# Experimental and Numerical Estimation of the Aerodynamic Forces Induced by the Wind Acting on a Fast-Erecting Crane

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Object

#### 2.2. Experimental Setup

_{0}, characteristic length L

_{0}and density ρ, were chosen. Only three parameters were considered for scaling parameters; because of mechanical problems, only three quantities could be chosen to be dimension-independent, and the others should fulfill the appropriate relationships.

- -
- Strouhal’s similarity criterion:

_{0}is the characteristic dimension (e.g., the lateral dimension of the flowing body by the air), V

_{0}is the characteristic velocity (e.g., the velocity of the non-disturbed airflow before a body), and f

_{0}is the frequency. The frequency scale for all considered frequencies is equal to

- -
- Reynold’s similarity condition:

- -
- Strouhal’s similarity criterion is satisfied for aeroelastic models and for vortex excitation;
- -
- Reynold’s similarity criterion in the laboratory environment does not have to lead to significant errors because in the case of cross-sections, which contain sharp edges or truss objects, the values of the aerodynamic coefficients do not, in fact, depend on the Reynold’s number in a wide range of Reynold’s number values.

#### 2.2.1. Truss of the Vertical Part of the Crane

_{T}= 157 mm and a height of h

_{T}= 560 mm. At both ends of the model, there are circular plates installed with a diameter equal to d = 400 mm and thickness of t = 4 mm to not disturb the airflow around the model. Additionally, the plates are chamfered, as shown in Figure 2. Inside the model, there is also a part of the ladder. The structural elements of the sections are connected by welding. The ladder, due to the small dimensions of the parts from which it is made, was assembled by sealing the previously rolled rungs, and then welded to the tower structure using connectors. The entire truss structure of the model was bolted to the circular plates. Finally, the model was degreased and protected against corrosion. In Figure 3, the real object and the CAD model used in the CFD simulations are presented.

#### 2.2.2. Truss of the Horizontal Part of the Crane

_{J}= 136 mm; and height h

_{J}= 104 mm. The section, similar to the tower model, is closed on both sides with aluminum circular plates with a diameter of d = 400 mm and thickness of t = 4 mm.

_{X}, M

_{Y}, and M

_{Z}and aerodynamic forces F

_{X}and F

_{Y}were measured using the five-component aerodynamic balance based on the electric resistance strain gauges. The orientation of the fixed x, y, z coordinate system is as follows: x—along the wind direction, y—across the wind direction, and z—vertical direction.

#### 2.3. Aerodynamic Tunnel

#### 2.4. The Wind Profile

_{V}(z) were calculated for the series of wind speeds in each measurement. The distribution of the wind parameters along the wind tunnel height for each case of flow was estimated using the least square method. The wind profiles are described by the power law formula:

_{ref}is the reference pressure; ρ is the actual density of the air; $\alpha $ is the power low exponent, i.e., roughness coefficient; and z is the height (spatial coordinate). The intensity of the turbulence can be estimated with the use of the following formula:

_{v}is the standard deviation of the measured wind speed, The standard deviation can be computed based on the dynamic component v

_{dyn}(t) of the instantaneous velocity V(t), where V(t) = V

_{ref}+ v

_{dyn}(t). The obtained wind profiles and the turbulence intensity for both investigated cases are depicted in Figure 9 and Figure 10, respectively.

#### 2.5. Measurement of the Force Components Acting on the Model and Corresponding Wind Speed

_{X}, the coefficient of the lateral aerodynamic force C

_{Y}, and the aerodynamic coefficient of torque C

_{Mz}:

_{X}is the aerodynamic drag (N); F

_{Y}is the lateral force (N); V is the average wind speed (m/s); ρ is the mass air density; A

_{ref}is the effective area of one of the supporting structures of the model, i.e., the area of the shadow normally projected by its members on a plane parallel to the wall; and b

_{ref}is the conventionally accepted characteristic dimension. Table 3 shows the appropriate reference values for the force coefficients.

#### 2.6. CFD Simulation

_{H}= 1.711 m). The model was placed 1500 mm from the inlet (blue wall in Figure 13) and 2000 mm from the outlet. The assumed dimensions of the aerodynamic tunnel precisely correspond to the geometric dimensions of the real aerodynamic tunnel. In the simulations, the standard air properties were assumed; therefore, ρ = 1.225 kg/m

^{3}, T = 15 °C, and p

_{0}= 101,325.25 Pa. The walls of the aerodynamic tunnel as well as the whole surface of the studied models were assumed to be stationary boundaries and a steady state of airflow was assumed.

