7 | 4.3 | 17.2 | 0.003 | 250 | 153,300 | 151,300 | 70 |

TK8 | 21.4 | 17.2 | 0.003 | 6187 | 417,400 | 407,300 | 350 |

TK9 | 42.0 | 17.2 | 0.003 | 23,830 | 598,700 | 578,800 | 686 |

Variant | D | ${\mathit{p}}_{\mathit{L}60}$ | ${\mathit{f}}_{\mathit{L}60}$ | ${\mathit{p}}_{\mathit{NWG}}$ | ${\mathit{f}}_{\mathit{NWG}}$ | $\frac{{\mathit{p}}_{\mathit{NWG}}-{\mathit{p}}_{\mathit{L}60}}{{\mathit{p}}_{\mathit{L}60}}$ | $\frac{{\mathit{f}}_{\mathit{NWG}}-{\mathit{f}}_{\mathit{L}60}}{{\mathit{f}}_{\mathit{L}60}}$ | ${\mathit{\sigma}}_{\mathit{L}60,\mathit{w}}$ | ${\mathit{\sigma}}_{\mathit{NWG},\mathit{w}}$ | ${\mathit{\sigma}}_{\mathit{L}60,\mathit{G}}$ | ${\mathit{\sigma}}_{\mathit{NWG},\mathit{G}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

Label | (m) | (kPa) | (Hz) | (kPa) | (Hz) | (%) | (%) | (MPa) | (MPa) | (MPa) | (MPa) |

TK1 | 4.3 | 22.500 | 39.111 | 16.850 | 28.035 | −25.111 | −28.321 | 67 | 139 | 15 | 62 |

TK2 | 21.4 | 1.750 | 16.640 | 1.500 | 16.038 | −14.286 | −3.614 | 16 | 31 | 6 | 17 |

TK3 | 42.0 | 0.640 | 11.187 | 0.650 | 12.016 | 1.563 | 7.404 | 18 | 19 | 5 | 14 |

Variant | D | ${\mathit{p}}_{\mathit{L}60}$ | ${\mathit{f}}_{\mathit{L}60}$ | ${\mathit{p}}_{\mathit{NWG}}$ | ${\mathit{f}}_{\mathit{NWG}}$ | $\frac{{\mathit{p}}_{\mathit{NWG}}-{\mathit{p}}_{\mathit{L}60}}{{\mathit{p}}_{\mathit{L}60}}$ | $\frac{{\mathit{f}}_{\mathit{NWG}}-{\mathit{f}}_{\mathit{L}60}}{{\mathit{f}}_{\mathit{L}60}}$ | ${\mathit{\sigma}}_{\mathit{L}60,\mathit{w}}$ | ${\mathit{\sigma}}_{\mathit{NWG},\mathit{w}}$ | ${\mathit{\sigma}}_{\mathit{L}60,\mathit{G}}$ | ${\mathit{\sigma}}_{\mathit{NWG},\mathit{G}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

Label | (m) | (kPa) | (Hz) | (kPa) | (Hz) | (%) | (%) | (MPa) | (MPa) | (MPa) | (MPa) |

TK4 | 4.3 | 8.700 | 10.874 | 7.350 | 10.407 | −15.517 | −4.295 | 96 | 128 | 12 | 94 |

TK5 | 21.4 | 0.585 | 5.362 | 0.594 | 6.430 | 1.496 | 19.909 | 29 | 34 | 5 | 17 |

TK6 | 42.0 | 0.200 | 3.762 | 0.205 | 4.905 | 2.500 | 30.382 | 16 | 17 | 3 | 8 |

Variant | D | ${\mathit{p}}_{\mathit{L}60}$ | ${\mathit{f}}_{\mathit{L}60}$ | ${\mathit{p}}_{\mathit{NWG}}$ | ${\mathit{f}}_{\mathit{NWG}}$ | $\frac{{\mathit{p}}_{\mathit{NWG}}-{\mathit{p}}_{\mathit{L}60}}{{\mathit{p}}_{\mathit{L}60}}$ | $\frac{{\mathit{f}}_{\mathit{NWG}}-{\mathit{f}}_{\mathit{L}60}}{{\mathit{f}}_{\mathit{L}60}}$ | ${\mathit{\sigma}}_{\mathit{L}60,\mathit{w}}$ | ${\mathit{\sigma}}_{\mathit{NWG},\mathit{w}}$ | ${\mathit{\sigma}}_{\mathit{L}60,\mathit{G}}$ | ${\mathit{\sigma}}_{\mathit{NWG},\mathit{G}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

Label | (m) | (kPa) | (Hz) | (kPa) | (Hz) | (%) | (%) | (MPa) | (MPa) | (MPa) | (MPa) |

