Determination of accidental forklift truck impact forces on drive-in steel rack structures

Transcription

1 Determination of accidental forklift truck impact forces on drive-in steel rack structures Author Gilbert, Benoit, J.R. Rasmussen, Kim Published 0 Journal Title Engineering Structures DOI Copyright Statement 0 Elsevier. This is the author-manuscript version of this paper. Reproduced in accordance with the copyright policy of the publisher. Please refer to the journal's website for access to the definitive, published version. Downloaded from Griffith Research Online

2 Determination of accidental forklift truck impact forces on drive-in steel rack structures Benoit P. Gilbert (), Kim J.R. Rasmussen () () School of Civil Engineering, The University of Sydney, Australia Now, School of Engineering, Griffith University, Australia () School of Civil Engineering, The University of Sydney, Australia Abstract: The paper addresses the problem of determining the accidental forklift truck impact forces on steel storage racks. Based on first principles of mechanics, simple models of a loaded forklift truck and a drive-in racking structure are presented. Model masses, as well as stiffness and damping coefficients are calibrated against experimental results obtained from tests of a forklift truck and a drive-in racking structure. Comparisons between experimental results and solutions obtained from the simple mechanical models show that the simple models accurately reproduce the static and dynamic behaviours of their associated structures. Based on the drive-in rack impact test results presented in a companion paper ([]) and the simple mechanical models for drive-in racks, actual impact forces are calculated and presented. Finally, using the impact test results and the simple mechanical models, the actual motion of the forklift truck body is calculated. This motion, being a common characteristic to all drive-in racking impacts, allows impact forces to be obtained for various pallet loads, impact elevations and rack characteristics. Thus, the paper concludes with a general method for calculating forces generated under forklift truck impact. Keywords: Steel storage rack, drive-in rack, impact force, forklift truck behaviour, drive-in rack behaviour. Introduction This paper addresses the problem of determining the accidental forklift truck impact forces on steel storage racks. As presented in the companion paper ([]), different types of forklift truck impact can occur on steel storage racks. This paper focuses on downaisle impacts occurring on drive-in racking structures but the methodology and methods presented herein can be easily extended to any type of rack or steel structure in general. Contrary to classical selective racks, by storing pallets on rail beams on the first-in, last-out principle drive-in racks minimise floor space allocation and are often a more economical alternative solution to selective racks. To allow the forklift truck passage, drive-in racks can only be braced at the back (spine bracing) and at the top (plan bracing) leading to slender uprights in the down-aisle direction, only restrained at the bottom by the base plate assembly and at the top by the portal beam. When subjected to a forklift truck impact in the down-aisle direction, the bowing of the upright may trigger progressive failure by allowing the pallets to drop through. The literature review shows that no investigation is available for determining the accidental forces that develop between a storage rack and a forklift truck during an

3 impact. International racking standards only consider arbitrary impact forces, the equations for which have no scientific justification. Furthermore these standards only deal with selective racks and are not applicable to drive-in racks. The present study is motivated by the high failure rate of drive-in racks compared to other types of racks and steel structures in general. Further details on the behaviour of drive-in racks and available literature for this type of structure are given in the companion paper ([]). Based on first principles of mechanics, the paper first develops a simple model of a loaded forklift truck. Model masses, as well as stiffness and damping coefficients are calibrated against experimental results obtained from isolated tests of an actual forklift truck. Subsequently, also based on first principles of mechanics, a simple model of a loaded drive-in rack is developed. Model masses, as well as stiffness and damping coefficients are calibrated against experimental results. Comparisons between experimental results and solutions obtained from the simple mechanical models show that the simple models accurately reproduce the static and dynamic behaviours of the forklift truck and the loaded drive-in rack. Using the result of experimental impact tests and the simple model of the drive-in rack, actual impact forces are calculated and presented. Finally, using the experimental impact test results and the two simple mechanical models, the actual motion of the forklift truck body is determined. This motion, being a common characteristic to all drive-in racking impacts, allows impact forces to be obtained for various pallet loads, impact elevations and rack characteristics. Thus, the paper concludes with a general method for calculating forces generated under forklift truck impact. Forklift truck behaviour. Static and dynamic tests on a loaded forklift truck To measure the static and dynamic behaviour of a forklift truck, tests have been performed on a NICIYU FB0 forklift truck model, tons load capacity. Three different loads (300 kgs, 775 kgs and 75 kgs) are placed sequentially on a pallet, with the centre line of the pallet located at a distance from the centre line of the bottom connection between the mast and the forklift truck, as shown in Fig.. is set to be equal to the impact elevation associated with the impact tests performed on a drive-in rack structure and presented in the companion paper ([]). A hydraulic jack with a quick release system applies a horizontal load in the impact direction at the location of the impact point with the rack. Two LVDTs (Linear Variable Displacement Transducer) record the horizontal displacements of the pallet at the jack location (LVDT 5) and of the forklift truck mast at the jack elevation (LVDT 53), as shown in Fig.. The location of the LVDTs allows the bending and torsion of the pallet/mast system to be measured. In the first phase, the jack pulls statically on the pallet at a low displacement rate and the static stiffness of the forklift truck is recorded. In a second phase, the load is quickly released leaving the system free to oscillate and the dynamic response of the system is recorded at a rate of 50 z. Three tests are performed for each pallet load.

