Numerical and Analytical Estimation of the Wind Speed Causing Overturning of the Fast-Erecting Crane—Part II

Featured Application

The presented results of the current work can be exploited to ensure safer exploitation of the tower cranes concerning the strong wind and the possibility of overturning them.

Abstract

The currently presented work is a continuation of the previous one, where the estimation of the forces induced by the wind flow acting on the fast-erecting crane. In that work, the values of the aerodynamic forces were determined experimentally and numerically for the sectional models of the tower and jib. Next, the obtained results were compared with the appropriate standards. Now, the main aim is to determine the critical wind speed causing the overturning of the whole structure. At the very beginning, the numerical analysis of the simplified model of the crane on the real scale is studied. The computations are performed with the use of the ANSYS FLUENT R22. The simulations are performed for three different wind profiles, namely: urban terrain, village terrain, and open terrain. Moreover, the various geometric configurations of the crane in the wind direction are studied. The k-ε model of turbulent flow is exploited. The obtained critical values of the wind speed are confronted with those that are obtained from standards and estimations based on the results obtained from previous investigations performed for sectional models. The influence of the load carried by the crane is also taken into consideration in the overturning of the structure.

1. Introduction

With rapid climate change, strong, unexpected winds or gusts of wind have become a real danger for the various high-slender structures. Examples of such structures are the tower cranes used practically at all construction sites. Moreover, these kinds of weather phenomena take place in countries like Poland, where they have very rarely been reported before. Two tragic accidents involving tower cranes have occurred in Cracow (the southern part of Poland) for the last two years. These accidents were caused by a strong, unexpected gust of wind. In the first case [1,2], two people died. In the second case [3], one person died and three were injured. It shows how important this problem is.

Generally, the wind is not a steady-state phenomenon. It is rather a dynamic phenomenon accompanied by turbulence and sudden changes in direction. Moreover, the gusts of wind can be even two times greater in comparison to the mean wind speed. The frequency of the gusts most frequently is about 1 Hz [4]. The impact of the wind on high-slender lattice structures, like various tower cranes, could be completely different depending on the wind speed and strength. The wind can cause the vibrations of the lifted load or crane itself. Modern slender steel lattice structures, like tower cranes, consist of members that carry high-tension forces. It causes solid profiles to possess low-damping properties. Therefore, the tower cranes are sensitive to wind-induced vibrations with increasing amplitude, so-called galloping vibrations. Such a phenomenon is reported in the case of the tension bars of slender tower cranes. It leads to fatigue damage to the individual elements or even total failure of the whole structure [5]. The analysis of the dynamic response of the tower cranes is more complicated due to wind excitation, which possesses rather random characteristics [6]. Next, Jiang and Li [7] investigated the dynamic reliability of the tower cranes with the wind load. They used a linear autoregressive model to simulate the time history of multidimensional fluctuating wind samples in conjunction with the finite element method. and concluded that for a wind speed of 20 m/s, the structural strength is relatively sufficient. Further detail analysis reveals that the most dangerous geometry configuration of the crane is when the horizontal jib is not strictly perpendicular to the wind direction [8]. Hechmi El Ouni et al. [9] perform a numerical analysis of the tower carne loaded by the turbulent wind with the use of the finite element method to design the active system of vibration damping, which prevents the crane from collapsing. Oliveira and Correia [10] investigated the vibration of two different tower cranes induced by seismic and wind excitation. They used the finite element method, namely SAP2000 software. Finally, the tower cranes are also installed in super-high-rise buildings. A separate issue is the phenomenon related to wind-affecting tower cranes used during the construction of these kinds of super-high-rise buildings [11,12].

The effect of the wind is not limited only to the crane itself but also to the load being lifted. Depending on the size, shape, and weight of the load, this influence may have a decisive impact on the safe operation of the whole structure. Skelton et al. [13] designed a special container of wing shape for lifted loads for tower cranes, which operate on the construction sites of high buildings. This container significantly reduces the aerodynamic drag force caused by the wind. Monteiro and Moreira [14] analyzed an offshore oil and gas platform module lifted by a crane. They assumed the pendulum-like displacement of the lifted load was caused by the wind. The pendulum-like displacement is determined via a finite difference scheme, whereas the drag coefficients of the platform module are estimated through CFD analysis. Cekus et al. [15,16] studied the dynamic behavior of the lifted load of cuboid shapes by crane. The lifted load is subjected to wind. Finally, Jin et al. [17] investigated the impact of the lifted load subjected to wind on the tower crane. The load under wind pressure is modeled as a pendulum. Nguyen [18] designed a special spring-damper device that can reduce the payload vibration caused by wind.