^{−5}m in height. Unfortunately, this leads to an enormous number of nodes and finite element volumes. Finally, the standard k-ε model with a standard wall function was used. It is worth noting that this model is still in use, for example, in the works of Zan et al. [19], Chen et al. [21], Lu et al. [32], and Augustyn et al. [57].

_{c}= 1.5 mm. The length of the edges of the faces of the elements that belong to the circular plates and the vertical rod is equal to l

_{c}= 3.4 mm. The maximal length of the rest cells is equal to l

_{c}= 108.867 mm. Figure 14 represents the cell mesh generated for the tower truss model.

_{v}= 9%. Table 5 shows the corresponding number of the nodes, faces, and cells. It should be noted here that the choice of the size of the cell was limited because for a cell size greater than l

_{c}= 0.002 m, the mesh on the ladder rungs would not generated properly, whereas for a cell size less than l

_{c}= 0.001 m, the number of the cells was extremely large. Moreover, for the k-ε turbulent model, and in the case where the cell size is very small, obtaining the convergent solution can be difficult. As is reported in Table 4, it is necessary to perform over 600 iterations to obtain a convergent solution. This problem is mainly connected to satisfying the continuity criterion. For other cell sizes, there were no problems with obtaining the convergent solution, and the number of necessary iterations did not exceed 50. Taking into consideration the results from Table 4, it seems that the choice of the cell size lc = 0.0015 m is reasonable. It is worth noting that for the tetrahedral elements, obtaining the converged solution would require several hundred iterations.

## 3. Results

#### 3.1. Experimental Results

_{v}= 3%; for tests using turbulent airflow, the average turbulent intensity was I

_{v}= 9% and 12% for the tower truss and jib truss models, respectively. In all investigated cases, the nominal wind speed was equal to V = 15 m/s. Moreover, it was assumed that the wind direction varied from 0° to 180° with a step of 15°. Figure 15, Figure 16, Figure 17 and Figure 18 show the profiles of the measured wind speed as a function of time. In the case of “quasi-laminar” flow, the wind speed did not change significantly. However, in turbulent flow, the wind speed varied considerably. Thus, in this case, precise wind measurements should be considered quite problematic.

_{X}force components for β = 45° and β = 135° are not equal to each other. Generally, together with increasing turbulent intensity, the values of these components of the aerodynamic forces should increase. Contrary to the aerodynamic force F

_{X}components, the F

_{Y}components as well as the values of the M

_{Z}moments are close to zero for both truss models.

#### 3.2. Comparison of the Numerical and Experimental Results

_{X}and C

_{Y}as a function of the angle β. As can be observed, the results reveal a relatively good agreement.

_{AVG}for coefficient C

_{X}for all investigated cases are shown in Table 10. These values were estimated according to the following formula:

_{X}

^{EXP}and C

_{X}

^{NUM}are the aerodynamic coefficients obtained from the experiment and numerical simulation, respectively.

_{X}, which plays an important role when the tip-over of the tower crane could happen. Moreover, the best match was obtained for the tower crane model in the case of turbulent flow. Finally, it is worth noting that in the case of the force coefficient C

_{Y}, the results from the experimental tests and numerical simulations reveal noticeably worse agreement in comparison with those of the C

_{X}coefficients. This could be explained by the fact that the F

_{Y}force component had values that are close to zero. Thus, it was measured during the experiment with a relatively large error.

## 4. Discussion

_{x0}(φ) [53] is as follows:

_{x0}(φ) is as follows:

_{e}is the enclosed area.

_{x0}is written as [53]

- For “quasi-laminar” flow, the value of the drag coefficient C
_{X0}obtained from the tests was 2.54 (Figure 23), and its value is between the curves given by Górski [53]. Note that the value of C_{X45}, which is equal to 3.22, fits the curve given by Cohen and Perrin [55]. This value is almost equal to C_{X0}^{t}, which is equal to 3.27. - For turbulent flow, the drag coefficient C
_{X0}^{t}obtained from the tests was 3.27, and C_{X45}^{t}was 4.11 (Figure 23). This proves the high impact of turbulence on the tested object. The C_{X0}^{t}coefficient is in line with the curve given by Cohen and Perrin [55] for a wind angle of attack of β = 45°.