4 Figure : Forklift truck test set-up The test results show that during the static phase of the test both LVDT 5 and 53 increase linearly with the load. During the dynamic phase of the test, LVDT 5 and 53 have similar displacement frequencies and proportional amplitudes, indicating that the bending and torsional responses of the mast are coupled. Detailed test results are reported in [].. Simple mechanical model of the forklift truck Typically, the mast of a forklift truck is attached to the forklift truck body at two distinct points. A bottom hinge corresponding to the front wheel axis allows the mast to rotate forward and backward while restraining the mast from rotating sideway. Two hydraulic jacks, referred to as tilt hydraulic jacks and positioned at a distance h above the bottom hinge and on each side of the mast control the tilting motion about the bottom hinge. A forklift truck has usually two wheels at the front and one or two wheels at the back linked to the body by the means of shock absorbers. Fig. shows a photo of the NICIYU forklift truck used in the experimental tests. Fig. 3 shows a simple mechanical model of the forklift truck, in which a vertical member at a horizontal distance l from the bottom hinge represents the mast and a horizontal member at a vertical distance from the base of the mast represents the forks. A rotational stiffness k B representing the rotational stiffness of the bottom hinge combined with the rotational stiffness provided by the forklift truck shock absorbers restrains the mast from rotating sideway about its base. Two horizontal springs with stiffness k J, located at a distance d J on each side of the mast, represent the tilt hydraulic jacks and restrain the mast against torsion. Due to the nature of their connections with the mast, the two tilt hydraulic jacks are considered to be pinned at theirs ends, unable to transfer shear. The damping of the forklift truck mast is found to be a drag force damping type, i.e. proportional to the square of the velocity. This type of damping is found when an oscillator is immersed into a fluid and is for the forklift truck related to the action of the tilt hydraulic jacks. Being significant, this torsional damping from the tilt hydraulic jacks is considered to be the predominant source of energy absorbtion in the system and all other sources of damping are ignored. A damping coefficient C J for each tilt hydraulic jack is considered in Fig. 3. A mass m representing the combined mass of the load, the pallet and forks is considered at the centre of gravity of the load, i.e. at a vertical distance CoG and at a horizontal distance l + l 3 from the bottom hinge of the mast. The pallet mass and the mass of the

5 forks are estimated to be 50 kgs each. A load F representing the impact load/jack load is applied on the forks at a horizontal distance l + l from the bottom hinge of the mast. Figure : NICIYU FB0 forklift truck Figure 3: forklift truck simple model The system shown in Fig. 3 is simplified to the statically equivalent model given in Fig. 4, in which the tilt hydraulic jacks are replaced by an equivalent single torsional spring with stiffness k T and associated damping coefficient C T.

