Stiffness Analysis of the Tracker Support Bracket and Its Bolt Connections

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1 October 25, 2000 Stiffness Analysis of the Tracker Support Bracket and Its Bolt Connections Tommi Vanhala Helsinki Institute of Physics 1. INTRODUCTION STIFFNESS ANALYSES ENVELOPE BRACKET BOLT CONNECTIONS BRACKET TO TUBE BRACKET TO FOOT FOOT TO HCAL CONCLUSIONS REFERENCES...12 APPENDIX

2 1. Introduction The Tracker support tube is supported at its ends by four aluminium brackets that locate at the horizontal symmetry plane. The outer end of the bracket is attached to HCAL. The brackets must bear the load caused by the mass of the Tracker (estimated to be 4t) and the dimensional changes of HCAL due to its thermal expansion. The space for the bracket is very limited due to the ECAL, cooling pipes, cables etc. Therefore, the shape of the bracket cannot be optimized as one would probably like. The support bracket is made of three entities, Figure 1. The entities: banana, bracket and the foot are connected together with stainless A4-70 steel. The bolt joint must provide stable connection in all circumstances. The bolt joints are needed, because the components are mounted to CMS in different phases of assembly. The joints are also needed because of required assembly clearances. Figure 1. The entities of the Tracker support bracket. The components are not in scale. 2. Stiffness analyses The stiffness of the support bracket was analysed with finite element (FE) models in Ansys. The analysis was done in two parts. The preliminary stiffness calculations were done with the bracket envelope model. Later, the displacement was calculated with the more detailed bracket model. The different approach to defining the model was used. The geometry of the envelope model was created with 3D-modeling software. Instead of that, the bracket model was defined with keypoints in Ansys. The two modelling methods should give equivalent results despite the different look of the model. The bracket was modelled in both analyses as a single structure. Therefore, the stiffness calculations of the bolt connection between pieces are presented separately later in this paper. 2.1 Envelope The FE-model of the bracket envelope was based on the Euclid database drawing TL012287MQ made by Mr. Pierre-Benoit Brouze. The bracket is according to drawing an assembly of two aluminium pieces. The connection between the bracket and the Tracker support tube was not yet designed. The bracket was modelled as one solid piece for the FE-analysis. The shape of the envelope was simplified by neglecting small details such as curvatures, rounds etc. 2

3 The geometry of the bracket was modelled in Pro/Engineer 3D-modelling software, Figure 2. The dimensions were given in millimetres. An IGES-file containing the envelope surfaces was exported from ProE. Figure 2. The bracket envelope model in Pro/Engineer. The IGES-file was then imported into Ansys 5.6. Ansys creates a solid model of the imported model. It deletes small areas and merges coincident keypoints. The element type was chosen (SOLID95). SOLID95 is a 3-D 20-node structural solid element. It is a higher order version of the 3- D 8-node solid element (SOLID45). According to Ansys element manual the chosen element type tolerates irregular shapes without as much loss of accuracy. SOLID95 elements have compatible displacement shapes and are well suited to model curved boundaries. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions. For the analysis, material properties for the aluminium bracket were defined (see Table 1) and the model was meshed, Figure 3. The rather coarse element mesh was used (smart element level 7). Table 1. Bracket material properties (aluminium) used in FE-model. Variable Symbol Young's modulus E [N/mm 2 ] Poisson's ratio Q 0.25 Density J 2700*10-9 [kg/mm 3 ] 3

Finally, the FE-model was analysed. The solution was done without error notices. The displacement of the bracket envelope with the given load is presented in figures 4 and 5.

4 Figure 3. A meshed bracket in Ansys. Then, boundary conditions, gravity and estimated load (a 10 kn load at the end of the bracket) were applied. All degrees of freedom (DOF) on the nodes at two attachment surfaces were fixed. The tube end of the bracket was free. The force was equally divided between 10 nodes at the end of the bracket. Finally, the FE-model was analysed. The solution was done without error notices. The displacement of the bracket envelope with the given load is presented in figures 4 and 5. The displacement is in millimetres. The nodal forces point to the negative Y-axis. The co-ordinate system of the bracket in the FE-analysis is illustrated in Figure 3. Please note that the system differs from the global co-ordinate system of the CMS Tracker. Figure 4. The total displacement of the bracket envelope in millimetres. 4

Figure 5. The vertical displacement of the bracket envelope in mm. 2.2 Bracket The further finite element study of the bracket stiffness was done with the more precise dimensions of the bracket.