_{X}for the jib truss model were calculated. Figure 24 shows the dependency C

_{X}= f(φ) for spatial trusses on the triangular base made of steel structural sections and tube, as recommended by French [54] and English [56] standards, and the results of the wind tunnel tests.

- For “quasi-laminar” flow, the resistance coefficient C
_{X90}obtained from the tests was 2.16 (Figure 24) and is close to the value from the curve given by the French standard for steel sections. This value is much higher than the values given for pipes by both the French [54] standard and the English standard [56], while C_{X60}has a value close to the curve given by the Polish standard for steel sections. - For turbulent flow, the value of the drag coefficient C
_{X90}^{t}obtained from the tests was 2.38 (Figure 24), and it indicates a high impact of turbulence on the tested object—a value above the values given by the French [54] and English [56] standards. It should be noted that the resistance coefficient C_{X60}^{t}is equal to 3.09.

## 5. Conclusions

_{X}, the average error between the numerical and experimental values did not exceed 8%.

_{X}and the coefficient of lateral aerodynamic force C

_{Y}was observed, as well as an increase in the value of the aerodynamic drag for the turbulent flow.

_{X}for laminar and turbulent flows for a wind angle of attack of β = 0° for the tower section model made of closed profiles was 28.7%. However, in the case of the sectional model of the jib (also made of closed profiles), the relative difference between the C

_{X}values of laminar flow and turbulent flow with a wind angle of attack of 60° was 13.6%, and for an angle of 90°, it was 10.2%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Fast-erecting Turmdrehkran 63K crane by Liebherr. (1) Stationary support, (2) tower crane chassis, (3) movable platform, (4) lifting mechanism, (5) rotation mechanism, (6) switch cabinet, (7) counterweight, (8) rope immobilizing the horizontal jib, (9) lower part of the tower, (10) upper part of the tower, (11) auxiliary drive, (12) lifting winch, (13), trolley-driving mechanism, (14) rear lashing, (15) guy support I, (16) crane jib—lashing rope II, (17) crane jib—lashing rope III, (18) crane jib—lashing rope I, (19) lifting rope, (20) jib—articulated element, (21) jib—assembly rope, (22) guy support II—head, (23) jib extension, (24) jib extension, (25) jib extension, (26) hook (2/4 strand). The dimensions of the crane are listed in Table 1.

**Figure 3.**Model of the truss of the tower of the crane: (

**a**) the experimental specimen; (

**b**) the CFD simulation.

**Figure 5.**Model of the truss of the horizontal jib of the crane: (

**a**) the experimental specimen; (

**b**) the CFD simulation.

**Figure 6.**The orientation of the model in the x, y, z coordinate system and the aerodynamic test conditions: the assumed wind direction W; the aerodynamic moments M

_{X}, M

_{Y}, and M

_{Z}; and aerodynamic forces F

_{X}and F

_{Y}.

**Figure 7.**Orientation of the sectional model of the tower during the test (β—wind attack angle, W—wind direction): (

**a**) model of the tower; (

**b**) model of the jib.

**Figure 8.**The view of the aerodynamic tunnel for (

**a**) “quasi-laminar” flow and (

**b**) turbulent flow. On the left side, there is a vertical rod with six thermo-anemometers for wind profile estimation.

**Figure 9.**The wind speed profile for (

**a**) flat open terrain (“quasi-laminar” flow) and (

**b**) for urban terrain (turbulent flow) according to the Eurocode standard [59].

**Figure 10.**The turbulent intensity profile for (

**a**) flat open terrain (“quasi-laminar” flow) and (

**b**) urban terrain (turbulent flow).

**Figure 12.**Diagram of the measuring system. 1—the model of the truss; 2—the five-component aerodynamic balance, 3—amplifier module with data acquisition systems, 4—the turntable with stepper motor, 5—the motor step control system, 6—the pressure sensor Pitot pipes located on the x, z plane, 7—the pressure scanner, 8—the wind sensor, 9—thermo-anemometer AMD 2000, 10—DaqBoard/2000 card, 11—PC.

**Figure 13.**Investigated model of the truss inside the volume filled with air. Blue wall is the inlet.

**Figure 14.**The mesh of the cells generated for the tower truss model (length of the cell edge l

_{c}= 0.0015 m).

**Figure 15.**Average speed and speed from the thermo-anemometer measured every 5 s for “quasi-laminar” airflow and wind direction β = 0° (tower truss model).