6 .3 Equations of motion Figure 4: Forklift truck equivalent simplified model For the model presented in Fig. 4 and for small displacements, the kinetic energy KE, potential energy PE and damping energy DE (from the generalised Rayleigh dissipation function) are given as, + ( l + l3 ). θ KE = m. θ () PE = k B θ + ktθ () = C T θ θ DE (3) 3 where θ and θ represent the bending and torsional angles respectively, as shown in Fig. 4. The equations of motion for θ and θ under external loading are obtained using the general Lagrange equation (Bangash [3]), d dt KE KE PE DE + + = M θi θi θ i θ i i (4) where M i represents the external bending or torsional moment. Eq. (4) applied to θ and θ gives, ( l + l ). θ = F m.. θ + kb. θ + m.. 3. (5)

7 m 3 T T 3. +.( l + l ). + k. θ + C. θ. θ + m..( l + l ). θ = F ( l l ) θ (6).4 Model calibration.4. Bending and torsional stiffness k B and k T The bending stiffness k B and torsional stiffness k T of the forklift truck mast are calculated from the static test results as, M B = kb θ = F (7) M ( l ) T kt = F + l = θ (8) where M B and M T are the bending and torsional moments induced by the jack force F respectively. θ and θ represent the bending and torsional angles respectively, as shown in Fig. 4. For small displacements, θ and θ are given by, θ = ( d53 θ. l ) (9) d5 d53 θ = (0) l where d 5 and d 53 are the recorded displacements of LVDTs 5 and 53 respectively. The stiffness k B and k T are hence calculated combining Eqs. (7) to (0) as, k B F. = () l l d53 + d5 l l k T ( l + l ) F. l = () d d 5 53 Table gives the values of k B and k T found for each experimental test using a linear interpolation during the static phase and the values of, l and l given in Table. It is observed from Table that while the bending rotational stiffness k B is essentially constant with increasing pallet load, the torsional stiffness k T increases with the pallet load. The increase in stiffness k T is related to the increase in oil pressure in the tilt hydraulic jacks required to prevent the mast from tilting forward when increasing the pallet load. The increase in oil pressure increases the stiffness k J and hence the stiffness k T. In view of the results shown in Table, throughout this paper k B is considered constant for all pallet loads and equal to its mean value kn.mm/rad. k T is considered to vary with the pallet load and linear interpolation is applied to the data shown in Table to obtain k T for pallet loads between 300 kg and 75 kg. For pallet load less than 300

8 kgs or greater than 75 kgs, k T is taken as kn.mm/rad and kn.mm/rad respectively..4. Damping coefficient The damping coefficient C T is calculated against experimental test results by numerically solving Eqs. of motion (5) and (6) using the commercial software FlexPDE [4]. The quick release system being not strictly perfect, the applied load F was found to decrease linearly from its initial value to zero between 0.6 and 0.6 sec. The decreasing rate of F is considered for each test when solving Eqs. (5) and (6) numerically. C T is found by best matching the numerical results to the experimental test results. The obtained values of C T are given in Table 3. It is observed from Table 3 that the damping coefficient C T increases with the pallet load which is probably related to the hydraulic pressure in the tilt jacks increasing with the pallet load. As for k T, C T can be linearly interpolated in Table 3 for different pallet loads..5 Validation of the forklift truck model Ignoring the damping of the underdamped system when calculating its natural angular frequencies, the natural frequencies ω i of the system shown in Eqs. (5) and (6) are evaluated using the determinant method as, ωi i. m. ω. m. + kb ωi. m.. ( l + l3 ) ( l + l ) ω. m. ( l + l ) + k. 3 i 3 T = 0 (3) Eq. (3) allows a unique solution given by, k. k ω = (4) m B T (. k + ( l + l ). k ) Table 4 compares the natural frequencies calculated using a Fourier Transform of the dynamic experimental tests with the natural frequencies calculated using Eq. (4). In Eq. (4), the values of k B and k T found in Table are used with the distances (l, l 3, CoG ) given in Table, in which CoG is the average value for the 3 different loads. Table 4 shows an excellent agreement between the experimental test results and solution obtained from the simple mechanical model introduced in Fig. 4, thus validating the simple mechanical model of the forklift truck. 3 Structural behaviour of drive-in rack 3. Static and dynamic tests on drive-in racks Static and forklift truck impact tests on a drive-in rack structure loaded with two different pallet loading configurations (namely Impact BC and Impact CD, see [] for details) have been performed using a hydraulic jack and the forklift truck analysed in Section respectively. In Impact BC, the pallets are not loaded in the vicinity of the T 3 B