5 Figure 5. The vertical displacement of the bracket envelope in mm. 2.2 Bracket The further finite element study of the bracket stiffness was done with the more precise dimensions of the bracket. The geometry of the bracket model differs from the envelope model at the tube end of the bracket. The bracket is to be connected to the tube with three via a stiff aluminium banana. The banana was not modelled, but only its reaction force with the bracket is defined. The bracket displaces in the Y-direction due to the mass of the Tracker. HCAL and the Tracker support tube are assumed to have infinite stiffness. Due to the symmetry of the Tracker, the tube end of the bracket cannot displace either in the X- or in the Z-direction. These boundary conditions did not exist in the envelope model. The foot and the bracket were modelled again as a single solid piece. The geometry of the model was created by first defining keypoints, then connecting these by lines and finally creating areas from the lines. The element type used was SHELL63. The thickness of the bracket was defined to be 55 mm at the foot, 45 mm in the inclined part and 35 mm at the end of the bracket. The mass of the bracket without banana was calculated to weigh about 30 kg. All DOFs on the nodes at two attachment surfaces were fixed. Then, the vertical load was applied and the displacement at the tube end of the bracket in the X- and Z-directions were fixed. The support forces at the tube end of the bracket are -1.1 kn in the X-direction and 0.9 kn in the Z- direction. The effect of the thermal expansion of the HCAL on the forces in the bracket was also analysed. The bracket should adapt dimensional changes of HCAL, because the Tracker support tube should not be 5

6 stretched or squeezed. HCAL is mostly made of bras, whose coefficient of the thermal expansion is 20 Pm/mqC. The temperature change of the HCAL was estimated to be r2.5qc, at maximum. The 2.5qC temperature change would increase or decrease the total length of HCAL (6.090 m) by 304 Pm. Therefore, one half of HCAL expands 152 Pm. Correspondingly, the temperature change mentioned above increases the inner radius of HCAL (1.836 m) by 92 Pm. The displacement of the tube end of the bracket with respect to the foot was defined with the displacement boundary conditions. A 2.5qC thermal expansion creates 3.9 kn in the X- and -3.0 kn in the Z-direction support forces at the end of the bracket. The displacement of the bracket with HCAL thermal expansion is presented in Figures 6 and 7. Figure 6. Total displacement of the bracket. Gravity, Tracker mass and HCAL thermal expansions included. Displacement is in mm. 6

Figure 7. Displacement of the bracket in the direction of gravity. Displacement is in mm. The stresses in the bracket structure were also calculated.

7 Figure 7. Displacement of the bracket in the direction of gravity. Displacement is in mm. The stresses in the bracket structure were also calculated. The computed stresses in the bracket structure are presented in Figure 8. The maximum stress (97 MPa) gives the safety factor to stresses more than 2. In addition, the stresses in the final bracket are slightly smaller than the computed values, thanks to round corners. Figure 8. Stresses in the bracket [MPa]. 7

