machine design, Vol.5(2013) No.4, ISSN 1821-1259 pp. 151 … · machine design, Vol.5 ... assemble a tower crane on a building site, the crane jib is ... applicability for particular

machine design, Vol.5(2013) No.4, ISSN 1821-1259 pp. 151-156

*Correspondence Author’s Address: University of Nis, Faculty of Mechanical Engineering, A. Medvedev 14, 18000 Nis, Serbia, [email protected]

Original scientific paper

BY STRUCTURAL DESIGN TO PERFORMANCE GROWTH Goran RADOIČIĆ1 - Miomir JOVANOVIĆ1,* - Danijel MARKOVIĆ1 - Vojislav TOMIĆ1 1 University of Niš, Faculty of Mechanical Engineering, Niš, Serbia

Received (05.06.2013); Revised (15.09.2013); Accepted (23.09.2013) Abstract: This paper considers the influence of geometry choice, actually the height choice to the change of own static and dynamic parameters of a heavy frame structure like tower crane. The research was performed numerically using the FEM technology in the form of more successive discrete analyses. On the basis of these analyses, a disposition area of eigenvalues was given. The paper also discovers – computes, what is gained by fastening a high tower to a building under construction. This is a topic which talks about linear and nonlinear analyses, small and large structure translations and design as causer of change in the main features of big and responsible structures. On the basis of these analyses, the geometry prediction to the required eigenfrequency of the structure is enabled. This is the essence of static and dynamic crane synthesis. Key words: structural design, eigenvalue, translations, modal analysis, frame structures 1. INTRODUCTION The tower cranes of frame support structure are designed to large hoisting heights (> 100 m) and range lengths (> 45 m). Before the jib erection, a crane height can be changed by adding the corresponding tower sections. To assemble a tower crane on a building site, the crane jib is installed so as to initially satisfy maximal nominal range. Then, the crane tower also is erected to maximal necessary hoisting height. It is interesting to observe the same crane with more different heights. Because it is very difficult to provide the experimental research for more of the same type real cranes with different heights at a time, the numerical FEM analysis alternatively is applied. By this analysis, the discrete values of own parameters for selected heights of a crane are computed. On the basis of these analyses, we can see how the change of design influences the static (translations) and dynamic (eigenvalues, vibration period) behaviour of high machines. In other words, how own static and dynamic parameters depend on geometry of these machines. Such structural analysis is an integral part of the support structures design because it often leads to the geometry redesign of structure in accordance with the values of own parameters. The external forces such as force of seismic shock, overload force, wind gust force etc. jeopardize the stability of high frame structures. Probability to lose the structural stability is higher if the structure height is increased. Such is the case with tower cranes as well. Therefore, in the civil engineering there is a practice to make the local safety connections (anchor devices) from a crane to a building under construction. These safety connections are usually realized in two points of a building at the same selected height. Manufacturers give recommendations to the prevention of incidents in the use of crane with large hoisting heights, which contain description of the safety connection. However, an

anchoring problem should be considered from the point of vibrating comfort. Hence, it should explore how the anchoring contributes to the vibrating comfort and if the anchor point height influence the quantity of strain energy (deformation work). 2. THEORETICAL SUPPORT TO

STRUCTURE ANALYSIS The static equilibrium equation of a discrete frame structure with n linear elements and m nodes by the linear matrix form is represented:

FuK , (1)

where, K is the stiffness matrix, {u} is the nodal displacement vector and {F} is the external force vector. The external forces which effect in the structure nodes are not variable at time i.e. they have constant intensity, course and direction and represent the static structure load. The component translations of all nodes by solving Eq. (1) are obtained. Based on that, the internal forces R in the ends of each structural element are determined. Normal modes analysis represents the basis of transient dynamic response analysis which should perform before dynamic analyses. The aim of normal modes analysis is to find eigenvalues (natural frequencies) and eigenvectors (mode shapes). For each eigenvalue, which is proportional to a natural frequency, there is a corresponding eigenvector, or mode shape. Natural frequencies are the frequencies at which a structure will tend to vibrate if subjected to a disturbance. Deformed shape at a specific natural frequency is called the mode shape. Normal mode analysis is also called real eigenvalue analysis. An assumed harmonic solution of the equation of motion (Eq. (2)), where M is mass matrix and {ü} is acceleration vector, for an un-damped free vibration is shown in Eq. (3) i.e.:

Goran Radoičić, Miomir Jovanović, Danijel Marković, Vojislav Tomić: By Structural Design to Performance Growth; Machine Design, Vol.5(2013) No.4, ISSN 1821-1259; pp. 151-156

