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13_041129_CLN_01.doc

***TO: DISTRIBUTION***

***FROM: C NEUMEYER***

***SUBJECT: LOADS ON FLAG BOX/HUB FRICTION INTERFACE***

Reference:

\[1\] "Update of NSTX Influence Matricies and EM Loads",
13_041123_CLN_01

\[2\] NSTX-CALC-13-020, "PF Coil Axial and Radial Force Calculation", R2

\[3\] NSTX-CALC-13-025, "TF Flag Box/Hub Connection", R1

This memo presents results from analyses performed to estimate loads on
the friction interface between the hub disks and flag boxes. In addition
to the FEA another analysis method is presented based on a
simplification which reduces the analysis to a statics problem. It is
assumed that the flag/box assembly is a simple rigid body, and that the
friction response of the interface can be modeled as point responses at
the radii of the box bolts. The latter assumption is based on the idea
that the pressure at the hub/box interface is concentrated in concentric
regions around the box bolts. Figure 1 shows a FEMLAB simulation of the
pressure distribution under a ½" stainless steel plate resulting from
the application of 5500lbf over the annular region of a 1" OD washer on
the other side of the plate. While this is a simplification of the
actual condition, it supports the idea that the box bolt reaction can be
modeled by a point response at the bolt radius.

![](media/image2.jpeg){width="4.76875in" height="3.296527777777778in"}

Figure 1 -- FEMLAB Model of Box Bolts at 5500lbf applied on 1" OD Washer

The static model is shown in figure 2. Moment on bundle is replaced by
F_bundle acting at bundle radius. F1, F2, F3 represent the box bolt
reactions, and F_lateral is the equivalent point force due to the
out-of-plane EM load. Reactions out through the flex links are ignored,
since that load path is relatively soft.

![](media/image3.jpeg){width="4.364583333333333in"
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Figure 2 -- Simplified Model of Load and Reactions on Flag Boxes

With the applied EM loads at bundle and on flag given, along with their
radius, and the radii of the three box bolts know, one can solve for F1,
F2, and F3 based on F=0 and M=0, assuming that F2 is linearly related to
F1 and F3 based on the spacing of the bolts.

$$F_{3} = \frac{\left\lbrack \frac{(F_{b} \ast r_{b} + F_{l} \ast r_{l})(r_{3} - r_{1})}{(r_{1}(r_{3} - r_{1}) + r_{2}(r_{3} - r_{2}))} - \frac{(F_{b} + F_{l})(r_{3} - r_{1})}{(2r_{3} - r_{1} - r_{2})} \right\rbrack}{\left\lbrack \frac{(r_{3} - 2 \ast r_{1} + r_{2})}{(2r_{3} - r_{1} - r_{2})} - \frac{(r_{3} \ast (r_{3} - r_{1}) + r_{2} \ast (r_{2} - r_{1}))}{(r_{1}(r_{3} - r_{1}) + r_{2}(r_{3} - r_{2}))} \right\rbrack}$$

$$F_{1} = \frac{\left\lbrack F_{3}(r_{3} - 2r_{1} + r_{2}) - (F_{b} + F_{l})(r_{3} - r_{1}) \right\rbrack}{(2r_{3} - r_{1} - r_{2})}$$

$$F_{2} = F_{1} - \frac{(F_{1} - F_{3})(r_{2} - r_{1})}{(r_{3} - r_{1})}$$

Table 1 gives values of the forces calculated using the procedure given
herein, and FEA results provided by I. Zatz, along with the friction
safety factors (out-of-plane loads only), for various operating states:
SOFT is the start of flat top, EOFT is the end of flat top, MAXMAX is
the case where all currents are simultaneously at their maximum values
in the same current direction, and OTHER is a solved for combination of
currents found to give the minimum friction safety factor while
respecting the constraint that the current ranges and polarities must be
consistent with realistic plasma operation. EM loads are calculated
using the influence matrices given in \[1\]. Bundle torque is assumed
taken 80% by the outer layer flags and 20% by the inners. Loads are
assumed to be equally shared between the friction interfaces at each
surface of the box.