If the wind or gust of wind is sufficiently strong, the whole tower crane can be overturned. This kind of accident is the most dangerous because of the potential deaths and relatively significant material losses. However, works devoted strictly to the problem of tipping over cranes of different kinds or other similar structures caused by a strong wind are very rare. Here can be quoted the following works concerning the overturning of the gantry container crane with payload [19], gantry cranes [20,21,22], or scissor lift [23].

Finally, it is worth noting that the surroundings (for example, buildings and other cranes) of the operating crane-like structures can have a significant impact on the wind load acting on the investigated structure. This phenomenon is known as an interference effect. As before in the case of the overturning of the crane-like structures due to wind load, the works devoted to this problem are also very rare. Here can be quoted the work by Wu et al. [24], where the system of three container cranes is studied. Each other’s impact as well as the dynamic response of the cranes are studied. Furthermore, the antenna system and its mutual impact are investigated by Holmes et al. [25], Martin et al. [26], and Carril et al. [27]. It has been found that the interference factor of microwave antenna dishes is greater than one for specific wind directions. It is worth noting that the lattice structure of the antenna is very similar to that of most crane structures. This phenomenon is also studied in the case of scaffolding [28], low-rise buildings [29,30,31,32], tall buildings [33], and cooling towers [34,35].

The currently presented work is a continuation of the problem discussed in the previous one [36]. In that work, we tried to determine the magnitude of the aerodynamic forces induced by the wind acting on the sectional model of the tower lattice and the horizontal jib lattice. The experimental tests in the aerodynamic tunnel and the CFD simulations were carried out. The obtained results were compared with appropriate standards and codes. The main aim of the current work is to investigate the impact of the wind on the fast-erecting crane, namely determining the most dangerous geometric configuration of the structure against the wind direction. We are also going to take into account the potential external load that is carried by the crane. To deal with this problem, we will exploit the results reported in the previous work as well as the CFD simulations of the whole structure on a real scale with the various wind profiles considered.

2. Materials and Methods

2.1. Object of Study

As in the previous work [36], the object of investigation is a fast-erecting 63 K crane by Liebherr. Its detailed description and the technical parameters are described in [36,37]. Here, it is worth noting that this structure is very similar to that that fell in Cracow in 2022 [1]. In Figure 1, the studied crane is shown. Here we only mention that the total height of the crane is equal to H = 31 m, and the length between the axis of rotation and the end of the jib is equal to L = 35 m.

To perform the CFD simulations of the airflow around the investigated crane, an appropriate simplified model of the real structure should be prepared. It is worth noting that the geometry model should enable the automatic generation of a good-quality mesh of the finite elements. Therefore, in such a complicated object as a lattice structure, some simplification must be introduced. Further, all simplifications will be discussed concerning the real structure. For the creation of the CAD model, the ANSYS R22 DesignModeler software is exploited. The simplifications that are introduced are as follows:

• The CAD model does not contain: rope immobilizing the horizontal jib, rear lashing, guy support I, crane jib—lashing rope II, crane jib—lashing rope III, crane jib—lashing rope I, jib—assembly rope, guy support II—head, lifting rope with hook.
• The jib extension at the end of the jib is made of the same lattice as other parts of the horizontal part of the crane.
• The wider and narrower parts of the crane tower truss are connected directly. These parts do not overlap each other like in the real structure.
• The dimensions of the transverse sections of some elements of the lattice are slightly changed to avoid problems and errors during automatic mesh generation.

In Figure 2, the details of the CAD model of the studied fast-erecting crane are depicted.