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0 uKuM , (2)

tu cos , (3)

where {} is eigenvector, ω is circular natural frequency and t is time. Substituting Eq. (3) into Eq. (2), the eigenequation can be written as:

02 MK . (4)

The most frequently used real eigenvalue extraction method in numerical analyses is the Lanczos method [1], followed by Modified Givens method and Sturm modified inverse power method. The Lanczos method is the best overall method due to its robustness, but the other methods (particularly the modified Givens method and the Sturm modified inverse power method) have applicability for particular cases of analysis. 3. CRANE STRUCTURE MODELING The dynamics research of construction machinery with frame structure by numerical methods requires theoretical models which faithfully represent a real structure. Particular attention in development of these models is dedicated to type choice, shape visualisation and finite elements orientation. Considering large mass of more tens of tons, the frame structure modelling by linear elements is performed more often. Besides the main masses, the modelling includes other present masses of smaller structure elements such as drums, motors etc. Fig. 1 and 2 show examples for the precise modelling of component real tower crane parts. This study considers the results of numerical analyses of the developed FEM model of the real crane frame structure POTAIN 744E. In the study, a discrete crane model with height of 17.6 m from the ground to the lower jib belt is initially used. The entire structure is represented by 1146 nodes and 1667 elements, 2. The model applies three types of finite elements. Thus, the beam element with 6 degrees of freedom in each node is used to describe the frame structure as well as the rod element to describe the support tie rods. Also, the 8-node solid element is used to represent the crane foundation of reinforced concrete. For the modelling of different tower heights, a group of elements – the tower section is used, Fig. 3. The addition of the tower sections corresponds to the procedure of real crane erection to a certain height. In the paper, two crane models are observed. First, the model of crane freely supported by its concrete foundation on the ground (free-standing model) 3. Second, the model of crane connected to a building under construction (anchor model), Fig. 4. The points of added connection - anchoring points are placed on the height h from ground and the normal distance l from the tower, Fig. 4. The crane connection to the building, in order to increase the crane stability, is performed using an anchor device. The anchor device is simplistically shown, Fig. 4, by a normal tie rod and a joint connection just placed on the building. The anchor connection length l depends on building geometry and working area availability. The anchor connection height h depends on the crane and building heights. Other solutions of the anchoring with different anchor connection heights can also be applied in practice.

Fig.1. Tower crane POTAIN 744E, the theoretical FEM model

Fig.2. Counterweight detail of the crane model

Fig.3. A part of tower model, the tower section


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Fig.4. Anchor model of a crane 4. CRANE HEIGHT INFLUENCE ON

TRANSLATIONS The geometrically linear numerical analysis is verified by the experimental results with the deviations of translation less than 1.5% in relation to the nonlinear analysis, 2. Hence, the static change of jib deflection in function of the tower height using the linear static analysis in software MSC FEMAP/NASTRAN 1 is further required. The change of free jib end deflection f to the different tower heights H (Fig. 4) under the effect of limit static load Q = 23.5 KN at the longest jib reach Lmax= 45m is required. The anchor connections are typical for buildings. The anchor height is constant i.e. h = 29.7 m. The length of anchor connection amounts l = 4 m. The tower height H is changed in the range from 17.6 to 51.3 m by adding the tower sections from Fig. 3. All tower heights in Fig. 5 are selected for the investigation.

Fig.5. Observed tower heights H in the research The obtained results are shown in the total translation diagram, Fig. 6. The vertical translation in z global direction here is the dominant translation. The curves a and b, Fig. 6, represent total translations of jib free end for the free-standing model (curve a) and the anchor model (curve b). By changing the tower height H, the curves a and b change their values. Three areas of solution for the applied tower heights in the total translation diagram, Fig.

6, are identified. From left to right, the first height area is applied to the free-standing model of the crane. The limit height of this area is H = 46.5 m (white area). It follows the anchor area (slightly darker). The beginning of this area corresponds to a crane manufacturers recommendation according to which the anchoring at tower heights H > 40 m is used. Greater heights (H > 46.5 m) than declared by manufacturer in the research are numericaly tested. Generally, the jib deflection f is higher with the free-standing crane model at all tower heights. According to the diagram in Fig. 6, more pronounced total translations in both models can be expected for the heights from the darkest area (H = 49÷53.5 m). This is also expected domain of the pronounced dynamical sensitivity with significant translation amplitudes and longer periods of vibration. The results of analysis indicate the reduction of static deflections of the structure to 15% by anchoring the crane at the height h 30 m. Although the anchoring of cranes to heights H < 40 m should not apply, the study includes the computation of jib translations to lower tower heights as well as translations to the heights which increase by adding a number of tower sections from Fig. 3.

