Design of a mounting bracket for electrical equipment Cimino Marta Mauriello Tommaso Optimisation de la tenue m ́ ecanique d’une structure par ́ el ́ ements finis 2A - 2021/2022 Abstract Nowadays FEM softwares allow to perform quick structural analyses of complex structures, yet an engineer is still needed in the design cycle to drive the optimization process. This brief work shows the optimization workflow employed to reduce the mass of a mounting bracket while respecting the assigned resistance and rigidity constraints. Contents 1 Specifications 2 2 Optimization process 3 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Load, constraints and mesh definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Case 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.5 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.6 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.7 Case 4: final solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Mesh size influence 8 4 Screws sizing 9 5 Conclusions and perspectives 10 1 1 Specifications The following are the requirements that the mounting bracket shall respect: • The piece will be realized in Al 2024 with the following characteristics: Young Module Elastic limit Density 70 GPa 300 MPa 2700 kg/m 3 Table 1: Mechanical properties of Al 2024 • The support will have to remain in the elastic domain for all the load conditions ( σ max < σ el ). • The maximum allowable displacement is 5 mm • The structure will have to withstand the following load cases (separately): – ˆ x acceleration: ± 10 g – ˆ y acceleration: ± 10 g – ˆ z acceleration: ± 8 g • The electrical equipment (25 kg) is considered homogeneous for the center of mass computation. • The connection with the electrical equipment shall leave the necessary space for its ventilation (a hole of 160). • The structure must not buckle. • The optimization criterion is to make the structure has lightweight as possible (considering a benchmark minimum value of 331 g obtained in previous studies). Figure 1: Scheme of the electric equipment to support and of the available space for the bracket. 2 2 Optimization process 2.1 Preliminary remarks A total of 6 analyses were conducted: for the loads applied on ˆ x , ˆ y , ˆ z in either one of the two directions (as the results are equivalent for loads of equal magnitude but opposite in sign) and for the associated buckling conditions. In order to simplify the computations, the analyses were not conducted on the whole assembly but only on the mounting bracket, defining appropriate boundary conditions and load cases to simulate its connections with the electrical equipment and with the fuselage beams. The design process has been carried out starting from simple structures, analysing stresses, deformations and buckling conditions under one of the load cases (usually the ˆ z one) and then iterating from the results. In order to speed up the process not all the load conditions have been analyzed for the first iterations (e.g. since the first structure clearly did not respect the rigidity constraint for one load case, the other load cases were not studied). Note that the structures shown in the screenshots do not all have the same scale factor (with respect to the deformation), as this has been selected at each time to improve the readability of the figure (either augmenting small deformations or decreasing too large ones). 2.2 Load, constraints and mesh definition Load transmission to the mounting bracket takes place at the four fixing holes. The loads resulting from the electrical equipment acceleration are applied at the centre of mass of the electrical equipment ( poignet ) in order to take into account not only the forces but also the moments transmitted to the mounting bracket. In the first analyses the bracket connections to the the fuselage beams have been modeled with fixation boundary conditions (3 translations and 3 rotations blocked) applied on the internal surface of four connection holes. These boundary conditions have been modified for the final solution in order to better approximate the real behavior of screwed connections. As shown in Fig. 2 in the final case the translations along ˆ y and ˆ z are blocked in the internal surface of the holes, while a constraint on the ˆ x translation is imposed at the connection plates to the beams, and no constraint has been imposed for the rotations. This also allowed to avoid the nonphysical stress concentrations that appear in the first analysis in correspondence of these holes. Eventually, the final analysis considered two holes in the upper connections, as employing two screws resulted necessary from the analysis of the reaction forces of this boundary condition (more details on the screws sizing in section 4). For all the upcoming analysis a mesh size of 2 mm with parabolic elements has been employed. This proved a good compromise in terms of accuracy of results and required computational time (about 40 s to run a static analysis). A study on the effect of the mesh size is presented in section 3. In some cases a mesh refinement has been enforced in correspondence of small features (like bevels). (a) Upper connections (b) Bottom connections Figure 2: Details on the boundary conditions for the final case. 3 2.3 Case 0 For the first case a simple structure with four beams with a rectangular full section has been considered. A square support (leaving the required space for the electrical equipment ventilation) connects the four beams at the extremity improving the overall rigidity. This structure weighs 676 g . The deformed meshed structure under the ˆ z load condition is shown in Fig. 3a. Fig. 3b shows the nodes translations for such a structure. As it can be seen the maximum displacement ( 125 mm ) is huge with respect to the design specification. Stress or on the other hand are in the correct range for the overall structure except for the holes at the support base, where there’s a large increase in stress due to the way the stress lines have to flow. A posteriori this image is not really representative of the stress distribution on the majority of the surface (only a blue color can be seen), while a lower maximum stress should have been set on the scale to improve the figure readability. This has been done for the following screenshots. (a) Deformed shape. (b) Nodes translation. (c) Von Mises stresses. Figure 3: Case 0, static analysis. As can be seen in Fig. 4b, the structure won’t undergo buckling as the buckling factor is close to two (in absolute value) for the first 10 modes. (a) Buckling deformation (b) Buckling modes. Figure 4: Case 0, buckling analysis. 