2.2. CFD Simulation

The CFD simulations of the airflow around the studied lattice structure of the studied tower crane were performed with the use of the ANSYS R22 Fluent with Meshing module. The computations were carried out for various configurations of the crane geometry concerning the wind direction as well as for three different wind profiles (for urban terrain, village terrain, and open terrain). In Figure 3, it is depicted how the geometry of the crane is defined concerning the wind direction. The assumed wind direction is parallel to the X-axis of the global Cartesian coordinate system. The geometry configuration is defined by the angle θ between the X-axis of the coordinate system and the horizontal jib. The computations were performed for the following angles, namely: θ = 0°, 15°, 30°, 45°, 60°, 75°, and 90° for each assumed wind profile.

The investigated structure is immersed in the cuboid volume filled with an air of geometrical dimensions: width × length equal to 125 m × 130 m, and height equal to 65 m. One wall is assumed to be an inlet (blue wall in Figure 4), and the opposite one is the outlet, as shown in Figure 4. The ground (the wall on which the crane is installed) is stationary, while the rest of the walls are assumed to be movable. The wind speed on these walls varies according to the predefined wind profile for the inlet. The external dimensions of the cuboid seem to be enough. The lattice structures, like tower cranes, cause a limited disturbance of the airflow due to the relatively small area of the whole structure. Standard air properties at sea level (temperature T = 15 °C, ambient pressure p₀ = 101,325.25 Pa) were assumed, namely: kinematic viscosity ν = 1.7894 × 10⁵ kg/(m·s), density ρ = 1.225 kg/m³.

For simulation of the turbulent flow of air, the standard k-ε model with a standard wall function is exploited. According to the authors’ experience so far [22,23,36], this model allows obtaining sufficiently precise results in a situation where the magnitude of the resultant aerodynamic forces (static pressure) is mainly determined by the drag force, while the lift force is several orders of magnitude smaller. Furthermore, the k-ε model of turbulent flow is still successfully exploited by other authors, for example [8,11,38]. Here, it is worth noting that the other available models of turbulent flow are rather difficult to use in this case. For example, the use of the Reynolds stress model [24] causes the convergence of the numerical solution to be unacceptably slow. Another possibility is to involve the k-ω turbulent flow [14,39,40]. In the ANSYS R22 Fluent software, this model is available in several different variants. Depending on the variant of this model used, the results in the form of aerodynamic force values may vary significantly. Moreover, when using the k-ω model, it is necessary to generate an appropriate FEM mesh in the boundary layer. This results in a significant increase in the number of elements, which seems unacceptable in the case of the currently considered structure.

2.2.1. Wind Profile

The investigated structure is subjected to a wind load. Generally, the wind profile of Davenport is described by the following formula [41]:

$$\left(z\right)=v_g{\left(\frac{z}{z_g}\right)}^{\alpha }$$

(1)

where v_g is the wind velocity measured at the height z_g over the ground level. The exponent α takes values of 0.4, 0.28, and 0.16 depending on the type of terrain, namely urban terrain, village terrain, or open terrain, respectively. In the current work, we arbitrarily assumed that in all studied cases v_g = 6 m/s at the height z_g = 10 m. In Figure 5, the assumed wind profiles are shown.

Based on the wind profiles depicted above, the reference wind velocity is also computed for the three wind velocity profiles according to the following formula:

$$v_{ref}=\frac{{{\displaystyle \int }}_0^{H_z}V\left(z\right)dz}{H_Z},$$

(2)

where H_Z is the total height of the crane, where H_z = 34.5 m. The values of the reference wind speed, computed according to Formula (2) for assumed wind profiles, are equal as follows: urban terrain v_ref = 7.0321 m/s, village terrain v_ref = 6.62928 m/s, and open terrain v_ref = 6.30482 m/s. These values will be used to compute the aerodynamic force and moment coefficients for the whole crane structure. Moreover, it is assumed that the turbulent intensity equals 9% with the turbulent length scale L_TURB = 3.5 m. These values are assumed to be arbitrary. The choice was made according to the parameters discussed in our previous work.

2.2.2. Finite Cell Mesh

In Figure 6, the finite cell mesh is generated automatically by the ANSYS Fluent Mesh module. It is assumed that the minimal length of the cell edge is equal to l_min = 0.028 m (value independent of the geometry configuration), and the maximal length of the cell edge is generally greater than l_max > 6 m, depending on the geometry configuration.