8 3. Bolt connections The strength and the margin of safety (MoS) of the bolt connections were calculated. A bolt joint is normally designed in such way that the friction force caused by the axial force of the bolt [1] transfers the shear force from a structure to another. Thus, the key property of the functional joint is the sufficient tension stiffness of the bolt. The coefficient of friction between the components to be connected is estimated to be P=0.3. Stainless steel will be used to connect bracket components together. The characteristics of the are presented in Table 2. The material properties of the bolt are presented in Table 3. The components that are squeezed together are made of aluminium whose Young's modulus is Ep=70000 Mpa. The stress in the bolt should not exceed the yield strength value. Table 2. Characteristics of the. M12 M14 M16 d (mm) P (mm) d 2 (mm) d 3 (mm) R min (mm) A 3 (mm 2 ) A s (mm 2 ) Table 3. Material properties of a stainless steel bolt. Young's modulus, Es [MPa] Yield strength Vty [MPa] 480 The formulas to dimension a bolt joint are given in [1]. To calculate bolt connection one must first describe the joint. The required parameters are introduced in Table 4. The parameters of the bolt connections are defined in Chapters Table 4. Parameters of the bolt connection. Variable Unit Description li mm Length of the bolt in the connection lk mm Height of connection Da mm Diameter of connection Db mm Diameter of hole for a bolt P - Coefficient of friction between the contact surfaces n - Dimensionless factor of the connection [1, Fig ] First, the spring coefficient of the bolt and the spring coefficient of the elements to be connected are calculated. The combined spring coefficient of the bolt and nut (ks) is calculated with the equation [1, ]. 8

9 ks ª ««0.8 d Es S d 2 4 ¹ 0.5 d Es A3 li Es S d 2 4 º»» ¼ 1 (1) The spring coefficient of the elements to be connected (kp) can be calculated with the equation [1, ]. ª «1 kp dk 2 Db 2 Da dk 2 ( ) x 2 º» ¼ x Ep S lk 4 (2), where x is the dimensionless variable of the joint defined by the dimensions of the joint. The value for x is calculated with the equation 3 [1, ] 3 lk dk x ( dk lk) 2 (3) The required normal force of the joint i.e. tension force of the bolt (Fkr) is calculated with the wellknown friction equation 4. If there is more than one bolt, the force is assumed to divide equally to all i.e. the joint is symmetrical. Thus, two in the joint divide the required tensions force of the bolt by two. Fkr Fq friction (4),where Fkr is the minimum tension force of the bolt and Fq the shear force at the joint The additional force ratio, I, for the definition of the maximum tension force of the bolt, is now defined [1, ]. I ks n ks kp (5) The required pre-strain of the bolt (Fv) is [1, ] Fa Fv Fkr 1 I (6), where Fa is the exterior axial load of the joint. The maximum force that will stretch the bolt is finally [1, ] Fsmax Fv I Fa (7) 9

10 The strength of the bolt i.e. the maximum force the bolt can bear (Fty) is calculated with the equation 8 Fty As Vty (8),where Vty is yield strength of the bolt. The margin of safety (MoS) can be calculated as a ratio between the maximum stretching force of the bolt and the strength of the bolt. MoS Fty Fsmax (9) The stiffness calculations of the bolt connections were done in Mathcad 2000 and they are all presented in Appendix Bracket to tube The bracket was planned to be attached to the banana with a single M12 bolt. The connection must bear the quarter mass of the Tracker (=1000kg) and a 3.9-kN horizontal force due to thermal expansion of HCAL. The total shear force at the connection would be thus 10.6 kn. In addition, there is a 3.0-kN force in the axial direction of the bolt. First, the joint with a single M12 bolt was calculated. The margin of safety was MoS=1.1. The margin was found too small, because it does not meet the earthquake safety requirements for example. An M16 bolt would yield MoS=2.0. However, an M16 bolt would require too much space. Therefore, the bracket should not be attached to the banana with a bolt joint that relies only on friction, but the shape of the components must reinforce the joint. The feasible joint could be made of a M12 bolt with a solid pin sticking out of the banana, Figure 9. Figure 9. An improved design of the banana. The banana is connected to tube with three M12. The margin of safety of the joint is MoS=3.2 assuming that the load is divided equally between the. The length of a thread in the insert should be 16 mm [1,Table ], at least. The sufficient thread guarantees that the thread does not rip of the bolt. 10