2.4 Case 1 Being clear that the first structure lacks in rigidity and weighs too much, square hollow section beams have been employed in the following analysis, and additional beams have been added on the two side faces. The static and buckling analysis of the structure for a load applied in the ˆ z direction 4 gave satisfactory results meeting all the requirements, as shown in Figs. 5 and 6. Nonetheless it was decided not to proceed with further analyses (ˆ x and ˆ y load direction) as the mass of the structure, 430 g , was still quite large. (a) Case 1, nodes translation. (b) Case 1, Von Mises stresses. Figure 5: Case 1, static analysis. Figure 6: Case 1, buckling modes. 2.5 Case 2 In case 2, the square hollow section beams were retained and tubular rods with a diameter of 2 mm were added to improve the overall structure rigidity. When the load is applied in the ˆ z direction, the results in terms of maximum stress and displacement are acceptable, whereas when the load acts in the other two directions the results are unsatisfactory. The results of the analysis with load applied along ˆ x are shown in Fig. 7. The value of the deflection, with a maximum of 3.13 mm, is within the prescribed limits. The Von Mises stresses, visible in Figure 7b, are beyond the yield stress in correspondence of the beams connection to the two supports. On the other hand the buckling factors are all less than one (Fig. 7c) for all the load cases, due to the small section inertia of the beams employed to improve the structure rigidity. Analogous considerations can be made for the ˆ y load case (Fig. 8). In this case there is a larger deflection value of 4.52 mm, but still within the limits. 2.6 Case 3 In case 3, the oblique beams have been replaced by four hollow tubular beams arranged in the shape of a square. The results of the analysis with the load along ˆ z are shown in Fig. 9. In this case the buckling shows no problems, yet the results in terms of deflection (with a maximum of 38 mm) and Von Mises stresses are largely unacceptable. This is due to the additional beams being 5 (a) Case 2, nodes translation. (b) Case 2, Von Mises stresses. (c) Case 2, buckling modes. Figure 7: Case 2, load on ˆ x (a) Case 2, nodes translation. (b) Case 2, Von Mises stresses. (c) Case 2, buckling modes. Figure 8: Case 2, analysis with load on ˆ y perpendicular to the longitudinal ones and hence not improving effectively the structure rigidity, proving that oblique ones are the correct choice. The structure, although lightweight, is totally unsuitable given the imposed requirements. 2.7 Case 4: final solution Taking into account all the strengths and weaknesses of the previous designs, and after some more tuning of the thicknesses and mass distributions, a final design has been obtained, which is shown in Fig. 10. This structure has the following characteristics: • It weighs 317 g , 14 g less than the previous minimum, resulting in a 4.3% mass reduction. • It employs hollow circular section beams (as they are more mass efficient) with different radius and thicknesses (around 3 mm of radius and 1 mm of thickness), depending on the associated stresses and buckling conditions. For example the bottom diagonal beams have a slightly larger thickness than the top ones to improve their buckling resistance, which happens to be critical when the load is applied on the ˆ z direction. The beams with the greatest radius are the four straight ones. • Since stress concentrations take place in proximity of the connections with the terminal sup- ports additional mass has been added in these regions in order to meet the structural require- ments (i.e. decrease the Von Mises stress). • In order to make the structure as lightweight as possible the square support is pierced where the stresses are lower. • Appropriate size bevels have been added in correspondence of sharp edges to improve the stress flow. 6 (a) Case 3, nodes translation. (b) Case 3, Von Mises stresses. (c) Case 3,buckling modes . Figure 9: Case 3, Analysis. • Each upper connection with the fuselage beams requires two M7 screws, while bottom ones only one M7 each. The graphic display of the results (displacement at the nodes, Von Mises stress and buckling coefficients) is shown, for the three different load conditions, in Figs. 11, 13 and 15. Figs. 12, 14 and 16 show details of the most stressed areas with the load respectively along ˆ x , ˆ y and ˆ z Figure 10: Final design for the mounting bracket. Load direction ˆ x ˆ y ˆ z Max displacement [mm] 2.05 3.34 2.17 Max stress [MPa] 294 289 272 First buckling factor 2.03 1.38 1.08 Table 2: Summary results It is observed that in all the three load cases the rigidity requirements are well respected (maxi- mum displacement much less than 5 mm), while the largest Von Mises stress is close to the maximum allowable value (300 MPa), but still inferior, as required. In terms of buckling the most critical load condition is the ˆ z one, and that’s why the bottom diagonal beams have a slightly greater cross section (if they were to be identical to the top ones the buckling factor would have been 0.99 instead of 1.08). 