# Assessment of the Effect of Wind Load on the Load Capacity of a Single-Layer Bar Dome

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**K**

**—linear stiffness matrix of the structure;**

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

**P**—vector of nodal load.

**K**

**—geometrical stiffness matrix.**

_{G}**K**

**is identified as the initial stress stiffness matrix. Eigenvalues µ are critical load multipliers. Eigenvector**

_{G}**q**illustrates the form of structure deformation. Running the analysis allows the user to quickly verify how close to instability the analyzed load condition is. However, analysis is still conducted in the field of linear physical and geometrical relationships.

**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 norm of

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

**R**→

**0**.

**K**

**of the structure arises as a result of the assembly of the stiffness matrices of the elements: ${\mathbf{K}}_{\mathbf{T}}^{\mathbf{e}}$**

_{T}**K**

**). A more effective proposition to describe these changes was proposed by Bergan and Soreide in [20]. They characterize the behavior of a multidimensional nonlinear system by using one scalar quantity, the current stiffness parameter (CSP), representing the ratio of two quadratic forms formulated for the tangent stiffness matrix at the initial ${\mathbf{K}}_{\mathbf{T}}^{\mathbf{0}}$ and current times ${\mathbf{K}}_{\mathbf{T}}^{\mathbf{i}}$:**

_{T}**K**

**during the motion in the N-dimensional, displacement solution space. Figure 1a shows a typical snap-through problem (load parameters versus some norm of displacement vector ||**

_{T}**q**||). The associated curve for CSP as a function of ||

**q**|| is traced in Figure 1b. It is noticeable that at the extreme points of the load-displacement curve, CSP has the value zero. In this situation, the incremental stiffness matrix

**K**

**is singular. CSP is positive for the stable branches of the load-displacement curve. The instable configurations are characterized by negative values of CSP.**

_{T}_{i}= η with the load parameter μ and chosen displacement q

_{i}being the controlling parameters. With the first of the two methods, subsequent points are the points of intersection between µ = η

_{i}and the equilibrium path, and with the second method—as the points of intersection between q

_{k}= η

_{i}with the same solution curve (Figure 2). Both procedures are inefficient at the vicinity of local extremum of the controlling parameter.

**q**—generalized coordinate vector; η − η

_{α}—parameter approximating the arc length; the dots appearing above the symbols indicate derivatives with respect to the arc length.

**q**

_{α}, µ

_{α}) and distant from the point (

**q**

_{α}, µ

_{α}) by (η − η

_{α}). It will intersect the equilibrium path almost perpendicular when the distance η − η

_{α}is sufficiently small.

#### 2.1. Truss Element Description

^{0}C and in the deformed configuration

^{t}C and

^{t+∆t}C (Figure 4). The following material and geometric constants were adopted: E—Young’s modulus; l

_{0}—the initial length; A

_{0}—the initial cross-sectional area; l—the length in the deformed configuration; A—the cross-sectional area in the deformed configuration. The deformation field in the deformed configuration

^{t}C is described by four components,

**q**= {u

_{1}, v

_{1}, u

_{2}, v

_{2}}, and the displacement increment between time t and t+∆t is described by vector ∆

**q**= {∆u

_{1}, ∆v

_{1}, ∆u

_{2}, ∆v

_{2}}. These displacement vectors correspond to nodal force vectors:

**Q**= {U

_{1}, V

_{1}, U

_{2}, V

_{2}}, and ∆

**Q**= {∆U

_{1}, ∆V

_{1}, ∆U

_{2}, ∆V

_{2}}.

_{11}= σ of the 2nd Piola–Kirchhoff symmetric stress tensor. The real axial force S’ = S·l/l

_{0}.

^{t+∆t}C are as follows:

^{t+∆t}

**q**=

^{t}

**q**+∆

**q**,

^{t+∆t}ε=

^{t}ε +∆ ε,

^{t+∆t}σ=

^{t}σ +∆ σ

^{t}

**q**,

^{t}ε,

^{t}σ—displacements, strain, and stress at time t; ∆

**q**, ∆ε, ∆σ—increments between time t and t + ∆t. Strain increment can be written as:

**F**

^{e}is the vector of internal forces in the element:

#### 2.2. Frame Element Description

^{0}C and in the deformed configurations

^{t}C and

^{t+∆t}C (Figure 5). The following material and geometric constants were adopted: E—Young’s modulus; l

_{0}—the initial length; A

_{0}—the initial cross-sectional area; I

_{0}—the initial moment of inertia; l—the length in the deformed configuration; A—the cross-sectional area in the deformed configuration; I—moment of inertia in the deformed configuration.

