Limit states of steel supporting structure for bridge cranes

Scientific Journal of Silesian University of Technology. Series Transport Zeszyty Naukowe Politechniki Śląskiej. Seria Transport Volume 108 2020 p-ISSN: 0209-3324 e-ISSN: 2450-1549 DOI: https://doi.org/10.20858/sjsutst.2020.108.13 Journal homepage: http://sjsutst.polsl.pl Article citation information: Mantič, M., Kuľka, J., Kopas, M., Faltinová, E., Hrabovský, L. Limit states of steel supporting structure for bridge cranes. Scientific Journal of Silesian University of Technology. Series Transport. 2020, 108, 141-158. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2020.108.13. Martin MANTIČ 1 , Jozef KUĽKA 2 , Melichar KOPAS 3 , Eva FALTINOVÁ 4 , Leopold HRABOVSKÝ 5 LIMIT STATES OF STEEL SUPPORTING STRUCTURE FOR BRIDGE CRANES Summary. This paper describes a question of evaluation necessity of bridge cranes using the method of limit deformation state and oscillation damping. The solution was performed by means of theoretical analysis and an experimental verification at the selected bridge crane. The final result sounds that in the case of a correct strength computing of given bridge crane, it is not necessary to also check deformation and damping of oscillation as well. Keywords: oscillation, damping, energy, experiment, deflection, time 1 Faculty of Mechanical Engineering, Technical University of Košice, Letná 9 Street, 042 00 Košice, Slovak Republic. Email: [email protected] 2 Faculty of Mechanical Engineering, Technical University of Košice, Letná 9 Street, 042 00 Košice, Slovak Republic. Email: [email protected] 3 Faculty of Mechanical Engineering, Technical University of Košice, Letná 9 Street, 042 00 Košice, Slovak Republic. Email: [email protected] 4 Faculty of Mechanical Engineering, Technical University of Košice, Letná 9 Street, 042 00 Košice, Slovak Republic. Email: e[email protected]k 5 Faculty of Mechanical Engineering, Institute of Transport, Technical University of Ostrava, 17. listopadu 15/2172 Street, 708 33 Ostrava-Poruba, Czech Republic. Email: leopo[email protected] 142 M. Mantič, J. Kuľka, M. Kopas, E. Faltinová, L. Hrabovský 1. INTRODUCTION The questions concerning vibrations of the bridge cranes are analysed in many professional works from various authors. For example, vibrations of the crane girder, which are occurring as a consequence of the crane travel on the crane track, are described by the authors in the works [1-3]. S cientific investigation of crane oscillations, together with the presentation of possibilities on how to eliminate these oscillations using a suitable form of the crane control strategy, is presented in [4-7]. Other methods, which are determined for the elimination of the crane oscillations by optimisation of the crane control components and mechanisms, are introduced in [8-10]. Different proposals of a dynamic model relating to the abovementioned problems are given in [11-14] and the corresponding experimental methods are in [15, 16]. Another interesting approach to the investigation of mechanical vibration, damage of transport machine parts and generation of failures is presented in [17-27]. Other experimental studies dealing with the mechanical properties of a layered beam, which is partially treated with a damping element based on a granular material, offers [28]. The author of [29], is focused on the assessment of vibration control performance using enhanced smart constrained layer damping treatment with edge elements. The non-linear dynamics of a crane is investigated in the contribution [30]. The commonly used or standard approach to the solution of a damping process represents the application of the logarithmic decrement, which enables estimation of the damping ratio from a time history of the oscillation process. Analysis of this method and optimisation of the parameters during the processing and evaluation of the results is presented in [31, 32]. A new algorithm OMI (Optimization in Multiple Intervals), which is intended for computation of the logarithmic decrement concerning the exponentially damped harmonic oscillations, is described as well as compared with the classic computational methods in the publications [33, 34]. According to these articles, it is possible to say that the OMI algorithm is proved to be the best solution in the computation of the logarithmic decrement and the resonant frequency for high damping levels. Moreover, it is possible to take into consideration, a typical causal relation between the wear process and wear damage of constructional parts, which is introduced for example in [35]. The existing STN 27 0103 [36] informs the design of steel crane structures