7 (a) Nodes translation. (b) Von Mises stresses. (c) Buckling modes . Figure 11: Final solution, load on ˆ x Figure 12: Details of maximum stress zones, load on ˆ x 3 Mesh size influence This section is devoted to the study of the evolution of results as a function of mesh size. Only the ˆ z load case has been considered, and the evolution of the maximum displacement and maximum Von Mises stress obtained for 6 different mesh sizes is reported in Table 3. Buckling results have not been taken into account as for smaller mesh sizes the required computational time became too large. Fig. 17 shows graphically the results, while Fig. 18 shows the mesh size on the actual structure. Mesh size [mm] 5 3.2 2.5 2 1.4 1 Max Displacement [mm] 2.09 2.12 2.14 2.17 2.2 2.17 Max Von Mises stress [MPa] 209 212 214 272 305 294 Table 3: Results of the study on the mesh size influence It’s noted that: • No asymptotic value has been obtained with the reduction of the mesh size; even finer mesh sizes should have been analyzed, but this resulted unpractical due to the large computational time required. • The behavior is not monotonic; it’s observed a decrease in the magnitude of the results when the mesh size decreases from 1.4 mm to 1 mm. It is hypothesised that a 1 mm sized mesh is able to catch more accurately the stresses in certain areas (fillets, rounded parts) as well as the reduction of stress concentrations that takes place in those regions, whereas larger mesh sizes performed rougher approximations and obtained larger stresses, and hence, displacements. As previously stated the analyses used for the optimization process were carried out using parabolic elements with a size of 2 mm. 8 (a) Nodes translation. (b) Von Mises stresses. (c) Buckling modes . Figure 13: Final solution, load on ˆ y Figure 14: Stress details, load on ˆ y 4 Screws sizing Once the structural analysis has been completed it is possible to retrieve the reaction forces exerted by the boundary conditions. These has been used to size the screws that will be used to connect the mounting bracket to the support beams. On the other hand, the sizing of the screws connecting the electrical equipment has not been carried out as 1) no data on the forces transmitted through these holes was available as no boundary condition was enforced there and 2) these connection are much less solicited. First the forces applied on the most solicited screw (the bottom ones in the ˆ z load condition) have been computed, obtaining: F x = 1360 N F y = − 449 N F z = − 1868 N (1) F x will produce a normal stress ( F n ) on the screw, while the resultant of the y and z components will generate a shear stress ( F s ). Hence the screw will have to withstand the following forces: F n = 1360 N F s = 1921 N (2) An M7 ISO screw has been selected, with an internal diameter d int = 5.77 mm (which will be used to compute the loaded area A int ). The selected screws are high resistance ones with a property class of 10.9 (corresponding to a yield stress of σ el = 900 MPa). The screw preload has been computed as a fraction (70%) of the maximum load that the screw can withstand: P 0 = 0 7 · A int · σ el = 16473 N (3) Then the maximum allowable shear force has been computed taking into account the preload, the normal force F n , a security factor υ = 1.25 and a (small) friction coefficient μ = 0.2 between the screw head/nut and the surfaces as follows: F s max = μ υ ( P 0 − 0 8 · F n ) = 0 2 1 25 (13096 − 0 8 · 1360) = 2462 N > F s (4) 9 (a) Nodes translation. (b) Von Mises stresses. (c) Buckling modes . Figure 15: Final solution, load on ˆ z Figure 16: Stress details, load on ˆ y The maximum allowable normal force is instead F n max = A int · σ el Taking into account the preload the condition to verify is hence: F n + P 0 = 17833 N < F n max = 23533 N (5) Both the condition on the normal and shear stress are verified by the selected screw, which has been employed for all the 6 connections of the bracket with the support beams. Notice that the top connections have two screws each, as only one screw wouldn’t have been able to withstand the total load (without being too large and possibly compromising the connection with the beams). Considering also a nut and washer and the thickness of the elements to connect the selected screw has a length of 25 mm, resulting in a 10.9 M7 × 25 hexagonal head. The screws that will be employed for the connection to the electrical equipment will instead be 10.9 M4 × 12 hexagonal head (considering the same property class for simplicity, and taking into account that the electrical equipment has connection holes of 4). 5 Conclusions and perspectives The employed optimization process proved effective in finding a structure that could respect the rigidity and resistance constraints while being as lightweight as possible. Nonetheless it’s possible to make some remarks and envision some possibilities of future work: • As the optimization criterion was the mass reduction, no attention has been put on the actual technological and economical feasibility of the final structure. In particular the several different dimensions of the beams and the various masses added to improve resistance make the structure difficult to realise with off-the-shelf components. A further step could be to try to design the bracket only with standardized aluminum rods. • On the other hand the design could be carried on by developing even more exotic shapes, possibly with variable sized cross section beams, to be then realized in additive manufacturing. 10 (a) Max displacement (b) Max Von Mises stress Figure 17: Evolution of results as a function of mesh size. (a) 5 mm (b) 3.2 mm (c) 2.5 mm (d) 2 mm (e) 1.4 mm (f) 1 mm Figure 18: Mesh visualisation. • No security coefficients have been considered in the overall design process (except for a 1.25 in the screw sizing, as it’s done in common practice). Implicitly it has been assumed that the maximum yield stress was assigned already taking into account some security coefficient. • The three load cases have been analyzed separately, as required in the specifications. It could be interesting to compare the obtained mass with that of a structure which is instead designed to withstand all the three load cases at the same time. 11