**Figure 16.**Average speed and speed from the thermo-anemometer measured every 5 s for turbulent airflow and wind direction β = 0° (tower truss model).

**Figure 17.**Average speed and speed from the thermo-anemometer measured every 5 s for “quasi-laminar” airflow and wind direction β = 0° (jib truss model).

**Figure 18.**Average speed and speed from the thermo-anemometer measured every 5 s for turbulent airflow and wind direction β = 0° (jib truss model).

**Figure 19.**Comparison of the force coefficients C

_{X}and C

_{Y}for the experimental and numerical results as a function of angle of wind attack β in the tower truss model with “quasi-laminar” airflow.

**Figure 20.**Comparison of the force coefficients C

_{X}and C

_{Y}for the experimental and numerical results as a function of angle of wind attack β in the tower truss model with turbulent flow.

**Figure 21.**Comparison of the force coefficients C

_{X}and C

_{Y}for the experimental and numerical results as a function of angle of wind attack β in the jib truss model with “quasi-laminar” flow.

**Figure 22.**Comparison of the force coefficients C

_{X}and C

_{Y}for the experimental and numerical results as a function of angle of wind attack β in the jib truss model with turbulent flow.

Technical Data | Value |
---|---|

Maximum permissible lifting capacity (kg) | 1750 |

Range of crane rotation angle around its own axis (°) | 360 |

Distance between end point of the jib and the axis of rotation (m) | 35 |

Maximum lifting angle of the entire jib (°) | 30 |

Maximum lifting angle of the second part of the jib (°) | 45 |

Total height of the tower crane (m) | 34.6 |

Spacing of the stationary supports (m) | 4.4 |

Total distance between the mobile platform and the axis of rotation (m) | 3.6 |

Distance between the rope immobilizing the jib to the axis of rotation (m) | 1.85 |

Height of the entire tower with chassis and stationary supports (m) | 31 |

Scale Name | Scale Mark | Value |
---|---|---|

Length scale | k_{L} | 1/7 |

Velocity scale | k_{V} | 1 |

Density scale (for air) | k_{ρ} | 1 |

Viscosity scale | k_{v} | 1 |

Gravity scale | k_{g} | 1 |

Sound speed scale | k_{a} | 1 |

Force scale | k_{P} | 1/49 |

Pressure scale | k_{p} | 1 |

Time scale | k_{t} | 1/7 |

Frequency scale | k_{f} | 7 |

Ratio of the mass density per element unit scale (structure) | k_{m} | 1/49 |

Ratio of the density of inertia moment per element unit length scale | k_{mb} | 1/2401 |

Longitudinal stiffness scale | k_{EA} | 1/49 |

Bending stiffness scale | k_{EI} | 1/2401 |

Torque stiffness scale | k_{GIs} | 1/2401 |

Model | A_{ref} (m^{2}) | b_{ref} (m) |
---|---|---|

Part of the tower truss | 0.021438 | 0.552 |

Part of the jib truss | 0.017304 | 0.138 |

**Table 4.**The values of the aerodynamic forces and corresponding coefficients for different mesh sizes. Wind speed V = 15 m/s, turbulent intensity I

_{v}= 9%.

Cell Size (mm) | F_{X} (N) | F_{Y} (N) | M_{Z} (Nm) | C_{X} | C_{Y} | C_{Mz} | Number of Iter |
---|---|---|---|---|---|---|---|