11 3.2 Bracket to foot The bracket was preliminarily designed to be attached to the foot with four parallel M12. Due to the shape of a bracket, both shear and axial forces load the joint. The distance between the four and the Tracker support tube is 550 mm in the X-direction and 350 mm in the Z-direction. The maximum shear force the M12 bolt is able to transfer with friction (12.14 kn) was calculated with equations presented in Appendix 1. The equilibrium of a four-bolt joint without structural support (Fig. 10 at left) was calculated first. Then, the forces were calculated with the five-bolt design. In case of five, the outermost remain their positions and the distance between the was decreased. The results of calculations with MoS of the connection are presented in Table 5. As one can notice, the joint cannot bear the load. The bracket would rotate until the bracket corner touches the foot. Table 5. Shear forces in the different bolt connections. Bolts Shear X (kn) Shear Y (kn) Shear total (kn) Axial force (kn) MoS Figure 10. Preliminary designed bolt connection between the bracket and the foot. If the structural support could be guaranteed in all circumstances with shim sheets for example (Fig. 10 at right) the margin of safety of the connection with four M12 would yield MoS=1.1 and five MoS=1.3. However, keeping the shim sheet at their places without extra components turned out to be complicated. Therefore, the two-directional bolt connection should be considered in this joint, Figure

12 Figure 11. An improved bolt connection between the bracket and the foot. The stiffness of the two-directional bolt connection was calculated. The forces in the joint due to thermal expansions of HCAL were also included. The margin of safety was calculated taking into account only the axial support the give. Therefore, the results of calculations are conservative. The 1 and 2 (Fig. 11) carry the moment around the Z-axis and 3 and 4 around the X-axis. The margin of safety of the connection is 1.4. The size of the can be increased if the safety margin is found insufficient. 3.3 Foot to HCAL The foot is to be connected to HCAL with two patterns of four M14. The distance between the centres of the patterns is 404 mm. The equilibrium was calculated with M12. Then, the margin of safety yields 1.8. M14 would yield the margin of safety more than Conclusions According to the bracket FE-analysis, the maximum displacement of the aluminium bracket equals 1.3 mm taking into account the mass of the Tracker and the loads caused by the thermal expansion of HCAL. The stresses in the bracket do not exceed the yield strength of the aluminium alloys. The envelope model yielded larger displacement. The results do not fully correspond. This is due to different boundary conditions at the end of Tracker support tube. The boundary conditions used in the bracket model are more realistic. The forces in the bracket due to thermal expansion of HCAL are relatively high. The forces can be diminished by reducing the thickness of the bracket at its bend. The bracket was modelled as a single solid piece instead of two pieces. Therefore, the screw connections between the components were analysed separately. All proposed bolt connections fulfil the load requirements. 5. References 1. Airila M., Ekman K., et al. Koneenosien suunnittelu. WSOY, Porvoo, Finland, p. ISBN

13 Appendix 1: Calculations of bracket bolt connections A ) Bracket to tube The bracket was planned to be connected to the banana by one M12-bolt. The joint between the bracket and the banana eliminates the moment in the bolt connection. Therefore, the bolt must bear the shear force caused by the mass of the Tracker (1000 kg), shear forces caused by the thermal expansion of the HCAL radius (app N) and the axial forces caused by the mass of the Tracker in an inclined tunnel and the thermal expansion of HCAL in longitudunal direction. The total axial force is 3009 N. Dimensions of the M12-bolt from the bolt standard SFS 4497 d := 12mm P := 1.75mm d2 := d3 := mm 9.853mm Rmin := 0.219mm A3 := 76.2mm 2 As := 84.3mm 2 dk := 17.5mm Diameter of the bolt's head (measured) Elastic modulus of the bolt and the pieces to be connected. Es := Pa Elastic modulus of the bolt Ep := Pa Elastic modulus of the connected materials Description of the connection := 1 Number of in the connection li := := 81.5mm Length of the bolt in the connection from Autocad drawing