Such a minimal cell edge length ensures that during the automatic mesh generation of tetrahedral elements, there are no degenerated or highly distorted cells. The existence of such cells makes it very difficult to obtain a convergence solution to the numerical problem. For considered geometry configurations, the number of nodes is over 2.5 million, and the number of cells is over 14 million. Depending on the geometry configuration, the number of nodes and cells varies not significantly.

To verify the precision of the numerical solutions, which are obtained for the assumed minimal length of the cell edge, the convergence test was performed. In

Table 1. The total number of nodes, faces, and cells.

Cell Size [m]	Faces 1	Nodes	Cells
0.030	750,922	2,290,762	12,676,871
0.028	852,216	2,593,299	14,342,502
0.020	1,705,888	4,813,382	26,393,269

¹ The number of faces generated on the crane surface.

and

Table 2. Results of the converged test.

Cell Size [m]	FX [N]	FY [N]	Mtip [Nm]	Number of Iter.
0.030	2227.649	−43.411	61,278.525	91
0.028	2221.892	−34.000	61,043.122	87
0.020	2196.509	−31.652	60,187.756	51

, the total number of nodes, faces, and cells generated for the following minimal lengths of the cell edges is reported: l_min = 0.056 m, 0.028 m, and 0.014 m. The convergence test was performed for the geometry configuration, for which the angle θ = 90°. The overturning moment M_tip is the moment that is measured concerning the tipping line, which is shown in Figure 7.

Taking into consideration the results presented above in Table 1 and Table 2, the assumed value of l_min = 0.028 m is quite reasonable. All other calculations will be performed for this value of l_min.

Finally, it is worth noting that l_e = 0.03 m is the largest element size, which enables automatic finite cell generation without warnings or errors. On the other hand, l_e > 0.02 m leads to an enormous, large number of nodes and cells. Moreover, considering the mentioned limitation connected with element size, the performed convergence test should be rather threaded as a sensitivity analysis.

3. Results of CFD Simulation

Figure 8a–c presents the results of the carried-out CFD simulations for different kinds of terrain and geometry configurations of the crane. It should be noted that F_Z aerodynamic force components are one order of magnitude less in comparison with the F_Y components. Therefore, they are not reported here. As can be observed in all investigated cases, the maximum of the F_X component of the aerodynamic force and the overturning moment C_M are observed for the angle θ = 75°. For larger values of the angle θ, these values decrease. The value of the F_Y aerodynamic force is one order of magnitude less in comparison with the F_X component. It is worth stressing here that for such defined wind profiles, the greatest values of the aerodynamic forces are obtained for urban terrain and the smallest ones for open terrain. Finally, in

Table 3. The values of aerodynamic coefficients for urban terrain.

	Urban Terrain vref = 7.03210 [m/s]			Village Terrain vref = 6.62928 [m/s]			Open Terrain vref = 6.340482 [m/s]
Angle α [°]	CX	CY	CM	CX	CY	CM	CX	CY	CM
0	1.517	0.000	0.900	1.511	0.002	0.831	1.530	0.03	0.762
15	2.077	−0.049	1.370	1.990	−0.029	1.229	1.939	−0.005	1.095
30	2.520	−0.295	1.733	2.395	−0.246	1.551	2.319	−0.202	1.385
45	2.956	−0.452	2.136	2.780	−0.393	1.896	2.636	−0.343	1.664
60	3.247	−0.458	2.463	3.007	−0.398	2.161	2.827	−0.352	1.888
75	3.272	−0.341	2.580	3.004	−0.308	2.252	2.774	−0.281	1.944
90	3.147	−0.048	2.506	2.856	−0.074	2.178	2.613	−0.111	1.870

A_ref = 23.31 m², B_ref = 34.5, m, ρ = 1.225 kg/m³.

, the aerodynamic force coefficients are collected, which are computed according to Formula (3), where v_ref is determined with the use of (1) and (2).