^{t}C is described by six components,

^{t}

**q**= {u

_{1}, v

_{1}, φ

_{1}, u

_{2}, v

_{2}, φ

_{2}}, and the displacement increment between time t and t + ∆t is described by vector ∆

**q**= {∆u

_{1}, ∆v

_{1}, ∆ φ

_{1}, ∆u

_{2}, ∆v

_{2}, ∆ φ

_{2}}. These displacement vectors correspond to nodal force vectors:

^{t}

**Q**= {U

_{1}, V

_{1}, Φ

_{1}, U

_{2}, V

_{2}, Φ

_{2}}, and ∆

**Q**= {∆U

_{1}, ∆V

_{1}, ∆Φ

_{1}, ∆U

_{2}, ∆V

_{2}, ∆Φ

_{2}}.

**σ**=

**E**·∆

**ε**

## 3. Results

#### 3.1. Example 1

#### 3.1.1. Von Mises Truss Analysis

_{y}= 235 MPa; Young modulus E = 210 GPa; and Poissons ratio v=0.3. The height-to-span ratio is H/L = 0.25, and the bar angle is γ = 14°. The lengths of both elements are l

_{0}= 412.3 cm. The values of the axial force in the bar and the vertical displacement of node 2, determined on the basis of LA (linear analysis) and GNA, do not differ significantly (Table 1).

**K**

**) stiffness matrix of the structure in the case of geometrically nonlinear analysis is 0.2851 × 10**

_{T}^{9}, while in the linear analysis, 0.2872 × 10

^{9}. The value of the current stiffness parameter (CSP) practically did not change. In the linear analysis, this value was 1.0, while in the geometrically nonlinear analysis, 0.9929. The critical load multiplier obtained from the LBA analysis is µ

_{cr}= 423.638. According to the recommendations of the PN-EN-1993-1-1 [3] standard, for a critical multiplier value µ > 10, a linear analysis is sufficient.

_{y}= 235 MPa; Young’s modulus E = 210 GPa; and Poissons ratio v = 0.3. The height-to-span ratio is H/L = 0.05 and the bar angle γ = 3°. The lengths of both elements are l

_{0}= 400.5 cm.

**K**

**) stiffness matrix of the structure for geometrically nonlinear analysis is 0.4977 × 10**

_{T}^{8}, while for linear analysis, 0.7999 × 10

^{8}. The value of the current stiffness parameter decreased significantly. In the linear analysis, this value was 1.0, while in the geometrically nonlinear analysis, 0.6223. The critical load multiplier obtained from the LBA is µ

_{cr}= 8.944. According to the recommendations of the PN-EN-1993-1-1 [3] standard, for a critical multiplier value µ < 10, a more accurate geometrically nonlinear structure analysis is recommended. The results obtained clearly confirm the entries in the Eurocode. In the second example, we will use them consistently.

#### 3.1.2. Discussion of Example 1

#### 3.2. Example 2

#### 3.2.1. Geometry and Loads

^{2}. The permanent load values for individual groups of nodes are summarized in Table 4.

^{2}. The snow load values for individual groups of nodes are summarized in Table 5.

_{n}—in order to specify the wind force deviation from the vertical axis “z”—δ

_{n}. The values of angles α

_{n}and δ

_{n}are shown in Figure 11. The dome is divided by meridians into 16 equal parts. The angle γ by which the wind force in the XY plane should be rotated is γ = 360/16 = 22.5°.

_{pe,A}, C

_{pe,B}, and C

_{pe,C}coefficients can be determined by linear interpolation of the data value in the graph in Figure 13 as a function of the f/d Equation (26).

_{pe,i}indices was determined:

_{b,0}= 25.04 m/s; while the exposure factor is: ${\mathrm{c}}_{\mathrm{e}}\left({\overline{\mathrm{z}}}_{\mathrm{e}}\right)=1.99$. The base value of the wind speed pressure was determined on the level q

_{b}= 0.392 kN/m

^{2}. The peak value of pressure wind speed is equal to q

_{p}$\left({\overline{\mathrm{z}}}_{\mathrm{e}}\right)$ = 0.392∙1.99 = 0.780 kN/m

^{2}.