TK7 | 4.3 | 4.375 | 4.841 | 3.883 | 5.116 | −11.249 | 5.679 | 114 | 122 | 19 | 119 |

TK8 | 21.4 | 0.360 | 2.426 | 0.333 | 3.317 | −7.639 | 36.709 | 28 | 36 | 5 | 19 |

TK9 | 42.0 | 0.112 | 1.721 | 0.114 | 2.569 | 1.786 | 49.288 | 14 | 15 | 3 | 8 |

Variant | D | ${\mathit{p}}_{\mathit{LBA},\mathit{L}60}$ | ${\mathit{p}}_{\mathit{GMNIA},\mathit{L}60}$ | $\frac{{\mathit{p}}_{\mathit{G}\mathit{M}\mathit{N}\mathit{I}\mathit{A},\mathit{L}60}}{{\mathit{p}}_{\mathit{L}\mathit{B}\mathit{A},\mathit{L}60}}$ | ${\mathit{p}}_{\mathit{LBA},\mathit{NWG}}$ | ${\mathit{p}}_{\mathit{GMNIA},\mathit{NWG}}$ | $\frac{{\mathit{p}}_{\mathit{G}\mathit{M}\mathit{N}\mathit{I}\mathit{A},\mathit{N}\mathit{W}\mathit{G}}}{{\mathit{p}}_{\mathit{L}\mathit{B}\mathit{A},\mathit{N}\mathit{W}\mathit{G}}}$ | ${\mathit{p}}_{\mathit{Eurocode}}$ | ${\mathit{p}}_{\mathit{ISO}}$ |
---|---|---|---|---|---|---|---|---|---|

Label | (m) | (kPa) | (kPa) | (kPa) | (kPa) | (kPa) | (kPa) | ||

TK1 | 4.3 | 25.835 | 22.500 | 0.871 | 25.572 | 16.850 | 0.659 | 14.404 | 19.717 |

TK2 | 21.4 | 2.613 | 1.750 | 0.670 | 2.637 | 1.500 | 0.569 | 1.297 | 1.776 |

TK3 | 42.0 | 1.065 | 0.640 | 0.601 | 1.082 | 0.650 | 0.601 | 0.471 | 0.646 |

TK4 | 4.3 | 9.110 | 8.700 | 0.955 | 8.973 | 7.350 | 0.819 | 4.790 | 6.557 |

TK5 | 21.4 | 0.839 | 0.585 | 0.697 | 0.845 | 0.594 | 0.703 | 0.431 | 0.591 |

TK6 | 42.0 | 0.312 | 0.200 | 0.640 | 0.317 | 0.205 | 0.647 | 0.156 | 0.215 |

TK7 | 4.3 | 4.470 | 4.375 | 0.979 | 4.374 | 3.883 | 0.888 | 2.395 | 3.279 |

TK8 | 21.4 | 0.408 | 0.360 | 0.883 | 0.409 | 0.333 | 0.812 | 0.215 | 0.295 |

TK9 | 42.0 | 0.150 | 0.112 | 0.748 | 0.151 | 0.114 | 0.754 | 0.078 | 0.107 |

TK1 | TK4 | TK7 | ||||
---|---|---|---|---|---|---|

$\mathit{u}{}_{\mathit{i}}$ | ${\mathit{P}}_{\mathit{L}\mathbf{60}}$ (kPa) | ${\mathit{P}}_{\mathit{NWG}}$ (kPa) | ${\mathit{P}}_{\mathit{L}\mathbf{60}}$ (kPa) | ${\mathit{P}}_{\mathit{NWG}}$ (kPa) | ${\mathit{P}}_{\mathit{L}\mathbf{60}}$ (kPa) | ${\mathit{P}}_{\mathit{NWG}}$ (kPa) |

$0.25\xb7{t}_{w}=0.75$ mm | 23.500 | 17.500 | 8.800 | 7.970 | 4.400 | 4.076 |

$0.50\xb7{t}_{w}=1.50$ mm | 22.500 | 16.850 | 8.700 | 7.350 | 4.375 | 3.884 |

$1.00\xb7{t}_{w}=3.00$ mm | 21.088 | 16.500 | 8.600 | 6.448 | 4.359 | 3.609 |

Conventional Girders | New Design of Girders |
---|---|

In some cases difficult to produce | Easier production |

Difficult adaptation to the tank | Easy adaptation to various tank sizes |

Need to select from available standardized profiles | Easy to change stiffness |

Uniform cross-section | Variable cross-section |

Lower mass | Larger mass (18% more compared to L60) |

Part of Girder | L60 | NWG |
---|---|---|

Mass per length of girders (kg/m) | 6.39 | 7.55 |

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## Share and Cite

**MDPI and ACS Style**

Drahorad, L.; Marsalek, P.; Hroncek, J.; Rybansky, D.; Sotola, M.; Poruba, Z.; Larys, M.
Design and Non-Linear Modeling of New Wind Girder Used for Bolted Tanks. *Buildings* **2023**, *13*, 2724.
https://doi.org/10.3390/buildings13112724

**AMA Style**

Drahorad L, Marsalek P, Hroncek J, Rybansky D, Sotola M, Poruba Z, Larys M.
Design and Non-Linear Modeling of New Wind Girder Used for Bolted Tanks. *Buildings*. 2023; 13(11):2724.
https://doi.org/10.3390/buildings13112724

**Chicago/Turabian Style**

Drahorad, Lukas, Pavel Marsalek, Juraj Hroncek, David Rybansky, Martin Sotola, Zdenek Poruba, and Michal Larys.
2023. "Design and Non-Linear Modeling of New Wind Girder Used for Bolted Tanks" *Buildings* 13, no. 11: 2724.
https://doi.org/10.3390/buildings13112724