From the obtained values of the F_X and F_Y forces, the aerodynamic coefficients C_X and C_Y were determined, which served as the basis for stability calculations of the 63 K crane. The appropriate values of the aerodynamic force and moment coefficients are determined with the use of the following formulas:

$$C_X=\frac{2F_X}{\rho \hspace{0.33em}{v}_{ref}^2{A}_{ref}}, C_Y=\frac{2F_Y}{\rho \hspace{0.33em}{v}_{ref}^2{A}_{ref}}, C_M=\frac{2M}{\rho \hspace{0.33em}{v}_{ref}^2{A}_{ref}{B}_{ref}}$$

(3)

where A_ref is the area of the shadow normally projected by the crane lattice members on a plane parallel to the wall; A_ref = 23.31 m²; and B_ref is the conventionally accepted characteristic dimension. In our case, B_ref is assumed to be equal to the total height of the crane; B_ref = H_Z. The convergence test is carried out for urban terrain, where v_ref = 7.0321 m/s. Other parameters are as discussed above. The values of the obtained aerodynamic coefficients are shown in Table 3.

3.1. Validation of Numerical Results Obtained from CFD Simulation with Experimental Results for Sectional Models

The results of the CFD simulations have been verified by comparing the overturning moments that act on the structure of the tower crane 63 K concerning one of the tipping edges for the two characteristic positions of the tower jib, namely for the angles θ = 0° and θ = ±90°. As a basis for the verification, it is assumed that the experimental results reported in [36]. These results concern the values of the aerodynamic forces F_X and F_Y obtained for the sectional models of the tower truss and jib truss of the tower crane 63 K. With the values of these forces, it is possible to estimate the overturning moment caused by the wind according to the appropriate standard [41]. In the case of standard, the most dangerous position of the crane jib concerning the wind direction is taken into consideration, namely, the jib position is perpendicular to the wind direction θ = 90°.

The overturning moments are equal to the sum of the moments caused by the masses of the particular parts of the crane jib G1–G4 without the external load but with the aerodynamic force generated by the wind taken into account, as shown in Figure 9. The geometrical dimensions as well as the masses of the structure elements are assumed according to the technical and commissioning documentation [37]. Therefore, it should be noted here that the mentioned parameters are estimated with a certain approximation, and they should not be treated as data provided by the producer. The test object itself is treated as an example for calculating the stability of a low-slewing crane.

The schema of the force distribution acting on the studied crane is shown in Figure 9 and

Table 4. Selected geometrical dimensions of the Liebherr 63 K crane and the values of moments used for calculations. The maximum load weight at a height of 21 m is 3050 kg [41].

Overturning Moments from Crane Weights Only ΣMO = 1228.90 kNm							Stability Moments from Crane Weights Only ΣMS = 1380.85 kNm
Jib					Max Load		Tower	Support Base of Tower	Counterweight
load
G1	G2	G3	G4	G5	Q1		G6	G7	G8
130	330	1440	1440	200	3050	kg	7600	7660	26,000
1275	3237	14,126	14,126	1962	29,921	N	74,556	75,145	255,060
distance from the tipping line
r1	r2	r3	r4	r5	rQ5		r6	r7	r8
39	33	22	6	21	21	m	0.87	2.10	4.54
moments
M1	M2	M3	M4	M5	MQ5		M6	M7	M8
49.17	106.75	313.54	88.52	41.29	629.63	kNm	64.67	157.80	1158.38

. The aerodynamic forces induced by wind are marked with the symbol W. The values of these forces are determined in the next step of the analysis according to the standard Eurocode for θ = 90° [41], experimental results for θ = 0° and θ = ±90° [36], and the results obtained from the CFD simulations for the θ range from −90° to +90° with steps equal to 15°.

