stiffness analysis of the tracker support bracket and...

October 25, 2000

Stiffness Analysis of the Tracker Support Bracket andIts Bolt Connections

Tommi VanhalaHelsinki Institute of Physics

1. INTRODUCTION ....................................................................................................................................................2

2. STIFFNESS ANALYSES.........................................................................................................................................2

2.1 ENVELOPE ..............................................................................................................................................................22.2 BRACKET................................................................................................................................................................5

3. BOLT CONNECTIONS ..........................................................................................................................................8

3.1 BRACKET TO TUBE................................................................................................................................................103.2 BRACKET TO FOOT................................................................................................................................................113.3 FOOT TO HCAL....................................................................................................................................................12

4. CONCLUSIONS.....................................................................................................................................................12

5. REFERENCES .......................................................................................................................................................12

APPENDIX 1. ...................................................................................................................................................................13

2

1. Introduction

The Tracker support tube is supported at its ends by four aluminium brackets that locate at thehorizontal symmetry plane. The outer end of the bracket is attached to HCAL. The brackets mustbear the load caused by the mass of the Tracker (estimated to be 4t) and the dimensional changes ofHCAL 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 wouldprobably like.

The support bracket is made of three entities, Figure 1. The entities: banana, bracket and the foot areconnected together with stainless A4-70 steel bolts. The bolt joint must provide stable connection inall circumstances. The bolt joints are needed, because the components are mounted to CMS indifferent 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. Theanalysis was done in two parts. The preliminary stiffness calculations were done with the bracketenvelope model. Later, the displacement was calculated with the more detailed bracket model. Thedifferent approach to defining the model was used. The geometry of the envelope model was createdwith 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 calculationsof 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 TL012287MQmade by Mr. Pierre-Benoit Brouze. The bracket is according to drawing an assembly of twoaluminium pieces. The connection between the bracket and the Tracker support tube was not yetdesigned. The bracket was modelled as one solid piece for the FE-analysis. The shape of theenvelope was simplified by neglecting small details such as curvatures, rounds etc.

3

The geometry of the bracket was modelled in Pro/Engineer 3D-modelling software, Figure 2. Thedimensions were given in millimetres. An IGES-file containing the envelope surfaces was exportedfrom 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 importedmodel. 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 typetolerates irregular shapes without as much loss of accuracy. SOLID95 elements have compatibledisplacement shapes and are well suited to model curved boundaries. The element is defined by 20nodes 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 themodel 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 SymbolYoung's modulus E 70000 [N/mm2]Poisson's ratio � 0.25Density � 2700*10-9 [kg/mm3]

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) wereapplied. All degrees of freedom (DOF) on the nodes at two attachment surfaces were fixed. The tubeend of the bracket was free. The force was equally divided between 10 nodes at the end of thebracket. 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. Thedisplacement is in millimetres. The nodal forces point to the negative Y-axis. The co-ordinatesystem of the bracket in the FE-analysis is illustrated in Figure 3. Please note that the system differsfrom the global co-ordinate system of the CMS Tracker.

Figure 4. The total displacement of the bracket envelope in millimetres.

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 dimensionsof the bracket. The geometry of the bracket model differs from the envelope model at the tube endof the bracket. The bracket is to be connected to the tube with three bolts via a stiff aluminiumbanana. The banana was not modelled, but only its reaction force with the bracket is defined. Thebracket displaces in the Y-direction due to the mass of the Tracker. HCAL and the Tracker supporttube are assumed to have infinite stiffness. Due to the symmetry of the Tracker, the tube end of thebracket cannot displace either in the X- or in the Z-direction. These boundary conditions did notexist in the envelope model.

The foot and the bracket were modelled again as a single solid piece. The geometry of the modelwas created by first defining keypoints, then connecting these by lines and finally creating areasfrom the lines. The element type used was SHELL63. The thickness of the bracket was defined to be55 mm at the foot, 45 mm in the inclined part and 35 mm at the end of the bracket. The mass of thebracket 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 appliedand the displacement at the tube end of the bracket in the X- and Z-directions were fixed. Thesupport 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. Thebracket should adapt dimensional changes of HCAL, because the Tracker support tube should not be

6

stretched or squeezed. HCAL is mostly made of bras, whose coefficient of the thermal expansion is20 �m/m�C. The temperature change of the HCAL was estimated to be �2.5�C, at maximum. The2.5�C temperature change would increase or decrease the total length of HCAL (6.090 m) by 304�m. Therefore, one half of HCAL expands 152 �m. Correspondingly, the temperature changementioned above increases the inner radius of HCAL (1.836 m) by 92 �m. The displacement of thetube end of the bracket with respect to the foot was defined with the displacement boundaryconditions. A 2.5�C thermal expansion creates 3.9 kN in the X- and -3.0 kN in the Z-directionsupport forces at the end of the bracket. The displacement of the bracket with HCAL thermalexpansion is presented in Figures 6 and 7.