Fig.6. Tower height influence to total translation With this approach the total translation trends are calculated. Thus, the curves a and b, Fig. 6, have been completely determined. The tower heights from the darkest area in the diagram, Fig. 6, are not recommended to use due to the significant enlargement of structural mass. The use of these heights can jeopardize the general structure stability in incidental situations or under the overload. A question of the optimal anchor height h of the lateral crane connection to a building in terms of crane behaviour at external excitation can also be interesting. Having in mind the recommendation for minimal tower height of an anchored crane, in this paper the anchor height h = 29.7 m, Fig. 5, is tested. The anchor height h (an anchoring point place) by the current built height of building is also practically conditioned. Hence, influence of the different heights h of kinematic couple of crane and building to static and dynamic crane parameters at the different tower heights H by future investigates can be defined.


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5. EIGENVALUES DISPOSITION AREA

The previously defined crane models, a – free-standing model and b – anchor model, further in normal modes analysis were used. To perform normal modal analysis seven tower heights H were selected. Five of them, from the range H = 17.6÷46.5 m, were used in order to determine the eigenvalues development trajectory of the model “a” as well as four heights from the range H = 42÷51.3 m in modal analysis of the model “b”. The

research of crane modal shapes to two tower heights, H = 42 m and H = 46.5 m, and both models (free-standing and anchor model) was performed. Modal analysis of both crane models using the software MSC FEMAP / NASTRAN 1 was performed. The Lanczos real eigenvalue extraction method to determine the eigenvalues was used. The first hundred mode shapes and circular frequencies of both models by this method were selected. The first ten eigenvalues of them, Table 1, for further analysis were taken.

Table 1. The first ten eigenfrequencies for several different tower heights H

Eigenfrequency i (Hz)

Mode shape

i

Tower height H (m)

Model “a” – Free-standing model Model “b” – Anchor model

17.6 26.5 35.4 42.0 46.5 42.0 46.5 49.0 51.3

1 0.1687 0.1417 0.1243 0.1148 0.1096 0.1596 0.1462 0.1436 0.1228

2 0.4749 0.3826 0.3026 0.2535 0.2257 0.2798 0.2439 0.2297 0.2136

3 0.5596 0.4647 0.3505 0.2821 0.2456 0.4299 0.3880 0.3737 0.3472

4 1.5444 0.9879 0.7552 0.6638 0.6210 0.8854 0.8632 0.8857 0.8456

5 1.6342 1.1370 0.9669 0.9149 0.8932 1.5556 1.1753 1.0593 0.9399

6 2.0274 2.0229 2.0190 2.0160 2.0139 1.8912 1.8821 2.0154 1.8783

7 2.5162 2.5064 2.5034 2.5021 2.5015 2.4827 2.4729 2.4795 2.3142

8 2.7747 2.7744 2.7725 2.7376 2.6081 2.6545 2.5558 2.5062 2.4855

9 3.0221 2.8901 2.8275 2.7821 2.7778 2.8441 2.8263 2.7771 2.7989

10 3.4109 3.3877 3.3677 3.3449 3.1596 3.4206 3.1187 3.0499 2.9500

Fig.7. Development trajectories of eigenfrequencies as functions of the tower height H (model “a”)


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Fig. 7 shows the development trajectories of the own circular frequencies for the model “a” (free-standing model) and different tower heights H. Starting from the lowest (curve 1) to the highest height H (curve 5), the frequency has non-linear decrease. The circular frequency 4, Fig. 7, has the largest total decrease for the entire range of height. This decrease takes value:

79561001

15

4

444 .

H

HH

%. (5)

Follow 3 = 56.11% and 2 = 52.46%. In the case of modal shapes with lower frequencies 2, 3 and 4, the strain energy is the most expressed. Thus, these frequencies are used to calculate the damping matrix which requires the choice of two dominant frequencies for the transient analysis, 4. Two interesting mode shapes with frequencies 6 and 7 whose values to all heights H are almost equal, in Fig. 7 can be seen. In these cases, the overall geometry has more important influence to the vibration than the tower height. For these mode shapes, with the increase of tower height, the total decreases of frequency 6 = 0.66% and 7 = 0.58% are calculated. Strain energy is mainly distributed to the vertical translations of hoisting jib in the case of mode 6 (Fig. 8), as well as to the torsion of hoisting and counter jib in the case of mode 7 (Fig. 9).