according to two groups of limit states. For the first group of limit states, which lead to the loss of loading capacity or the loss of position stability, the following criteria are central to the design of steel supporting structure:  strength and stability,  fatigue strength,  position stability. For the second group of limit states, the following criteria are crucial:  static deformation (deflection, displacement, and twisting),  dynamic structure response (steel structure frequency, amplitude, and damping). Limit states of steel supporting structure for bridge cranes 143. The second group of limit states is outlined in the Article X of the standard only in section 66, which states as follows: “Deformation and oscillation damping must not affect operational safety and must not interfere with proper crane function with regard to its work specifications”. In Annex VIII, the standard recommends how to approach the bridge crane evaluation. To a great extent, the STN 27 0103 [36] overlaps with the German DIN standard 15018 “Krane. Grundsätze für Stahlwerke. Berechnung“ [37] in the calculation procedures. This DIN, however, does not specify checking for the limit state of deformation and damping. This article lays out the theoretical analysis which consequently ties into the experimental analysis on a particular bridge crane. 2. OSCILLATION DAMPING OF A STEEL SUPPORTING STRUCTURE IN A BRIDGE CRANE A degree of freedom in a flexible system is conceived as a number of independent coordinates which determine the location of all the system’s masses. If the continuous distribution of the flexible elements is small in comparison with the masses distributed in the individual points, then the flexible element mass (m) can be downplayed and only the coordinates of the masses distributed in the points examined. Let us propose that the resistance of the surroundings is directly proportional to the speed in the following relation: Fresist = – k.v , where k > 0, (1) k – the damping coefficient (kg.s-1). The resistance force counters the speed which is expressed by the minus sign in this formulation. If the acting restoring force is directly proportional to the displacement, the equation of motion is as follows: , (2) c – spring stiffness (N.m-1). Once the substitution is applied: and 2b = k/m, (3) where b is the damping coefficient and the ω0 constant is the natural angular frequency, that is, the angular frequency of the undamped harmonic oscillator. After the adjustment, we get the following equation of motion: . (4) 144 M. Mantič, J. Kuľka, M. Kopas, E. Faltinová, L. Hrabovský The solution to the homogenous second order differential equation points to the following formulation: x = eλt, , . (5) After we apply the function to Equation 4, we get the following characteristic equation: , (6) with this solution: . (7) The two values correspond to the general solution to Equation 4 in the linear combination: (8) provided that λ1 ≠ λ2. According to the extent of the damping, the following scenarios can play out:  The damping is large and b2 – > 0, then both solutions of the characteristic equation are real numbers, and x has no periodical element. Theoretically, in time t →the body gets back to the equilibrium position x = 0. If the damping is this large, the motion is called aperiodic. Oscillation does not occur at all.  The damping is such that b2 – = 0. In such a case, mathematics informs the solution to Equation 4 as the function x = e-bt as well as the function x = t.e-bt. The general solution is their linear combination in the following form: . (9) This motion is called critical aperiodic motion.  Damped oscillatory motion occurs only when damping is small, if b2 – < 0. Then: where . (10) The general solution to the equation of motion is expressed as follows: . (11) Along the lines of this procedure similar to the harmonic oscillator case, substitution and adjustment bring forward the real formulation of the general solution: Limit states of steel supporting structure for bridge cranes 145. . (12) The angular frequency is smaller than the angular frequency at the undamped oscillation of the same system, and the amplitude which also changes, exponentially decays away over time: . (13) Damped oscillatory motion cannot be considered periodic because the oscillating point does not reach its original displacement. The motion here is quasiperiodic, and the T period can only be conceived as a time interval, past which a mass point passes the equilibrium position. The damped oscillation period is: . (14) It is true that T > T0, where T0 is a period of natural oscillations. If the damping is small, the period practically