14 lk := 81.5mm Length of the connecion Da := 30mm Outer diameter of the connection Db := 13.5mm Diameter of the hole friction := 0.3 n := 0.5 Coefficient of friction Dimensionless factor of the connection (Figure ) Forces in the connection Fq := Fa := 10575N 3009N Shear force in the connection Axial force in the connection There isn't momentum in the bolt connection due joint Spring coefficient of the bolt ks := 0.4 d Es π d li Es π d ks kg = s 2 Spring coefficient of the pieces to be connected Dimensionless factor x: x := lk Da 0.2 ( ) 1 kp := dk 2 Db 2 x = dk ( Da dk ) ( x + 2) x Ep π lk 4 kp kg = s 2 Required normal force of the joint (tension force of the bolt) Fkr := Fq friction Fkr = N "Additional force ratio" phi := ks n ks + kp phi = 0.212

15 Pre-strain required: Fa Fv := Fkr + ( 1 phi) Fv = N Maximum force in the bolt connection Fa Fsmax := Fv + phi Fsmax = N Maximum force the bolt can bear (Yield strength of stainless steel bolt A4-70 is 480N/mm 2 ) N Fty12 := As 480 mm 2 Fty12 = N Margin of safety (MoS) of the bolt connection. MoS := Fty12 Fsmax MoS = Margin of Safety does not meet the common engineering requirements. Therefore, either the size of the bolt should be increased or more bolt should be used in the connection. The corresponding joint with M16 bolt Dimensions of the M16-bolt from the bolt standard SFS 4497 d := 16mm P := 2mm d2 := d3 := mm mm Rmin := 0.250mm A3 := 144mm 2 As := 157mm 2 dk := 21.5mm Diameter of the bolt's head

16 Spring coefficient of the bolt ks := 0.4 d Es π d li Es π d ks kg = s 2 Spring coefficient of the pieces to be connected Dimensionless factor x: x := lk Da 0.2 ( ) 1 kp := dk 2 Db 2 x = dk ( Da dk ) ( x + 2) x Ep π lk 4 kp kg = s 2 "Additional force ratio" phi := ks n ks + kp phi = Pre-strain required: Fa Fv := Fkr + ( 1 phi) Maximum force in the bolt connection Fv = N Fa Fsmax := Fv + phi Fsmax = N Maximum force the bolt can bear (Yield strength of stainless steel bolt A4-70 is 480N/mm 2 ) N Fty16 := As 480 mm 2 Fty16 = N Margin of safety (MoS) of the bolt connection. MoS := Fty16 Fsmax MoS = 1.97

17 B ) Bracket to foot The bracket was designed to attach to the foot with four parallel M12. The forces in the joint is calculated with moment equilibrium equation around z-axis that is the most crirical one. The shear force is transferred to the foot with the axial force of a bolt. All four are assumed to carry equal shear force. b1 ) no structural support Forces Fy := Fx := Fz := 9810N 3950N 3009N 4 as designed := 4 Moment around z-axis Fsx ( ( mm mm ) := 9810N 550mm Fsx := ( 9810N 550mm) 360mm Fsx = N Additional horizontal shear force due to Fx Fsxe := Fx Fsxe = 987.5N Vertical shear force Fsy Fsy := Fy Fsy = N Total shear force Fs Fstot := ( Fsx + Fsxe) 2 + Fsy 2 Fstot = N Axial force of the bolt Fa := Fz Fa = N

18 Margin of Safety of the connection Required normal force of the joint (tension force of the bolt) Fkr := Fstot friction Fkr = N Maximum force in the bolt connection Fmax := Fkr + Fz Fmax = N Mos := Fty12 Fmax Mos = The outermost bolt maintain their positions. The distance between the is 67.5 mm := 5 Moment around z-axis Fsx ( ( mm mm ) := 9810N 550mm Fsx := ( 9810N 550mm) 405mm Fsx = N Additional horizontal shear force due to Fx Fsxe := Fx Fsxe = 790N Vertical shear force Fsy Fsy := Fy Fsy = N Total shear force of the bolt, Fs Fstot := ( Fsx + Fsxe) 2 + Fsy 2 Fstot = N Axial force of the bolt