Fig.8. Model „a“, H= 46.5 m, mode 6, 6= 2.0139 Hz The first lowest circular frequency 1 corresponds to the rotation of entire structure around the vertical z-axis for all heights H and models. Larger vertical deflections f correspond to the modal shapes of frequencies 2 and 4 for the model “a”, i.e. 3 and 5 for model “b”. These are shapes which imply the highest axial force and stress in the structure. One such forward-back shape of vibration for the crane model with anchor connection is shown in Fig. 10. The anchor connection of crane in Fig. 10 on the height h = 29.7 m and the distance l = 4 m from the tower in two points of building (A1 and A2) is achieved. The anchor connection allows three degrees of freedom in the connection points A1 and A2 as three rotations around global axes (in software 1: pinned).

Fig.9. Model „a“, H= 46.5 m, mode 7, 7 = 2.5015 Hz

Fig.10. Model „b“, H = 46.5 m, mode 5, 5 = 1.1753 Hz Fig. 11 shows another interesting own shape of vibration, mode 4, for the tower height H = 42 m of the anchor model “b”. Strain energy of this mode shape causes the lateral translation of structure. This mode shape corresponds to any other height H of the model “b”.

Fig.11. Model „b“, H = 42 m, mode 4, 4 = 0.8854 Hz


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Two crane models show significantly different eigenvalues and periods of vibration. We can see the differences of circular frequencies for two same heights of models from the range H = 42÷46.5 m in Fig. 12. The eigenvalues of anchor model, Fig. 12 on the right, generally are higher in relation to the free-standing model, Fig. 12 left. For example, the lowest frequency of tower rotation 1 is increased by 39% to the anchor model

which leads to the faster damping of structure vibration (i.e. periods of vibration have shorter duration). Fig. 12 on the right shows the darker area of frequency change trend (trend-line area) with increasing the tower height. This is the area of non-recommended tower heights to use. Actually, it should avoid the tower heights formed by adding the tower sections, in this area.

Fig.12. Decrease of eigenfrequency with the tower height growth

6. CONCLUSION 1) Including the largest number of discrete masses as

well as smaller masses in the modelling of the heavy frame structures, lead to the higher accuracy of static, modal and transient analysis results.

2) Change of design, such as the increase of height and the anchoring, significantly influences the eigenvalues of heavy lifting and moving equipment.

3) By anchoring a tower crane, the static deflections of structure in the most expressed deformation areas are reduced to 15%.

4) The kinematic anchor connections are at the same time elements which preserve the stability and the design solutions which contribute to the vibration comfort quality. By the theoretic solution of connection, depending on two selected crane tower heights of 42 and 46.5 m, the period of vibration is decreased to 40% in relation to the free-standing crane model (calculated from Table 1 in which the period of vibration is T = 1/).

5) By numericaly calculating the eigenvalues, two principles are defined: 1) “higher height” = “lower frequency”, 2) “anchoring” = “less strain energy”. The first principle indicates the increase of elastic translations and periods of vibration depending on the increase crane height. The second principle indicates that smaller deformation work and faster vibration damping of entire system are achieved by anchoring.

6) On the basis of previously conducted simulations we can formulate another principle which reads as follows: “higher anchor place” = “less strain energy”.

7) The adding of tower sections is limited by the permissible own mass of a crane and structural

stresses. Increasing the number of tower sections leads to general instability of a structure.

ACKNOWLEDGEMENT The paper is a part of research carried out under project TR35049. The authors would like to thank the Ministry of Education and Science of the Republic of Serbia for their support.

REFERENCES [1] *** (2004). Basic Dynamic Analysis, MSC Nastran

Version 68, MSC Software Corporation, Santa Ana, USA, Available from: http://www.mscsoftware.com, Accessed: 2004

[2] Jovanović, M.; Radoičić, G.; Petrović, G. & Marković, D. (2011). Dynamical models quality of truss supporting structures. Facta Universitatis – Series Mechanical Engineering, Vol. 9, No. 2, 2011, pp. 137-148, ISSN 0354-2025, UDC 624.07 519.711 515.3

[3] Radoičić, G. & Jovanović, M. (2013). Experimental identification of overall structural damping of system. Strojniški vestnik – Journal of Mechanical Engineering, Vol. 59, No. 4, 2013, pp. 260-268, ISSN 0039-2480, DOI:10.5545/sv-jme.2012.569,

[4] Rose, T. (2002). An approach to properly account for structural damping, frequency-dependent stiffness/damping, and to use complex matrices in transient response. Worldwide Aerospace Conference & Technology Showcase, presentation no. 6, from:www.mscsoftware.com/events/aero2002/partner/index.cfm, accessed on 2002-04-08.

machine design, Vol.5(2013) No.4, ISSN 1821-1259 pp. 151 … · machine design, Vol.5 ... assemble a tower crane on a building site, the crane jib is ... applicability for particular

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