#### 3.2.2. Case 1. Load Combination: 1.15G + 1.5S

_{y}= 235 MPa and Young’s modulus E = 210 GPa; Poisson’s ratio v = 0.3. In order to dimension the designed structure, three groups of rods were modeled: meridians, parallels, and diagonals.

_{cr}= 2.11199. Figure 17 shows the buckling modes for four successive eigenvalues. According to PN EN 1993-1-1/5.2.2 (5)B [3], when the lowermost critical load multiplier µ

_{cr}< 3.0, it is necessary to carry out a more accurate second order (GNA) analysis for the structure.

_{cr}= 1.562. Exceeding this multiplier results in a peculiar stiffness matrix. GNA is a mathematical tool with which it is possible to generate complete equilibrium paths. Tracking the equilibrium path is inherently related to the analysis of post-critical states. The problem presented in example 2 concerns the design of a real shallow steel lattice structure. These types of structures are susceptible to large displacement gradients, which can only be considered by using geometrically nonlinear relationships. GNA in this case is used to determine the internal forces, displacements, and critical load multiplier. In the case of the analyzed shallow lattice dome, the critical load is related to the phenomenon of the node snap-through. After the critical load is exceeded, only unstable equilibrium states occur. From an engineering point of view, the part of the equilibrium path after reaching the critical load multiplier does not cause significant changes to the structure design process. Figure 18 shows the equilibrium path of the considered bar dome for case 1.

#### 3.2.3. Case 2. Load Combination 1.15 × G +1.5 × S+0.9 × W

_{cr}= 2.35506. Figure 20 shows the buckling modes for four successive eigenvalues.

_{cr}= 1.390. Figure 21 shows the equilibrium path of the considered bar dome for case 2.

#### 3.2.4. Discussion of Example 2

_{cr}= 2.11199, while in case 2 it increases to µ

_{cr}= 2.35506. Both values indicate that a second order analysis is necessary. The critical load multiplier values for geometrically nonlinear analysis are, respectively, for case 1, µ

_{cr}= 1.562 and for case 2, µ

_{cr}= 1.390.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Dependence of load parameter μ on the norm of ||

**q**||; (

**b**) dependence of the current stiffness parameter (CSP) on the norm of ||

**q**||

**.**

**Figure 4.**Truss element in the undeformed (initial),

^{0}C, and deformed

^{t}C and

^{t+∆t}C configurations.

**Figure 5.**Frame element in the undeformed (initial),

^{0}C, and deformed

^{t}C and

^{t+∆t}C configurations.

**Figure 8.**(

**a**) Equilibrium path of shallow von Mises truss, (

**b**) CSP-q path of shallow von Mises truss.

**Figure 12.**Diagram of the structure division into wind zones acc. [39].

**Figure 13.**Values of Cpe,i (i = A—red; B—green; C—blue) coefficients for dome structure acc. [39].

**Figure 20.**Buckling modes of the structure included wind suction: (

**a**) 1st, (

**b**) 2nd, (

**c**) 3rd, (

**d**) 4th.

Force/Displacement/Analysis | LA | GNA |
---|---|---|

Axial force | 20.616 kN | 20.660 kN |

Nodal displacement of node No. 2 | 0.236 cm | 0.237 cm |

Force/Displacement/Analysis | LA | GNA |
---|---|---|

Axial force | 100.125 kN | 116.6 kN |

Nodal displacement of node No. 2 | 2.236 cm | 2.795 cm |

Group Name | Numbers of Nodes |
---|---|

R1 | 6, 16, 26, 36, 46, 56, 66, 76, 7, 17, 27, 37, 47, 57, 67, 77 |

R2 | 5, 15, 25, 35, 45, 55, 65, 75, 8, 18, 28, 38, 48, 58, 68, 78 |

R3 | 4, 14, 24, 34, 44, 54, 64, 74, 9, 19, 29, 39, 49, 59, 69, 79 |

R4 | 3, 13, 23, 33, 43, 53, 63, 73, 10, 20, 30, 40, 50, 60, 70, 80 |

R5 | 2, 12, 22, 32, 42, 52, 62, 72, 11, 21, 31, 41, 51, 61, 71, 81 |

Nodes Group | Own Weight Load Value—G (kN) |
---|---|

R1 | 3.564 |

R2 | 7.129 |

R3 | 10.683 |

R4 | 14.229 |

R5 | 8.878 |

Node No. 1 | 7.180 |

Nodes Group | Snow Load Value—S (kN) |
---|---|

R1 | 4.059 |

R2 | 8.118 |

R3 | 12.165 |

R4 | 16.203 |

R5 | 10.110 |

Node No. 1 | 8.176 |

Pressure Coefficients | x_{A} | x_{1} | x_{2} | x_{3} | x_{B} | x_{4} | x_{5} | x_{6} | x_{C} |
---|---|---|---|---|---|---|---|---|---|