3.2. The Wind Force Estimation According to the Standard

The estimation of the wind force W according to the standard [41] is based on the reference area A_ref of the whole supporting structure of the tower crane and the air pressure, which acts on the structure under assumed wind speed v_g = 6 m/s at the height H = 10 m above the ground level for the urban terrain. Due to the significant changes in the wind force together with the increasing height of the tower crane, the reference area A_ref was split into three parts, namely: A_ref1 = 11.04 m² of the crane jib, A_ref2 = 12.35 m² of the tower, and A_ref3 = 3.91 m² of the counterweight. It should be stressed that according to various standards [39,40,41,42,43,44,45,46], the load induced by the wind, which acts on structures, is treated as a static normal pressure subjected to the external surface of a designed structure. Therefore, it is assumed that the wind pressure p = 245 N/m² corresponds to the wind speed equal to v = 20 m/s. Thus, for a particular reference area and wind speed v_g = 6 m/s at height h = 10 m the following forces are obtained, namely: for A_ref1, where h₁ = 33.81 m, W₁ = 359.6 N; for A_ref2, where h₂ = 20.33 m, W₂ = 341.84 N; and for A_ref3, where h₃ = 2.83 m, W₃ = 57.5 N. Regarding the fact that the value of W3 is relatively small in comparison with the other ones (as well as the small height and large mass of the counterweight), in further computations, the overturning moment induced by this wind force was neglected. The obtained results M_{O ECode90°} as a sum of the overturning moments M_O (Table 4) and the moments induced by wind M_{W ECode90°} [41] with the wind force taken into account are shown in Figure 10 as a constant value, which is independent of the rotation of the jib.

3.3. The Wind Force Estimation Based on the Experiment

In Figure 10, there are three points that correspond to the overturning moments M_{O exp 0°; 90°.} These moments are the sum of the values of the moments estimated based on the wind force for the position of the crane jib, corresponding to θ = 0° and θ = ±90°. The appropriate values are determined with the use of force coefficients C_X obtained from experimental analysis [36] of the sectional models and the moments M_O (Table 4). In the experimental investigations, two sectional models were analyzed, namely part of the crane tower and part of the crane jib. At the very beginning of the current analysis, the reference area of the tower A_refT and the reference area of the jib A_refJ should be determined with the model scale considered. It should be noted here that the determination of the total wind force acting on the crane jib is relatively easy because all parts of the jib are located at the same height, namely h₉ in Figure 11. However, the crane tower consists of eight parts—sectional models. Thus, it is necessary to determine the value of the wind force that acts on each section (it is assumed that the force is applied in the middle of each section) and compute the appropriate moment. It results in eight forces acting on the eight heights. The resultant overturning moment induced by the wind is equal to the sum of the product of these eight pairs of force W₁–W₈ and corresponding height h₁–h₈, which is shown in Figure 11.

The values of the wind speed at the assumed height can be determined according to Formula (1) for urban terrain. Next, the total wind force acting on the crane jib is determined using the following equation:

$$W_J=\hspace{0.33em}0.5·\left(v_9(h_9\right){)}^2·A_{refJ}·\rho ·C_{XJ},$$

(4)

In the case of the tower the particular wind forces are estimated for i = 1, 2, …, 8 as:

$$W_{Ti}=\hspace{0.33em}0.5·\left(v_i(h_i\right){)}^2·A_{refT}·\rho ·C_{XT},$$

(5)

and, finally, moments for jib M_WJ and tower M_WT:

$$M_{WJ}=W_J·h_9, M_{WT}={{\displaystyle \sum }}_{i=1}^8{W}_{Ti}·h_i$$

(6)

The numerical data and values of the calculated forces and moments for both angles are shown in

Table 5. Turbulent flow [36].

i	v(i) [m/s]	h(i) [m]	Aref [m2]	J Jib/ T Tower	WCx0(i)	MWCx0(hi)	WCx90(i)	MWCx90(hi)
9	9.77	33.81	8.08	J	0	0	907.01	30,664.68
					ΣMWJ	0	ΣMWJ	30,664.68
						-		CxT90 = 1.92
8	9.42	30.85	10.20	T	1579.08	48,717.11	1534.82	47,351.64
7	8.92	26.93		T	1416.42	38,146.37	1376.72	37,077.18
6	8.37	23.01		T	1248.94	28,739.89	1213.93	27,934.35
5	7.77	19.09		T	1075.61	20,535.08	1045.47	19,959.51
4	7.09	15.18		T	895.25	13,587.66	870.16	13,206.82
3	6.29	11.25		T	704.62	7927.99	684.87	7705.78
2	5.30	7.33		T	500.19	3667.17	486.17	3564.39
1	3.90	3.41		T	271.23	925.31	263.63	899.37
vref*	7.13				ΣMWT	162,246.57	ΣMWT	157,699.04
						CxT0 = 2.85		CxT90 = 2.77