equals the period of undamped oscillations. The period increases with growing damping. The ratio of amplitudes of two consequent maximum displacements are denoted as and called damping. . (15) The natural damping logarithm is the logarithmic damping decrement δ. δ = ln = bT. (16) From the dependence of amplitude on time (13), it is apparent that the oscillation amplitude decreases e-times over a time interval which equals 1/b. The inverted value of the logarithmic damping decrement expresses the number of oscillations, during which the oscillation amplitude changes e-times. The greater the logarithmic damping decrement, the fewer the number of oscillations necessary for a particular decrease of the amplitude. The total mechanical energy of the oscillating oscillator is proportional to the square of the amplitude. If the energy of the oscillator with damping in the point in time t = 0 equalled E0, then the mechanical energy of the oscillator decreases with increasing time according to the equation: . (17) Friction causes energy dissipation; mechanical energy of the oscillatory motion changes to thermal energy and the motion gradually decays. For motion to be maintained in the oscillating system, then energy must be supplied to it in a suitable way. Experiments confirm that for most mechanical materials, the value of the dissipated energy during a single oscillation cycle does not depend on frequency but that it is just the function of the oscillation 146 M. Mantič, J. Kuľka, M. Kopas, E. Faltinová, L. Hrabovský amplitude. On the other hand, it is damping that is used in technical practices to eliminate undesirable vibrations. Fig. 1. Dependence of amplitude on time at damped oscillation 3. COMPUTER AND EXPERIMENTAL APPLICATION ON THE REAL BRIDGE CRANE Basic data about the electrical bridge crane under examination: Capacity Q = 500,000 N Span of the crane L = 28,200 mm Crane travel speed vtravel = 0.416 m/s Lifting speed vlift = 0.0333 m/s Trolley mass mtrolley = 6,680 kg Main girder gravity G1 = 294,800 N Main cross beam gravity G2 = 41,240 N The calculated static values of the cross-section for the main girder without stiffeners and the rails are illustrated in Fig. 2. 3.1. Calculating damping for the steel supporting crane structure According to STN 27 0103 titled “Design of crane steel structures” – Calculation by limit states, Article X – Steel supporting structure calculation based on the II. group of the serviceability limit states – deformation: deformation and oscillation damping must not affect operational safety and must not interfere with proper crane function in regard to its work specifications”. The recommended approach to the bridge crane evaluation is outlined in Annex VIII of the referenced standard. The new shape of the girder caused by deformation is not detrimental to the operation if the deflections are smaller than the recommended values outlined in [28] and are caused by a random nominal loading, which includes as follows:  rated loading,  constant loading, Limit states of steel supporting structure for bridge cranes 147.  loading brought about by natural mass, and the movable parts, which move in relation to the movement of the load. The new shape of the girder caused by deformation is not detrimental to the operation if the deflections are smaller than the recommended values outlined in [28] and are caused by a random nominal loading, which includes as follows:  rated loading,  constant loading,  loading brought about by natural mass, and the movable parts, which move in relation to the movement of the load. zT = 812.5 mm A = 55 556 mm2 Jy = 2.2396 x 1010 mm4 Jz = 4. 396 x 109 mm4 Fig. 2. Cross-section view of the main girder in the bridge crane For the electrical bridge crane (50t x 28.2 m) with two main girders, along the top of which the trolley travels, the ratio between the maximum deflection from random nominal loading in relation to length, based on [37] can reach L/700 at most, that is, maximum deflection can reach: (18) Such a deflection is not detrimental to the crane operation. A steel supporting structure ought to meet the conditions of damping while oscillating. For cranes with box girders, it is recommended that the amplitude, after the nominal load of the oscillating bridge is set aside, sank in the middle of the bridge within 15 s to 0.5 mm at most. The damping time for a single-mass substitution system is determined by the following relation: (19) 148 M. Mantič, J. Kuľka, M. Kopas, E. Faltinová, L. Hrabovský (20) where is the maximum deflection of the girder from rated loading in (mm), f is the frequency of natural oscillations in the girder (s-1), is the