x | 0.00 | 3.125 | 6.25 | 9.375 | 12.50 | 15.625 | 18.75 | 21.875 | 25.00 |

C_{pe} | −1.14 | −1.00 | −0.85 | −0.71 | −0.56 | −0.47 | −0.38 | −0.29 | −0.20 |

Node Number | Wind Load Value (kN) | Node Number | Wind Load Value (kN) | Node Number | Wind Load Value (kN) | Node Number | Wind Load Value (kN) |
---|---|---|---|---|---|---|---|

1 | −2.588 | 21 | −1.385 | 41 | −2.245 | 61 | −3.391 |

2 | −5.445 | 22 | −4.06 | 42 | −2.675 | 62 | −1.815 |

3 | −8.726 | 23 | −6.506 | 43 | −4.286 | 63 | −2.909 |

4 | −6.551 | 24 | −4.885 | 44 | −3.218 | 64 | −2.184 |

5 | −4.372 | 25 | −3.26 | 45 | −2.148 | 65 | −1.457 |

6 | −2.186 | 26 | −1.63 | 46 | −1.074 | 66 | −0.729 |

7 | −0.383 | 27 | −0.729 | 47 | −1.074 | 67 | −1.63 |

8 | −0.767 | 28 | −1.457 | 48 | −2.148 | 68 | −3.26 |

9 | −1.149 | 29 | −2.184 | 49 | −3.218 | 69 | −4.885 |

10 | −1.5311 | 30 | −2.909 | 50 | −4.286 | 70 | −6.506 |

11 | −0.955 | 31 | −1.815 | 51 | −2.675 | 71 | −4.06 |

12 | −4.7766 | 32 | −3.391 | 52 | −2.245 | 72 | −1.385 |

13 | −7.654 | 33 | −5.434 | 53 | −3.597 | 73 | −2.22 |

14 | −5.746 | 34 | −4.08 | 54 | −2.701 | 74 | −1.666 |

15 | −3.835 | 35 | −2.723 | 55 | −1.802 | 75 | −1.112 |

16 | −1.917 | 36 | −1.361 | 56 | −0.901 | 76 | −0.556 |

17 | −0.556 | 37 | −0.901 | 57 | −1.361 | 77 | −1.917 |

18 | −1.112 | 38 | −1.802 | 58 | −2.723 | 78 | −3.835 |

19 | −1.666 | 39 | −2.701 | 59 | −4.08 | 79 | −5.746 |

20 | −2.22 | 40 | −3.597 | 60 | −5.434 | 80 | −7.654 |

- | 81 | −4.776 |

**Table 8.**Values of the internal forces and the maximum horizontal and vertical displacement of node 79 for LA.

Internal Forces/Displacement | LA | ||
---|---|---|---|

Meridian Bar no 30 | Parallel Bar no 114 | Diagonal Bar no 193 | |

N_{Ed} (kN)—axial force | 563.694 | 162.79 | 1.977 |

N_{c,Rd} (kN)—design capacity of the section under uniform compression | 1543.95 | 552.250 | 212.910 |

N_{b,Rd} (kN)—design buckling resistance of the compressed element | 1486.389 | 275.405 | 60.072 |

M_{y,Ed,Max} (kNm)—design bending moment with respect to y-y axis | 28.537 | 0.569 | 0.316 |

M_{y}_{,c,Rd} (kNm)—design bending resistance with respect to y-y axis | 102.827 | 16.511 | 4.892 |

M_{z,Ed,max} (kNm)—design bending moment with respect to z-z axis | −2.300 | 0.572 | −0.090 |

M_{z,c,Rd} (kNm)—design bending resistance with respect to z-z axis | 102.827 | 16.511 | 4.892 |