. Similar computations were also carried out in the case of the laminar flow of the air. The results are presented in

Table 6. Laminar flow [36].

i	v(i) [m/s]	h(i) [m]	Aref [m2]	J Jib/ T Tower	WCx0(i)	MWCx0(hi)	WCx90(i)	MWCx90(hi)
9	9.77	33.81	8.08	J	0	0	628.32	21,242.64
					ΣMWJ	0	ΣMWJ	21,242.64
						-		CxT90 = 1.31
8	9.42	30.85	10.20	T	1141.37	35,212.87	1169.12	36,069.18
7	8.92	26.93		T	1023.79	27,572.30	1048.69	28,242.81
6	8.37	23.01		T	902.73	20,773.27	924.69	21,278.44
5	7.77	19.09		T	777.46	14,842.81	796.36	15,203.76
4	7.09	15.18		T	647.09	9821.20	662.83	10,060.03
3	6.29	11.25		T	509.30	5730.37	521.68	5869.72
2	5.30	7.33		T	361.54	2650.64	370.33	2715.10
1	3.90	3.41		T	196.05	668.81	200.81	685.08
vref*	7.13				ΣMWT	117,272.28	ΣMWT	120,124.12
						CxT0 = 2.06		CxT90 = 2.11

.

Concerning the above analysis, the total overturning moment, together with the impact of the wind M_{O exp 0°; 90°} considered, can be computed. This moment is a sum of the moment Mo presented in Table 4, induced by the mass of the particular parts of the structure, and the moments M_WJ and M_WT shown in Table 5.

$$M_{\mathrm{O} \exp 0°; 90°}=M_o+M_{WJ}+M_{WT}.$$

(7)

The values of the obtained overturning moments are shown in Figure 10. The symbol v_ref* denotes the arithmetic average wind speed, computed according to Figure 11.

3.4. The Wind Force Estimation According to the CFD

Based on the above-determined coefficients C_X and C_Y (as a result of the CFD simulation), the values of the aerodynamic forces are evaluated. These quantities are a function of the wind speed. Next, it is assumed that the wind forces are applied and computed at the height, equal to jib height h1. Because in the case of the real structure, the measurement of the wind speed is performed at the top of the tower crane, for the determination of the wind forces W_X, WY, it is assumed that the height is h₁. It is worth noting that W_X and W_Y act relative to the tipping edge of the investigated crane (63 K). These forces are defined as a function of wind speed.

$$W_X=F_X\cos \left(\theta \right), W_Y=F_Y\sin \left(\theta \right), W=\sqrt{{\left(W_X\right)}^2+{\left(W_Y\right)}^2}$$

(8)

In the next step, the reduced forces W and overturning moment M_oCFD63K are described as a function of the rotation angle theta of the jib position.

$$M_{\mathrm{O} \mathrm{CFD} 63\mathrm{K}}=M_O+W·h_1.$$

(9)

As is shown in Figure 10, the three isolated points from the experiment (turbulent and laminar flow). which correspond to the jib position for jib rotation angle theta equal to 0° and 90°, almost overlap each other with the characteristic of the overturning moment. The overturning moment is based on the values of the wind forces obtained from CFD simulations.

The determination of the wind forces in all cases was performed according to the standards [41,46].

4. Discussion

4.1. The Maximum Wind Force

A comparison of overturning and stabilizing moments obtained from numerical tests with calculations based on M_{o ECode90°} [41] for different wind speeds acting on the jib height at h₁ = 33.81 m is shown in Figure 12 and Figure 13.

Taking into account the above results, it is possible to determine the characteristics of the overturning moment as a function of wind speed for an angle of θ = 15° with the maximum load on the boom and for an angle of θ = 45° with no load on the boom. The results obtained are shown in Figure 14.

Using the data shown in Figure 14 and Figure 15, we can determine the maximum wind speed at which crane toppling will occur (

Table 7. The maximum wind speed v for which overturning moments cause loss of stability of the 63 K crane.

v  [m/s]	With Max Load on jib [kNm]		v  [m/s]	Without Max Load on jib [kNm]
h1 = 33.8 m	MO ECode 90°	MO CFD 63K +/−15°	h1 = 33.8 m	MO ECode 90°	MO CFD 63K +/−45°
14.88	-	1380.67	28.28	-	1380.81
23.39	1380.67	-	48.57	1380.52	-

).