9 impacted upright allowing this upright to freely deform under impact. In Impact CD, the pallets are loaded in the vicinity of the impacted upright allowing the investigation of the restraining effect of the pallets on the upright deformation. During the static tests, a hydraulic jack pulls on the impacted upright in the down-aisle direction at the impact elevation, while during the impact tests, the forklift truck impacts the rack in the downaisle direction with its load raised at the impact elevation. Displacements of the impacted upright and overall rack are recorded in the down-aisle direction. The complete test results are presented and detailed in the companion paper. 3. Simple mechanical model for drive-in racks When subjected to an impact force, a drive-in racking structure resists the impact by an overall displacement of the structure and by bending of the impacted upright. The stiffness k o associated with the overall displacement of the rack and the stiffness k u associated with the bending of the impacted upright at the impact point were determined from the previously mentioned static experimental tests. A simple mechanical model of a drive-in racking system is given in Fig. 5 where m o and m u represent the associated masses of the overall rack displacement and the impacted upright respectively. C o represents the damping coefficient of the overall rack and F u represents the damping of the impacted upright in form of a friction force. While the overall damping of the rack may be assumed to be linear due to classical energy loss in the system, the damping associated with the impacted upright is non-linear and is generated by the friction between the pallets and the rail beams. In Fig. 5, F imp represents the impact force on the rack, x F represents the displacement of mass m u and x o represents the displacement of mass m o. Figure 5: Rack simple mechanical model with frictional damping In order to simplify the previous mechanical model, the frictional damper of the impacted upright will be replaced by a linear damper without significant loss of accuracy as shown in []. The new simple mechanical model is shown in Fig. 6 where C u is the linear damping coefficient of the impacted upright. Figure 6: Rack simple mechanical model

10 3.3 Equations of motion The equations of motion for x F and x o of the system presented in Fig. 6 are, m u xf + Cu xf xo + ku = ( xf xo ) Fimp (5) 3.4 Model calibration m o x o + Co xo + ko xo Cu xf xo ku ( xf xo ) = 0 (6) The values of k u, k o, m o, m u, C o and C u are obtained from the experiments detailed in []. The values of k u and k o are calculated from static tests performed on the drive-in rack structure and reported in Table 5. m o and C o are obtained from the impact test results using the recorded free oscillation of the overall rack displacement x o after impact. m o is calculated using the natural frequency of the overall rack in Eq. (7), while C o is obtained from the logarithmic decrement given in Eq. (8) over one period T o. Calculated values for m o and C o are reported in Table 5. To m o = ko (7) π C o m = T o o log x o xo ( experimental at t) ( experimental at t + T ) o (8) C u is obtained using Eq. (6) and the values of k o, m o and C o calculated previously by averaging the values of C u calculated for each experimental recording step Δt over the first phase of the impact tests, in which the overall rack essentially stays stationary while the impacted upright bends. m u cannot satisfactory be obtained from previous equations and has to be estimated. In Impact BC configuration only the impacted upright, part of the adjacent frame bracing and part of the adjacent rail beams are associated with the relative displacement x F - x o. This leads to a light mass m u, estimated to be about 50 kgs. In Impact CD configuration part of the adjacent pallets are also associated with the relative displacement of x F - x o. This leads to a heavier mass m u, estimated to be about 400 kgs. It is observed that the average damping ratio ζ o of the overall rack defined by Eq. (9), is equal to 0.07% which is close to the typical value of damping ratio of 0.% to 0.9% for steel structures (Bangash [3]). C o ς o = (9) moko