Utilization (%) | 67 | 65 | 8 |

Maximum vertical displacement (mm)—for node 79 | 42.52 | ||

Allowable vertical displacement (mm)—D/300 | 83.33 | ||

Maximum horizontal displacement (mm)—for node 79 | 3.93 | ||

Allowable horizontal displacement (mm)—H/150 | 6.67 |

Group Name | Nodes Nos | Cross Section |
---|---|---|

Parallel | 2 to 81 | RO 219.1 × 10 |

Meridian | 82 to 161 | RO 101.6 × 8 |

Diagonal | 162 to 224 | RO 76.1 × 4 |

**Table 10.**Values of the internal forces and the maximum horizontal and vertical displacement of node 54 for GNA.

Internal Forces/Displacement | GNA | ||
---|---|---|---|

Meridian Bar no 20 | Parallel Bar no 107 | Diagonal Bar no 180 | |

N_{Ed} (kN)—axial force | 596.243 | 208.020 | 1.61 |

M_{y,Ed,Max} (kNm)—design bending moment with respect to y-y axis | 39.097 | 0.825 | 0.431 |

M_{z,Ed,max} (kNm)—design bending moment with respect to z-z axis | −2.774 | −0.628 | −0.119 |

Utilization (%) | 81 | 85 | 9 |

Maximum vertical displacement (mm)—for node 54 | 53.54 | ||

Allowable vertical displacement (mm)—D/300 | 83.33 | ||

Maximum horizontal displacement (mm)—for node 54 | 5.51 | ||

Allowable horizontal displacement (mm)—H/150 | 6.67 |

**Table 11.**Values of the internal forces and the maximum horizontal and vertical displacement of node 9 for LA.

Internal Forces/Displacement | LA | ||
---|---|---|---|

Meridian Bar no 10 | Parallel Bar no 106 | Diagonal Bar no 165 | |

N_{Ed} (kN)—axial force | 518.05 | 164.801 | 19.766 |

N_{c,Rd} (kN)—design capacity of the section under uniform compression | 1543.95 | 552.25 | 212.910 |

N_{b,Rd} (kN)—design buckling resistance of the compressed element | 1486.389 | 275.405 | 43.369 |

M_{y,Ed,Max} (kNm)—design bending moment with respect to y-y axis | 29.862 | 0.693 | 0.122 |

M_{y}_{,c,Rd} (kNm)—design bending resistance with respect to y-y axis | 102.827 | 16.511 | 4.892 |

M_{z,Ed,max} (kNm)—design bending moment with respect to z-z axis | −2.235 | −0.617 | −0.019 |

M_{z,c,Rd} (kNm)—design bending resistance with respect to z-z axis | 102.827 | 16.511 | 4.892 |

Utilization (%) | 65 | 67 | 49 |

Maximum vertical displacement (mm)—for node 9 | 48.42 | ||

Allowable vertical displacement (mm)—D/300 | 83.33 | ||

Maximum horizontal displacement (mm)—for node 9 | 4.52 | ||

Allowable horizontal displacement (mm)—H/150 | 6.67 |

**Table 12.**Values of the internal forces and the maximum horizontal and vertical displacement of node 9 for GNA.

Internal Forces/Displacement | GNA | ||
---|---|---|---|

Meridian Bar no 10 | Parallel Bar no 106 | Diagonal Bar no 165 | |

N_{Ed} (kN)—axial force | 555.081 | 214.545 | 36.408 |

M_{y,Ed,Max} (kNm)—design bending moment with respect to y-y axis | 43.965 | 1.210 | 0.173 |

M_{z,Ed,max} (kNm)—design bending moment with respect to z-z axis | −2.704 | −0.720 | −0.008 |

Utilization (%) | 83 | 92 | 90 |

Maximum vertical displacement (mm)—for node 9 | 68.72 | ||

Allowable vertical displacement (mm)—D/300 | 83.33 | ||

Maximum horizontal displacement (mm)—for node 9 | 6.52 | ||

Allowable horizontal displacement (mm)—H/150 | 6.67 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Opatowicz, D.; Radoń, U.; Zabojszcza, P.
Assessment of the Effect of Wind Load on the Load Capacity of a Single-Layer Bar Dome. *Buildings* **2020**, *10*, 179.
https://doi.org/10.3390/buildings10100179

**AMA Style**

Opatowicz D, Radoń U, Zabojszcza P.
Assessment of the Effect of Wind Load on the Load Capacity of a Single-Layer Bar Dome. *Buildings*. 2020; 10(10):179.
https://doi.org/10.3390/buildings10100179

**Chicago/Turabian Style**

Opatowicz, Dominika, Urszula Radoń, and Paweł Zabojszcza.
2020. "Assessment of the Effect of Wind Load on the Load Capacity of a Single-Layer Bar Dome" *Buildings* 10, no. 10: 179.
https://doi.org/10.3390/buildings10100179