1.000 | 7.765 | 0.083 | 0.015 | 2.628 | 0.028 | 0.009 | 609 * |

1.500 | 7.748 | 0.088 | 0.017 | 2.622 | 0.030 | 0.010 | 48 |

2.000 | 7.808 | 0.059 | 0.018 | 2.643 | 0.020 | 0.011 | 42 |

Cell Size (mm) | Faces (Truss *) | Nodes | Cells |
---|---|---|---|

1.000 | 17,458,489 (317,697) | 12,446,064 | 3,088,976 |

1.500 | 9,977,802 (160,973) | 7,076,642 | 1,775,847 |

2.000 | 6,854,749 (92,731) | 4,859,870 | 1,219,894 |

Angle β (°) | F_{X} (N) | F_{Y} (N) | M_{Z} (Nm) |
---|---|---|---|

0 | 6.086 | −0.832 | 0.277 |

15 | 7.061 | −0.034 | 0.245 |

30 | 7.682 | −0.418 | 0.310 |

45 | 7.711 | −0.654 | 0.391 |

60 | 7.593 | −0.595 | 0.457 |

75 | 6.854 | −0.832 | 0.457 |

90 | 6.234 | 0.143 | 0.391 |

105 | 7.357 | 0.468 | 0.261 |

120 | 8.420 | 0.409 | 0.408 |

135 | 8.952 | 0.025 | 0.408 |

150 | 8.568 | −0.418 | 0.440 |

165 | 7.475 | −0.595 | 0.408 |

180 | 6.618 | −0.093 | 0.310 |

Angle β (°) | F_{X} (N) | F_{Y} (N) | M_{Z} (Nm) |
---|---|---|---|

0 | 8.420 | −1.009 | 0.261 |

15 | 9.306 | −0.743 | 0.245 |

30 | 10.370 | −1.245 | 0.326 |

45 | 10.577 | −1.482 | 0.408 |

60 | 10.104 | −1.275 | 0.424 |

75 | 8.716 | −1.600 | 0.440 |

90 | 8.184 | −1.157 | 0.408 |

105 | 8.834 | −0.507 | 0.391 |

120 | 9.809 | −0.861 | 0.457 |

135 | 9.041 | −0.684 | 0.473 |

150 | 9.306 | −1.245 | 0.522 |

165 | 8.243 | −1.304 | 0.489 |

180 | 7.091 | −0.891 | 0.359 |

Angle β (°) | F_{X} (N) | F_{Y} (N) | M_{Z} (Nm) |
---|---|---|---|

0 | 5.580 | 0.021 | 1.079 |

15 | 4.769 | 1.332 | 0.214 |

30 | 4.579 | 0.474 | 0.688 |

45 | 5.103 | 0.283 | 0.645 |

60 | 4.865 | 0.307 | 0.704 |

75 | 4.221 | −0.409 | 0.293 |

90 | 3.863 | 0.378 | 0.625 |

105 | 4.078 | 1.022 | 0.685 |

120 | 5.032 | 0.021 | 0.790 |

135 | 5.294 | −0.146 | 0.471 |

150 | 5.008 | −0.337 | 1.116 |

165 | 4.793 | −1.267 | 0.253 |

180 | 4.984 | −0.099 | 0.661 |

Angle β (°) | F_{X} (N) | F_{Y} (N) | M_{Z} (Nm) |
---|---|---|---|

0 | 7.822 | −0.671 | 0.299 |

15 | 6.963 | 0.665 | 0.504 |

30 | 6.725 | −0.575 | 0.425 |

45 | 7.774 | −0.981 | 1.251 |

60 | 7.369 | −0.885 | 0.918 |

75 | 6.987 | −1.696 | 0.306 |

90 | 5.676 | −0.480 | 0.230 |

105 | 6.439 | 0.021 | 0.346 |

120 | 6.916 | −0.623 | 0.375 |

135 | 7.369 | −1.076 | 0.290 |

150 | 7.035 | −1.553 | 0.244 |

165 | 7.464 | −2.579 | 0.257 |

180 | 6.916 | −1.243 | 0.247 |

Model | Airflow | Average Error ε_{AVG} (%) |
---|---|---|

Part of the tower truss | “Quasi-laminar” | 4.956 |

Turbulent | 7.928 | |

Part of the jib truss | “Quasi-laminar” | 6.550 |

Turbulent | 6.659 |

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## Share and Cite

**MDPI and ACS Style**

Augustyn, M.; Barski, M.; Chwał, M.; Stawiarski, A.
Experimental and Numerical Estimation of the Aerodynamic Forces Induced by the Wind Acting on a Fast-Erecting Crane. *Appl. Sci.* **2023**, *13*, 10826.
https://doi.org/10.3390/app131910826

**AMA Style**

Augustyn M, Barski M, Chwał M, Stawiarski A.
Experimental and Numerical Estimation of the Aerodynamic Forces Induced by the Wind Acting on a Fast-Erecting Crane. *Applied Sciences*. 2023; 13(19):10826.
https://doi.org/10.3390/app131910826

**Chicago/Turabian Style**

Augustyn, Marcin, Marek Barski, Małgorzata Chwał, and Adam Stawiarski.
2023. "Experimental and Numerical Estimation of the Aerodynamic Forces Induced by the Wind Acting on a Fast-Erecting Crane" *Applied Sciences* 13, no. 19: 10826.
https://doi.org/10.3390/app131910826