11 3.5 Validation of the rack model In order to validate the simple mechanical model introduced in Fig. 6, x o obtained from the equation of motion (6) is compared to experimental test results by using x F from experimental tests as input data. The equation is solved by using FlexPDE [4] and the mass, stiffness and damping values given in Table 5. Fig. 7 compares x o for a typical experimental test and the corresponding x o calculated from FlexPDE [4]. The results show excellent agreement between the two values of x o both during and after the impact, thus validating the simple mechanical model of the rack. Comparisons between the two values of x o for all impact tests are reported in [], all comparisons show excellent agreements. Figure 7: Comparison between x o from numerical methods and x o for Test 07 - Impact BC configuration 4 Impact force The impact force F imp developed between the forklift truck and the drive-in rack during the experimental impact tests introduced in Section 3 and detailed in the companion paper ([]) is now calculated by solving Eq. (5) using FlexPDE in which values of x F and x o obtained from experimental test are introduced as input data. Fig. 8 and Fig. 9 show the impact force F imp for 5 tests for Impact BC configuration and 5 tests for Impact CD configuration, calculated using the values of m u, C u and k u given in Table 5. While performing the impact tests, it was observed that the pallet under the 300 kgs load has a tendency to slide on the forks leading to two distinct sequences of impact. This phenomenon is detailed in [5], and is observed in Fig. 8 and Fig. 9 where the impact load increases to a plateau corresponding to the pallet sliding on the forks. The sliding lasts about 0.3 sec, then the impact force increases again until the forklift truck loses contact with the rack.

12 Figure 8: Impact forces for Impact BC configuration Figure 9: Impact forces for Impact CD configuration If the pallet sliding effect is omitted, it is observed from Fig. 8 and Fig. 9 that the impact load is rather consistent for each impact configuration and reaches a maximum between 0. sec and 0. sec for Impact BC configuration and between 0.08 sec and 0. sec for Impact CD configuration. 5 Combining the models for the rack and forklift truck In order to combine the mechanical model of the forklift truck with any rack model representing different rack configurations, pallet loads and impact elevations, a common characteristic to all impacts on drive-in racking structures has to be found. Typically the impact of a forklift truck on a drive-in storage rack is induced by the rotation of the forklift truck body about its front wheel axis as shown in Fig. 0. The rotation of the forklift truck body will essentially remain the same for all impacts on drive-in racking structures and can be taken as a common input for combined forklift truck and rack model. ence, impact forces and displacement at the impact point can be calculated from the combined model.

13 Figure 0: Forklift impact on drive-in racking structure Typically, the front wheel axis of a forklift truck corresponds to the bottom hinge connection of the mast with the forklift truck body. Fig. shows the simple mechanical model of the forklift truck introduced in Fig. 4 rotated about the front wheel axis by an angle α. The rotation θ is the additional rotation produced by the extension and compression of the tilt jacks. For small displacements, the kinetic energy KE of the forklift truck in Eq. () is rewritten as, KE = m ( 3 ) θ + l + l α + θ (0) Figure : Rotating motion of the forklift truck mechanical model The potential energy PE and damping energy DE in Eqs. () and (3) respectively are unchanged. The equations of motion for α, θ and θ under external loading are obtained using the Lagrange equation given in Eq. (4) as, ( l + l ). θ + m..( l + l ). α = F m.. θ + kb. θ + m imp. ()