19 Fz Fa := Fa = N Margin of Safety of the connection Required normal force of the joint (tension force of the bolt) Fkr := Fstot friction Fkr = N Maximum force in the bolt connection Fmax := Fkr + Fz Fmax = N Mos := Fty12 Fmax Mos = b2 ) structural support Forces Fy := Fx := Fz := 9810N 3950N 3000N 4 as designed := 4 Moment equilibrium around z-axis Fsx ( ( )mm := 9810N 550mm N 150mm Fsx := ( 9810N 550mm N 150mm) ( )mm Fsx = N Additional horizontal shear force due to Fx Fsxe := Fx Fsxe = 987.5N Vertical shear force Fsy

20 Fy Fsy := Fsy = N Total shear force Fs of each bolt Fstot := ( Fsx + Fsxe) 2 + Fsy 2 Fstot = N Axial force of the bolt Fa := Fz Fa = 750 N Margin of Safety of the connection Required normal force of the joint to transfer shear force (tension force of the bolt) Fkr := Fstot friction Fkr = N Maximum axial force of the bolt Fmax := Fkr + Fz Fmax = N Mos := Fty12 Fmax Mos = := 5 Moment equilibrium around z-axis Fsx ( ( )mm := 9810N 550mm N 150mm Fsx := ( 9810N 550mm N 150mm) ( )mm Fsx = N Additional horizontal shear force due to Fx Fsxe := Fx Fsxe = 790N Vertical shear force Fsy

21 Fy Fsy := Fsy = N Total shear force Fs of each bolt Fstot := ( Fsx + Fsxe) 2 + Fsy 2 Fstot = N Axial force of the bolt Fa := Fz Fa = 600 N Margin of Safety of the connection Required normal force of the joint to transfer shear force (tension force of the bolt) Fkr := Fstot friction Fkr = N Maximum axial force of the bolt Fmax := Fkr + Fz Fmax = N Mos := Fty12 Fmax Mos = b3 ) 2 dimensional bolt connection Forces Fy := Fx := Fz := 9810N 3950N 3000N 2 in the 2 directions The joint is calculated to be carried only with the axial forces of the. := 4

22 Moment equilibrium (z-axis) Fa1 ( ) mm := Fy 550mm + Fx 150mm Fa1 := ( Fy 550mm + Fx 150mm) 300mm Fa1 = N Moment equilibium (x-axis) Fa2 ( ) mm := Fy 350mm + Fz 150mm Fa2 := ( Fy 350mm + Fz 150mm) 300mm Fa2 = N Margin of Safety of the connection Required normal force of the joint (tension force of the bolt) Shear force in the Y-direction Fsy := Fkr := Fy Fsy friction Fsy = N Fkr = N Maximum tension forces in the bolt connection The axis of the bolt in the direction of x-axis Fmax1 := Fkr + Fa1 Fmax1 = N Mos := Fty12 Fmax1 Mos = The axis of the bolt in the direction of x-axis Fmax2 := Fkr + Fa2 Fmax2 = N Mos := Fty12 Fmax2 Mos = B ) Foot to HCAL The foot is connected to HCAL with two patterns of four M14. The

23 distance between the centres of the patterns is 404 mm. The shear force is transferred to HCAL with the axial force of a bolt. All are assumed to carry equal shear force. The stiffness was calculated with M12. Forces Fy := Fx := Fz := 9810N 3950N 3000N Moment equilibrium (x-axis), shear forces Fsm ( ( mm ) := Fy 350mm 350mm Fsm := Fy ( 8 202mm) Fy Fsy := 8 Fz Fsz := 8 Fsm = N Fsy = N Fsz = 375N Total shear force Fs of each bolt Fstot := ( Fsm + Fsz) 2 + Fsy 2 Fstot = N Moment equilibrium (z-axis), axial forces Fas ( ( 4 15 mm mm ) := Fx 350mm + Fz 550mm Fas := ( Fx 350mm + Fz 550mm) 220mm Fas = N Margin of Safety of the connection Required normal force of the joint to transfer shear force (tension force of the bolt) Fkr := Fstot friction Fkr = N Maximum axial force of the bolt := +

24 Fmax := Fkr + Fas Fmax = N Fty12 Mos := Mos = Fmax

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