Figure 6. Total displacement of the bracket. Gravity, Tracker mass and HCAL thermalexpansions included. Displacement is in mm.

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 bracketstructure are presented in Figure 8. The maximum stress (97 MPa) gives the safety factor to stressesmore than 2. In addition, the stresses in the final bracket are slightly smaller than the computedvalues, thanks to round corners.

Figure 8. Stresses in the bracket [MPa].

8

3. Bolt connections

The strength and the margin of safety (MoS) of the bolt connections were calculated. A bolt joint isnormally 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 isthe sufficient tension stiffness of the bolt. The coefficient of friction between the components to beconnected is estimated to be �=0.3.

Stainless steel bolts will be used to connect bracket components together. The characteristics of thebolts are presented in Table 2. The material properties of the bolt are presented in Table 3. Thecomponents that are squeezed together are made of aluminium whose Young's modulus isEp=70000 Mpa. The stress in the bolt should not exceed the yield strength value.

Table 2. Characteristics of the bolts.

M12 M14 M16d (mm) 12 14 16P (mm) 1.75 2.0 2.0d2 (mm) 10.863 12.701 14.701d3 (mm) 9.853 11.546 13.546Rmin (mm) 0.219 0.250 0.250A3 (mm2) 76.2 105 144As (mm2) 84.3 115 157

Table 3. Material properties of a stainless steel bolt.

Young's modulus, Es [MPa] 210000Yield strength �ty [MPa] 480

The formulas to dimension a bolt joint are given in [1]. To calculate bolt connection one must firstdescribe the joint. The required parameters are introduced in Table 4. The parameters of the boltconnections are defined in Chapters 3.1 - 3.3.

Table 4. Parameters of the bolt connection.

Variable Unit Descriptionli mm Length of the bolt in the connectionlk mm Height of connectionDa mm Diameter of connectionDb mm Diameter of hole for a bolt� - Coefficient of friction between the contact surfacesn - Dimensionless factor of the connection [1, Fig. 3.2.5-7]

First, the spring coefficient of the bolt and the spring coefficient of the elements to be connected arecalculated. The combined spring coefficient of the bolt and nut (ks) is calculated with the equation[1, 3.2.5-13].

9

ks0.8 d�

Es ��d2

4�

�

� �

0.5d

Es A3��

li

Es ��d2

4�

��

��

1�

��

(1)

The spring coefficient of the elements to be connected (kp) can be calculated with the equation [1,3.2.5-20].

kp dk2 Db2�� 1

2Da dk�( )� x 2�� x��

��

��

Ep ��

lk 4��

(2)

, where x is the dimensionless variable of the joint defined by the dimensions of the joint. The valuefor x is calculated with the equation 3 [1, 3.2.5-15]

x3 lk dk�

dk lk�( )2��

(3)

The required normal force of the joint i.e. tension force of the bolt (Fkr) is calculated with the well-known friction equation 4. If there is more than one bolt, the force is assumed to divide equally to allbolts i.e. the joint is symmetrical. Thus, two bolts in the joint divide the required tensions force ofthe bolt by two.

FkrFq

bolts friction��

(4)

,where Fkr is the minimum tension force of the bolt and Fq the shear force at the joint

The additional force ratio, �, for the definition of the maximum tension force of the bolt, is nowdefined [1, 3.2.5-32].

� nks

ks kp��

(5)

The required pre-strain of the bolt (Fv) is [1, 3.2.5-33]

Fv Fkr 1 �� Fa

bolts��

(6)

, where Fa is the exterior axial load of the joint. The maximum force that will stretch the bolt isfinally [1, 3.2.5-24]

Fsmax Fv �Fa

bolts��

(7)

10

The strength of the bolt i.e. the maximum force the bolt can bear (Fty) is calculated with theequation 8

Fty As �ty�� (8)

,where �ty is yield strength of the bolt. The margin of safety (MoS) can be calculated as a ratiobetween the maximum stretching force of the bolt and the strength of the bolt.

MoSFty

Fsmax��

(9)

The stiffness calculations of the bolt connections were done in Mathcad 2000 and they are allpresented in Appendix 1.