4.2. Trace of the Center of Gravity

The trace of the center of gravity is a perpendicular projection of the center of gravity (in this case, tower crane 63 K) on the rectangular plane limited by the tipping edges. The geometrical dimensions of this plane are determined by the four supports of the investigated crane. In the current analysis, it is assumed that the overturning of the crane is possible only concerning the edge, which is oppositely located in the wind direction (Figure 3). It should be noted that the wind direction is constant to the contrary of the crane jib, which can rotate about its axis by the angle theta and the value of the wind speed. The two cases have been investigated. In the first case, the crane does not carry an external load; Q1 = 0. In the second case, the crane carries the maximal load Q1 = 3050 kg (Figure 9).

The trace of the center of gravity, depending on the angle θ of rotation of the crane jib, generates the closed contour. This curve can be described with the help of the guiding ray. Below are the formulas describing the guiding ray in the case of a lack of wind r_c and in the case where the wind is present r_cw. The guiding ray is defined by its coordinates x_c, y_c and x_cw, y_cw:

$$x_c\left(\theta \right)=-r_c·\cos \theta , y_c\left(\theta \right)=-r_c·\sin \theta ,$$

(10)

where:

$$r_c=\frac{M_7+M_7-Q1-M_1-M_2-M_3-M_4-M_5}{Q1+G1+G2+G3+G4+G5+G6+G7+G8},$$

(11)

and the trace of the center of gravity at the time of the wind r_cw:

$$x_{cw}\left(\theta \right)=-r_c·\cos \theta +r_c+r_{cw}\left(\theta \right), y_{cw}\left(\theta \right)=y_c\left(\theta \right),$$

(12)

where:

$$r_{cw}\left(\theta \right)=\frac{M_7+M_7-Q1-M_1-M_2-M_3-M_4-M_5-M_{oCFD63K}\left(\theta \right)}{Q1+G1+G2+G3+G4+G5+G6+G7+G8+F_{xCFD63K}\left(\theta \right)}.$$

(13)

The obtained contours are presented in Figure 15 for the trace of the center of gravity in the case of maximal load, and the second case with no load is shown in Figure 16.

5. Conclusions

The presented work is devoted to the problem of the loss of stability of the tower cranes caused by strong wind or gusts of wind. In the first step of the analysis, the numerical simulations where the CFD ANSYS R22 Fluent software is exploited were performed for three different wind profiles. It occurred that the most dangerous conditions are met in urban terrain. It is caused by the significant gradient of the wind speed value along the vertical direction. Next, the obtained results, mainly the values of the aerodynamic force coefficients, are confronted with those obtained from previously performed experimental investigations. This experimental analysis was reported in our previous work concerning the studies of the sectional models of the tower crane truss and the jib truss performed in the aerodynamic tunnel. The results are also verified with the data provided by the appropriate standards. A relatively good agreement is observed in the case of the jib position, which corresponds to the values of the theta angle at 0° and 90°. However, the maximal values of the aerodynamic forces are observed for θ = 15° or 30° degrees. Finally, the influence of the external load is also taken into account. The new analytical formula is provided, which enables estimation of the trace of the center of gravity in the case when the crane is subjected to external load and without the load. The presence of the wind causes the characteristic “heart-like” shape of the curves, which describes the trace of the center of gravity as a function of the jib rotation.

Finally, it is worth noting that the most dangerous configuration of the tested crane is the one for which the angle θ = 15° with load and θ = 45° without load. These configurations correspond to a critical wind speed of v = 14.9 m/s (53.6 km/h) and v = 28.3 m/s (101.9 km/h), respectively. The obtained values agree with the wind speed that caused the crane disaster in Cracow (Poland) in 2023 [1,2]. This is particularly important because during the safe configuration of the crane (the boom is aligned parallel to the wind direction), a sudden, strong gust of wind is capable of toppling the entire crane, as can be seen in the figures, which show the trace of the center of gravity.