TABLE 1 -- Loads and Friction Safety Factors

+---------------+-------+-----+-------+-----+--------+-------+-----+
|               | SOFT  | FEA | EOFT  | FEA | MAXMAX | OTHER |     |
|               |       |     |       |     |        |       |     |
|               |       | S   |       | E   |        |       |     |
|               |       | OFT |       | OFT |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| M_bundle      | 26317 |     | -     |     | -30025 | -     | ft- |
|               |       |     | 26330 |     |        | 27873 | lbf |
+---------------+-------+-----+-------+-----+--------+-------+-----+
|               | 3     |     | -3    |     | -      | -3    | in- |
|               | 15809 |     | 15955 |     | 360299 | 34481 | lbf |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| R_bundle      | 3.85  |     | 3.85  |     | 3.85   | 3.85  | in  |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| F_bundle      | 82071 |     | -     |     | -93633 | -     | lbf |
|               |       |     | 82109 |     |        | 86923 |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| #Inner Turns  | 12    |     | 12    |     | 12     | 12    |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| #Outer Turns  | 24    |     | 24    |     | 24     | 24    |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| f_Inner       | 0.20  |     | 0.20  |     | 0.20   | 0.20  |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| f_Outer       | 0.80  |     | 0.80  |     | 0.80   | 0.80  |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| F_Bund        | 2736  |     | -2737 |     | -3121  | -2897 | lbf |
| le_Outer_Flag |       |     |       |     |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| M_Later       | -2621 |     | 13775 |     | 65160  | 35549 | in- |
| al_Outer_Flag |       |     |       |     |        |       | lbf |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| F_Later       | -367  |     | 1141  |     | 5077   | 2765  | lbf |
| al_Outer_Flag |       |     |       |     |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| F_n           | 2368  |     | -1596 |     | 1956   | -132  | lbf |
| et_Outer_Flag |       |     |       |     |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| R_Lateral     | 7.14  |     | 12.07 |     | 12.83  | 12.86 | in  |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| R_bolt1       | 6.51  |     | 6.51  |     | 6.51   | 6.51  | in  |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| R_bolt2       | 9.14  |     | 9.14  |     | 9.14   | 9.14  | in  |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| R_bolt3       | 12.89 |     | 12.89 |     | 12.89  | 12.89 | in  |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| Numerator     | -826  | #   | 1278  | #   | 3239   | 2135  |     |
|               |       | DIV |       | DIV |        |       |     |
|               |       | /0! |       | /0! |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| Denominator   | -0.51 | #   | -0.51 | #   | -0.51  | -0.51 |     |
|               |       | DIV |       | DIV |        |       |     |
|               |       | /0! |       | /0! |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| F1            | -2927 | -1  | 3225  | 1   | 4395   | 3793  |     |
|               |       | 424 |       | 318 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| F2            | -1057 | -   | 869   | 218 | -21    | 513   |     |
|               |       | 154 |       |     |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| F3            | 1615  | 552 | -2497 | -   | -6330  | -4174 |     |
|               |       |     |       | 428 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| ∑F            | 0     | -1  | 0     | 1   | 0      | 0     |     |
|               |       | 026 |       | 108 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| ∑M            | 0     | 0   | 0     | 0   | 0      | 0     |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| COF           | 0.60  | 0   | 0.60  | 0   | 0.60   | 0.60  |     |
|               |       | .60 |       | .60 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| Bolt Force    | 5500  | 5   | 5500  | 5   | 5500   | 5500  | lbf |
|               |       | 500 |       | 500 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| Number of     | 2     | 2   | 2     | 2   | 2      | 2     |     |
| Surfaces      |       |     |       |     |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| Resisting     | 6600  | 6   | 6600  | 6   | 6600   | 6600  | lbf |
| Force/bolt    |       | 600 |       | 600 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| SF1           | 2.3   | 4.6 | 2.0   | 5.0 | 1.5    | 1.7   |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| SF2           | 6.2   | 4   | 7.6   | 3   | 311.3  | 12.9  |     |
|               |       | 2.9 |       | 0.3 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| SF3           | 4.1   | 1   | 2.6   | 1   | 1.0    | 1.6   |     |
|               |       | 2.0 |       | 5.4 |        |       |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+
| SF_MIN        | 2.3   | 4.6 | 2.0   | 5.0 | 1.0    | 1.6   |     |
+---------------+-------+-----+-------+-----+--------+-------+-----+

Load currents corresponding to the OTHER (practical worst case)
condition are given in Table 2.