3.1 Bracket to tube

The bracket was planned to be attached to the banana with a single M12 bolt. The connection mustbear the quarter mass of the Tracker (=1000kg) and a 3.9-kN horizontal force due to thermalexpansion 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. Themargin was found too small, because it does not meet the earthquake safety requirements forexample. 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 onfriction, but the shape of the components must reinforce the joint. The feasible joint could be madeof 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 bolts. The margin of safety of the joint is MoS=3.2assuming that the load is divided equally between the bolts. The length of a thread in the insertshould be 16 mm [1,Table 3.2.4-1], at least. The sufficient thread guarantees that the thread does notrip of the bolt.

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3.2 Bracket to foot

The bracket was preliminarily designed to be attached to the foot with four parallel M12 bolts. Dueto the shape of a bracket, both shear and axial forces load the joint. The distance between the fourbolts and the Tracker support tube is 550 mm in the X-direction and 350 mm in the Z-direction. Themaximum shear force the M12 bolt is able to transfer with friction (12.14 kN) was calculated withequations 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 bolts, the outermost boltsremain their positions and the distance between the bolts was decreased. The results of calculationswith MoS of the connection are presented in Table 5. As one can notice, the joint cannot bear theload. 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) MoS4 15.0+1.0 2.5 16.2 0.8 0.745 13.3+0.8 2.0 14.3 0.6 0.84

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 bolts would yield MoS=1.1 andfive bolts MoS=1.3. However, keeping the shim sheet at their places without extra componentsturned out to be complicated. Therefore, the two-directional bolt connection should be considered inthis joint, Figure 11.

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 tothermal expansions of HCAL were also included. The margin of safety was calculated taking intoaccount only the axial support the bolts give. Therefore, the results of calculations are conservative.The bolts 1 and 2 (Fig. 11) carry the moment around the Z-axis and bolts 3 and 4 around the X-axis.The margin of safety of the connection is 1.4. The size of the bolts can be increased if the safetymargin is found insufficient.

3.3 Foot to HCAL

The foot is to be connected to HCAL with two patterns of four M14 bolts. The distance between thecentres of the patterns is 404 mm. The equilibrium was calculated with M12 bolts. Then, the marginof safety yields 1.8. M14 would yield the margin of safety more than 2.

4. Conclusions

According to the bracket FE-analysis, the maximum displacement of the aluminium bracket equals1.3 mm taking into account the mass of the Tracker and the loads caused by the thermal expansionof HCAL. The stresses in the bracket do not exceed the yield strength of the aluminium alloys. Theenvelope model yielded larger displacement. The results do not fully correspond. This is due todifferent boundary conditions at the end of Tracker support tube. The boundary conditions used inthe bracket model are more realistic. The forces in the bracket due to thermal expansion of HCALare 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 screwconnections between the components were analysed separately. All proposed bolt connections fulfilthe load requirements.

5. References

1. Airila M., Ekman K., et al. Koneenosien suunnittelu. WSOY, Porvoo, Finland, 1995. 796 p.ISBN 951-0-20172-3

:=

Length of the bolt in the connection from Autocad drawingli 81.5mm:=

Number of bolts in the connectionbolts 1:=

Description of the connection

Elastic modulus of the connected materialsEp 70000 106⋅ Pa:=

Elastic modulus of the boltEs 210000 106⋅ Pa:=

Elastic modulus of the bolt and the pieces to be connected.

Diameter of the bolt's head (measured) dk 17.5mm:=

As 84.3mm2:=

A3 76.2mm2:=

Rmin 0.219mm:=

d3 9.853mm:=d2 10.863mm:=

P 1.75mm:=d 12mm:=Dimensions of the M12-bolt from the bolt standard SFS 4497

The bracket was planned to be connected to the banana by one M12-bolt. The joint betweenthe 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. 3950 N) and theaxial 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.

A ) Bracket to tube

Appendix 1: Calculations of bracket bolt connections

phi 0.212=phi nks

ks kp+⋅:=

"Additional force ratio"

Fkr 3.525 104× N=Fkr

Fq

bolts friction⋅:=

Required normal force of the joint (tension force of the bolt)

kp 3.739 108×

kg

s2

=kp dk2

Db2−( ) 1

2dk⋅ Da dk−( )⋅ x 2+( )⋅ x⋅+

Ep π⋅lk 4⋅

⋅:=

x 1.221=xlk

Da

0.2

:=

Dimensionless factor x:

Spring coefficient of the pieces to be connected

ks 2.752 108×

kg

s2

=ks0.4 d⋅

Es π⋅d

2

4⋅

li

Es π⋅d

2

4⋅

+

1−:=

Spring coefficient of the bolt

There isn't momentum in the bolt connection due joint

Axial force in the connectionFa 3009N:=

Shear force in the connectionFq 10575N:=

Forces in the connection

Dimensionless factor of the connection (Figure 3.2.5-7)n 0.5:=

Coefficient of frictionfriction 0.3:=

Diameter of the holeDb 13.5mm:=

Outer diameter of the connectionDa 30mm:=

Length of the connecionlk 81.5mm:=

Diameter of the bolt's head dk 21.5mm:=

As 157mm2:=

A3 144mm2:=

Rmin 0.250mm:=

d3 13.546mm:=

d2 14.701mm:=

P 2mm:=d 16mm:=

Dimensions of the M16-bolt from the bolt standard SFS 4497

The corresponding joint with M16 bolt

Margin of Safety does not meet the common engineering requirements. Therefore, eitherthe size of the bolt should be increased or more bolt should be used in the connection.

MoS 1.058=MoSFty12

Fsmax:=

Margin of safety (MoS) of the bolt connection.

Fty12 4.046 104× N=Fty12 As 480⋅

N

mm2

:=

Maximum force the bolt can bear (Yield strength of stainless steel bolt A4-70 is 480N/mm2)

Fsmax 3.826 104× N=Fsmax Fv phi

Fa

bolts⋅+:=

Maximum force in the bolt connection

Fv 3.762 104× N=Fv Fkr 1 phi−( )

Fa

bolts⋅+:=

Pre-strain required:

MoS 1.97=MoSFty16

Fsmax:=

Margin of safety (MoS) of the bolt connection.

Fty16 7.536 104× N=Fty16 As 480⋅

N

mm2

:=

Maximum force the bolt can bear (Yield strength of stainless steel

bolt A4-70 is 480N/mm2)

Fsmax 3.826 104× N=Fsmax Fv phi

Fa

bolts⋅+:=


Fv 3.747 104× N=Fv Fkr 1 phi−( )

Fa

bolts⋅+:=

Pre-strain required:

phi 0.263=phi nks

ks kp+⋅:=

"Additional force ratio"

kp 4.314 108×

kg

s2

=kp dk2

Db2−( ) 1

2dk⋅ Da dk−( )⋅ x 2+( )⋅ x⋅+

Ep π⋅lk 4⋅

⋅:=

x 1.221=xlk

Da

0.2

:=

Dimensionless factor x:

Spring coefficient of the pieces to be connected

ks 4.804 108×

kg

s2

=ks0.4 d⋅

Es π⋅d

2

4⋅

li

Es π⋅d

2

4⋅

+

1−:=

Spring coefficient of the bolt

Fa 752.25 N=FaFz

:=

Axial force of the bolt

Fstot 1.616 104× N=Fstot Fsx Fsxe+( )

2Fsy

2+:=

Total shear force Fs

Fsy 2.453 103× N=Fsy

Fy

bolts:=

Vertical shear force Fsy

Fsxe 987.5N=FsxeFx

bolts:=

Additional horizontal shear force due to Fx

Fsx 1.499 104× N=Fsx

9810N 550⋅ mm( )

360mm:=

Fsx 2 135⋅ mm 2 45⋅ mm+( )⋅ 9810N 550⋅ mm:=Fsx 2 135⋅ mm 2 45⋅ mm+( )⋅

Moment around z-axis

bolts 4:=

4 bolts as designed

Fz 3009N:=

Fx 3950N:=

Fy 9810N:=

Forces

b1 ) no structural support

The bracket was designed to attach to the foot with four parallel M12 bolts. 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 bolts are assumed to carry equal shear force.

B ) Bracket to foot



2Fsy

2+:=

Total shear force of the bolt, Fs

Fsy 1.962 103× N=Fsy

Fy

bolts:=


Fsxe 790N=FsxeFx

bolts:=


Fsx 1.332 104× N=Fsx

9810N 550⋅ mm( )

405mm:=

Fsx 2 135⋅ mm 2 67.5⋅ mm+( )⋅ 9810N 550⋅ mm:=Fsx 2 135⋅ mm 2 67.5⋅ mm+( )⋅

Moment around z-axis

bolts 5:=

5 bolts. The outermost bolt maintain their positions. The distance between the bolts is 67.5 mm

Mos 0.741=MosFty12

Fmax:=

Fmax 5.463 104× N=Fmax Fkr

Fz

bolts+:=


Fkr 5.387 104× N=Fkr

Fstot

friction:=


Margin of Safety of the connection

bolts


Fsxe 987.5N=FsxeFx

bolts:=


Fsx 9.98 103× N=Fsx

9810N 550⋅ mm 3950N 150⋅ mm+( )

15 105+ 195+ 285+( )mm:=

Fsx 15 105+ 195+ 285+( )mm⋅ 9810N 550⋅ mm 3950N 150⋅ mm+:=Fsx 15 105+ 195+ 285+( )mm⋅