14 m 3 T T 3 3 imp. +.( l + l ). θ + k. θ + C. θ. θ + m..( l + l ). θ + m. ( l + l ). α = F ( l l ) () Using the geometrical relationship between x F, θ and θ, ( l + )( θ α ) x F. θ + l + (3) =. the above equations of motion can be rewritten for θ and α as, m.. B ( l + l ) ( l + l ) θ ( l + l ) 3 + m.. k θ + m k x kb + x B ( l + l ) ( l + l ) α ( l + l ) α = F. 3 F imp F (4) m + m ( l + l ) ( l + l ) ( l + l ) α = F.( l + l ) 3 3 ( l + l3 ) ( l + l3 ) ( l + l ) θ + kt. θ + CT. θ. θ + m ( l + l3 ) imp x F (5) α is obtained by solving Eqs. (4) and (5) using the software FlexPDE [4] with x F taken from experimental test results. The impact force F imp is calculated as per Section 4 and the values given in Table to Table 3. Involving simpler equations, it is worth noting that convergence in FlexPDE [4] is more easily achieved by solving Eqs. (4) and (5) for α+θ and θ instead of α and θ. α is then retrieved by subtracting θ from α+θ. Fig. and Fig. 3 plot α calculated for all tests shown in Fig. 8 and Fig. 9 respectively. α is only plotted in the domain of validity of Eqs. (4) and (5), i.e. when there is contact between the pallet and the rack, which is considered to be until the impact force reaches zero in Fig. 8 and Fig. 9. If the sliding of the pallet is omitted, results show that α is consistent for all tests and reaches its maximum value at about 0.5 sec. The maximum bay opening occurs at this point as detailed in the companion paper ([]). It is observed that when the maximum impact load is reached between 0.08 sec and 0. sec in Fig. 8 and Fig. 9, α is still increasing. The maximum average value of α at 0.5 sec is equal to 0.03 rad. α is approximated as a multi-linear curve by averaging the results shown in Fig. and Fig. 3 over the first critical 0.5 sec (see Fig. 4), the multi-linear approximation can be used as input in Finite Element models or in solving the equations of motion (5), (6), (4) and (5) to obtain the impact force and rack displacement under impact. The analysis proceeds through the steps: ) Determine the forklift characteristics (k T, k B, C T ), the rack characteristics (m u, C u, k u, m o, C o and k o ) using FE for instance. ) Determine the impact characteristics (, CoG, l, l, l 3 and m). 3) Obtain F imp by solving the four equations of motion (5), (6), (4) and (5) in x F, x o, F imp and θ using α as input.

15 Figure : Forklift truck body rotation for Impact BC configuration Figure 3: Forklift truck body rotation for Impact CD configuration Figure 4: Multi-linear approximation of α

16 6 Conclusions Based on first principles of mechanics, this paper proposes simple models for a loaded forklift truck impacting with a loaded steel drive-in rack. Model masses, as well as stiffness and damping coefficients are calibrated against experimental results obtained from tests of a forklift truck and a drive-in rack structure. The simple models are found to accurately reproduce the static and dynamic behaviour of the forklift truck and drivein rack. The impact forces developed between the forklift truck and the drive-in rack are calculated using the simple rack model and impact tests on a drive-in rack structure presented in a companion paper ([]). Finally, combining the two simple mechanical models and using the impact test results, the actual forklift truck body rotation during impact is calculated. This motion, assumed to be a common characteristic to all drive-in racking impacts, allows impact forces to be obtained for various pallet loads, impact elevations and rack characteristics. Thus, the paper provides a general method for calculating forces generated under forklift truck impact. 7 References [] Gilbert BP, Rasmussen KJR. Impact tests and parametric impact studies on drive-in steel storage racks. Submitted to Engineering Structures. [] Gilbert BP, Rasmussen KJR. Determination of accidental forklift truck impact forces on drive-in steel rack structures: School of Civil Engineering, The University of Sydney, Australia; 009. Research Report R90. [3] Bangash MY. Impact and explosion, analysis and design. Boca Raton, Florida, USA: CRC Press Inc [4] FlexPDE. FlexPDE 5..0s, User guide: PDE Solutions Inc [5] Gilbert BP, Rasmussen KJR, Zhang. Impact tests and parametric impact studies on drive-in steel storage racks: School of Civil Engineering, The University of Sydney, Australia; 009. Research Report R903.

17 Load = 300 kg Load = 775 kg Load = 75 kg k B k T k B k T k B k T (kn.mm/rad) (kn.mm/rad) (kn.mm/rad) (kn.mm/rad) (kn.mm/rad) (kn.mm/rad) Test Test Test Average Table : Static test results stiffness (mm) CoG (mm) l (mm) l (mm) l 3 (mm) d W (mm) Table : Dimension values for tests Load = 300 kg Load = 775 kg Load = 75 kg C T (kn.mm.s²/rad²) C T (kn.mm.s²/rad²) C T (kn.mm.s²/rad²) Test Test Test Average Table 3: Calculated damping coefficients C T Load = 300 kg Load = 775 kg Load = 75 kg f (hz) f (hz) f (hz) Test Test Test Average Eq. (4) Difference (%) Table 4: Natural frequency of forklift truck k o (kn/mm) m o (kg) C o (kn/(mm/s)) k u (kn/mm) m u (kg) C u (kn/(mm/s)) Impact BC Impact CD Table 5: Drive-in rack simple model characteristics