TABLE 2 -- Current Combination for

Worst Case Box/Hub Friction Loading

  ---------------------------------------------------------- ------------
  Ioh                                                        24.0

  Ipf1a                                                      5.0

  Ipf1b                                                      0.0

  Ipf2                                                       0.0

  Ipf3                                                       20.0

  Ipf4                                                       0.0

  Ipf5                                                       20.0

  Itf                                                        71.2
  ---------------------------------------------------------- ------------

Comparison between the FEA for the SOFT and EOFT cases is given in Table
3.

TABLE 3 -- Comparison Between Bolt Load Calculations

  --------------------------------- ------------------ ------------------
                                    SOFT               EOFT

  F_1                               -1463              1612

  F_1_FEA                           -712               659

  Ratio_1                           49%                41%

  F_2                               -528               434

  F_2_FEA                           -77                109

  Ratio_2                           15%                25%

  F_3                               807                -1248

  F3_3_FEA                          276                -214

  Ratio_3                           34%                17%
  --------------------------------- ------------------ ------------------

The above suggests that the actual load may be as much as 49% of that
calculated using the simplified analysis. Taking credit for the FEA
result, and applying the realistic maximum condition, the friction
safety factors including the vector sum of in-plane and out-of-plane
loads are given in Table 4 (following the calculation technique used in
\[2\]).

TABLE 4 -- Box Bolt Safety Factor Calculations

  --------------------------------- ---------- --------- -------- --------
  COF for Coated Plates             0.600      0.600     0.600    

  At 6 kG                           6          6         6        kG

  In Plane EM Moment                70653      70653     70653    in-lbf

  \% In Plane Load to Hub/Box Shear 42%        42%       42%      

  Moment Arm to Interface           3.25       3.25      3.25     in

  In Plane Shear Load per Interface 4569       4569      4569     lbf

  In Plane Shear Load per Bolt      1523       1523      1523     

  Max Out of Plane EM Load per Bolt 2087       2087      2087     

  \% Out Of Plane Shear Load per    49%        49%       49%      
  Bolt                                                            

  Out of Plane Shear Load per Bolt  1023       1023      1023     

  Field Strength kG                 6.00       5.00      4.00     kG

  In Plane Load                     1523       1058      677      lbf

  Out Of Plane Load                 1023       852       682      lbf

  Resultant Load                    1834       1358      961      lbf

  Min Bolt Tension                  3900       3900      3900     lbf

  Avg. Bolt Tension                 5500       5500      5500     lbf

  Max. Bolt Tension                 6500       6500      6500     lbf

  Min Resistance to Load            2340       2340      2340     lbf

  Avg Resistance to Load            3300       3300      3300     lbf

  Max Resistance to Load            3900       3900      3900     lbf

  Min. FS Friction OOP              2.29       2.75      3.43     

  Avg. FS Friction OOP              3.23       3.87      4.84     

  Max. FS Friction OOP              3.81       4.58      5.72     

                                                                  

  Min. FS Friction IP               1.54       2.21      3.46     

  Avg. FS Friction IP               2.17       3.12      4.88     

  Max. FS Friction IP               2.56       3.69      5.76     

                                                                  

  Min. FS Friction Resultant        1.3        1.7       2.4      

  Avg. FS Friction Resultant        1.8        2.4       3.4      

  Max. FS Friction Resultant        2.1        2.9       4.1      
  --------------------------------- ---------- --------- -------- --------

*Summary*

The analysis predicts a safety factor of at least 1.3 at 6kG. However,
the considering the complexity of the structural system, and the
simplifications inherent not only in the statics analysis but also in
the FEA, we need to proceed cautiously. A technique to inspect for
relative displacement between box and hub needs to be developed.

L Dudek M Kalish M Ono J Schmidt A Von Halle M Williams

R Woolley IZatz