Moment equilibrium around z-axis

bolts 4:=

4 bolts as designed

Fz 3000N:=

Fx 3950N:=

Fy 9810N:=

Forces

b2 ) structural support

Mos 0.841=MosFty12

Fmax:=


Fz

bolts+:=


Fkr 4.749 104× N=Fkr

Fstot

friction:=



Fa 601.8 N=FaFz

bolts:=


Fsxe 790N=FsxeFx

bolts:=


Fsx 7.984 103× N=Fsx

9810N 550⋅ mm 3950N 150⋅ mm+( )

15 82.5+ 150+ 217.5+ 285+( )mm:=

Fsx 15 82.5+ 150+ 217.5+ 285+( )mm⋅ 9810N 550⋅ mm 3950N 150⋅ mm+:=Fsx 15 82.5+ 150+ 217.5+ 285+( )mm⋅

Moment equilibrium around z-axis

bolts 5:=

5 bolts

Mos 1.059=MosFty12

Fmax:=


Fz

bolts+:=

Maximum axial force of the bolt

Fkr 3.746 104× N=Fkr

Fstot

friction:=

Required normal force of the joint to transfer shear force (tension force of the bolt)


Fa 750 N=FaFz

bolts:=



2Fsy

2+:=

Total shear force Fs of each bolt

Fsy 2.453 103× N=Fsy

Fy

bolts:=

bolts 4:=

The joint is calculated to be carried only with the axial forces of the bolts.

2 bolts in the 2 directions

Fz 3000N:=

Fx 3950N:=

Fy 9810N:=

Forces

b3 ) 2 dimensional bolt connection

Mos 1.324=MosFty12

Fmax:=


Fz

bolts+:=


Fkr 2.997 104× N=Fkr

Fstot

friction:=



Fa 600 N=FaFz

bolts:=



2Fsy

2+:=


Fsy 1.962 103× N=Fsy

Fy

bolts:=

The foot is connected to HCAL with two patterns of four M14 bolts. The

B ) Foot to HCAL

Mos 1.916=MosFty12

Fmax2:=

Fmax2 2.112 104× N=Fmax2 Fkr Fa2+:=

The axis of the bolt in the direction of x-axis

Mos 1.438=MosFty12

Fmax1:=

Fmax1 2.813 104× N=Fmax1 Fkr Fa1+:=

The axis of the bolt in the direction of x-axis

Maximum tension forces in the bolt connection

Fkr 8.175 103× N=Fkr

Fsy

friction:=

Fsy 2.453 103× N=Fsy

Fy

bolts:=

Shear force in the Y-direction



Fa2 1.294 104× N=Fa2

Fy 350⋅ mm Fz 150⋅ mm+( )

300mm:=

Fa2 25 275+( )⋅ mm Fy 350⋅ mm Fz 150⋅ mm+:=Fa2 25 275+( )⋅ mm

Moment equilibium (x-axis)

Fa1 1.996 104× N=Fa1

Fy 550⋅ mm Fx 150⋅ mm+( )

300mm:=

Fa1 75 225+( )⋅ mm Fy 550⋅ mm Fx 150⋅ mm+:=Fa1 75 225+( )⋅ mm

Moment equilibrium (z-axis)

+:=


Fkr 9.281 103× N=Fkr

Fstot

friction:=



Fas 1.378 104× N=Fas

Fx 350⋅ mm Fz 550⋅ mm+( )

220mm:=

Fas 4 15⋅ mm 4 40⋅ mm+( )⋅ Fx 350⋅ mm Fz 550⋅ mm+:=Fas 4 15⋅ mm 4 40⋅ mm+( )⋅

Moment equilibrium (z-axis), axial forces

Fstot 2.784 103× N=Fstot Fsm Fsz+( )

2Fsy

2+:=


Fsz 375N=FszFz

8:=

Fsy 1.226 103× N=Fsy

Fy

8:=

Fsm 2.125 103× N=Fsm Fy

350mm

8 202⋅ mm( )⋅:=

Fsm 8 202⋅ mm( )⋅ Fy 350⋅ mm:=Fsm 8 202⋅ mm( )⋅

Moment equilibrium (x-axis), shear forces

Fz 3000N:=

Fx 3950N:=

Fy 9810N:=

Forces

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 bolts are assumed to carry equal shear force. The stiffness was calculated with M12 bolts.

Fmax Fkr Fas+:= Fmax 2.306 104× N=

MosFty12

Fmax:= Mos 1.754=

stiffness analysis of the tracker support bracket and...

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