logarithmic decrement of oscillation damping which depends on the ratio between the girder height and length. For the 50t x 28.2 m crane, the bridge girder height is h = 1650 mm (Fig. 2), hence, the ratio h/L = 1650/28200 = 0.0585. Welded plate box girders are braced with a compression boom, with a height to length ratio greater than 1:20. Based on [36], the logarithmic damping decrement . A girder spring constant can be determined by the following relation: (Nm-1) (21) where E = 2.1 x 105 MPa is the elastic modulus, L = 28 200 mm is the bridge length, = 2.2396 x 1010 mm4 is the axial quadratic cross-sectional moment in relation to the neutral axis. Then c0 = 10066619.79 Nm-1. The reduced mass of the girder, trolley, and all mass oscillating with the girder, once the rated load is set aside, is determined by the following relation: (kg) (22) where q – girder mass per length (kg.mm-1), i – number of girders along which the trolley travels, mtrolley – the trolley’s natural mass without load, including the mass which the trolley retains once the load is set aside. To determine the reduced mass of the girders, the gravity of the individual crane components was relayed from the static calculation of the equipment, according to which the girder mass where there are no crane trolley units: (kg) . (23) The second girder’s mass includes the mass of the crane travel units, of the switchgear, of the walkway, and of the trolley wire (relayed from the static calculation): (kg). (24) Limit states of steel supporting structure for bridge cranes 149. If we consider that the drawing documentation for the trolley mass mtrolley = 6680 kg, then the reduced mass of the girder without the travel units and the walkway is: (kg) (25) and for the girder with the travel units: (kg) . (26) The relationships (20) and (21) are used to determine natural frequencies of the respective beams f1 = 4.847 Hz, f2 = 3.142 Hz. Damping period of the respective girders established from the value of the maximum static deformation from the rated load is as follows: (mm) . (27) The damping period for the first and the second girder equals: , . (28) Because the damping period in both cases is shorter than 15 s, the steel supporting crane structure meets the conditions for the oscillation damping according to [36]. 3.2. Experimental evaluation of damping in a steel supporting crane structure Fig. 3 shows the strain-gauge measurement that was proposed and made on the electrical bridge crane with box girders. The proposed methodology enables us to determine the stresses and to identify the extent of the deflection caused by the suspended load, the amplitude of the oscillating bridge in the middle of its span after unloading or lifting the load, and the time of amplitude damping required for reaching a 0.5 mm oscillation. Furthermore, Fig. 3 shows where the sensors were applied in a diagram. Strain-gauge sensors were placed on the edges of the top beam flange along the axis of the bridge symmetry. To measure, we used HBM strain-gauge sensors, bonding cement (X60), measuring apparatus (Spider8) and the evaluation software (Catman) also from HBM. Strain-gauge measurement was made under the following loading:  loading of the crane with a trolley without load,  loading with a 48,000 kg load. The measuring apparatus was balanced in the empty trolley mode in the middle of the bridge length. The measured incremental values of relative deformation were used by the Catman software to evaluate and visualise time changes of normal stress increments in the points illustrated in Fig. 3. The measured time sequences of the relative deformation were used to calculate normal stresses in the points of the single-axis stress of the main girder. Fig. 4 illustrates the time changes of incremental stress during intermittent lifting of the 48,000 kg load in the points of measurements 1, 2, 3 and 4. 156 M. Mantič, J. Kuľka, M. Kopas, E. Faltinová, L. Hrabovský 15. 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[In Slovak: STN 27 0103: 1989. Design of Steel Crane Structures. Calculation According to Limit States. Praha. Czech Office for Standardisation and Measuring]. 158 M. Mantič, J. Kuľka, M. Kopas, E. Faltinová, L. Hrabovský 37. DIN 15 018: 1984. Krane. Grundsätze für Stahltragwerke. Berechnung. Berlin. Deutsches Institut für Normung. [In German: DIN 15 018: 1984. Cranes. Steel Supporting Structures. Calculations. Berlin. German Institute for Standardisation]. Received 11.03.2020; accepted in revised form 12.05.2020 Scientific Journal of Silesian University of Technology. Series Transport is licensed under a Creative Commons Attribution 4.0 International License