Support Tube FEA Modeling Summary

1.0 Introduction

This document describes the initial vibration results for FEA analysis
performed on the Pixel Support Tube (PST) composite installation
structure. The structure is initially analyzed without additional mass
(i.e. Pixel and services structures) as these are thought to be
primarily mechanically independent (due to fixation schemes). Later, the
models are examined with a distributed service mass incorporated into
the shell. The PST is modeled with and without several mechanical
structures that are attached to it, in order to quantify the stiffening
effects of these components. Different material properties are also
investigated. For instance, a fiberglass forward tube section is
investigated in response to concerns about loads the PST could apply to
surrounding detectors (namely the SCT). In all analyses, the four lowest
modes were computed using ANSYS FEA software.

#  Geometry

The PST consists of three individual segments that are bolted together
through stiffening flanges: 1 barrel, and 2 forwards. For modeling
purposes, the flange connections are assumed to be perfect, and no bolt
holes are included. All bonded joints (including flange to tube
connections) are modeled as perfectly welded interfaces, with no glue
properties or elements. A schematic of the PST model is shown in Figure
1.

![](media/image1.png){width="5.586111111111111in"
height="3.4402777777777778in"}

Figure 1. Important features in the PST FEA model (longitudinal ribs
have not been shown).

#  Constraints

The PST is modeled with different constraint conditions in order to
evaluate the need for specific types of fixation to the surrounding
detectors (outside of the pixel detector). The baseline fixation scheme
is shown in Figure 2, along with the support conditions that were
examined during analysis. In particular, the effect of end-fixation at
the forward tubes (on extreme sides A and C) is analyzed by removing the
vertical freedom on both sides, and the X direction freedom on one side
of the PST. The effect of fixing across the diameter of the barrel tube
is also examined. Also, it should be noted that all constraints are
applied to small areas or lines on the PST flanges. These constraints do
not allow free rotation, and thus cannot be considered "simple" in the
classical beam sense. This support condition more closely models the
proposed mounting flexures than constraints that allow free rotation.

![](media/image2.png){width="6.034027777777778in"
height="3.6277777777777778in"}

Figure 2. PST baseline support conditions showing constraints that were
varied in the FEA model (in red).

# 4.0 Model Properties

The PST model consists of either 2 or 3 types of materials. The barrel
shell, or tube, structure is assumed to be constructed of high modulus
(HM) graphite fiber (E=420 GPa) in a cyanate ester resin matrix, with 8
plies arranged in a quasi-isotropic manner (\[0/90/+45/-45\]~S~). This
layup achieves an isotropic modulus of approximately 98 GPa. The forward
shells are composed of either HM graphite or quartz glass fiber; the
later gives a quasi-isotropic modulus of 21 GPa given a fiber modulus of
91 GPa. Stiffeners and ribs in the forward sections are assumed to be HM
graphite in all cases. The flange is first investigated as a molded and
machined PEEK (polyethyletherketone) structure, and then later as a
carbon structure comprised of quasi-isotropic ultra high modulus (UHM)
graphite (E=560 GPa) in a cyanate ester matrix, resulting in a bulk
modulus of 126 GPa. In all cases, materials are modeled as either shells
or solids, with isotropic properties and linear behavior. For the cases
where service mass is included in the analysis, the material densities
of the forward shells are increased in order to account for the added
mass. A summary of all model material properties is given in Table 1.

Table 1. FEA Material Properties.

+-------------+-------------+-------------+-------------+-------------+
| **Material  | **Material  | **Fiber     | **Bulk      | **Density** |
| Model**     | Location**  | Modulus     | Modulus     |             |
|             |             | (GPa)**     | (GPa)**     | **          |
|             |             |             |             | (kg/m^3^)** |
+-------------+-------------+-------------+-------------+-------------+
| High Mod.   | Shell, Hat  | 420         | 98          | 1650        |
| Carbon      | Stiffeners  |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| Ultra High  | Flanges     | 560         | 126         | 1650        |
| Mod. Carbon |             |             |             |             |
|             | (Option 1)  |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| PEEK        | Flanges     | N/A         | 3.5         | 1500        |
| (graphite   |             |             |             |             |
| filled)     | (Option 2)  |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| Glass       | Forward     | 91          | 21          | 1800        |
| (Quartz)    | Shells      |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| HM Carbon   | Forward     | 420         | 98          | 12300       |
| (incl.      | Shells      |             |             |             |
| service     |             |             |             |             |
| mass)       |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| Glass       | Forward     | 91          | 21          | 12450       |
| (incl.      | Shells      |             |             |             |
| service     |             |             |             |             |
| mass)       |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+

# 

**5.0 Analysis Cases**

The primary free variables in this analysis study were as follows:

-   Flange Material (PEEK or Carbon Fiber)

-   Longitudinal Ribs

-   Hoop Stiffeners (in Forwards)

-   End Plug (PP1 panel)

-   Flange Shape

-   Barrel Constraints (ties across SCT diameter)

-   Service Mass (incorporated in forward shell material)

-   Fiberglass Forward Shells

-   Forward End Constraints\*

\*The issue of using constraints at the forward ends of the PST reflects
a fundamental change in support scheme, so *most* model cases were run
with both fixed and free forward ends.

6.  **Results**

The results are presented in a stepwise fashion, illustrating the
progression of the PST design. Each result section presents the
inclusion of one of the above design elements, showing why it was either
retained or rejected in subsequent iterations.

**6.1 Case 1 - Flange Material (PEEK or Carbon Fiber)**

The initial baseline PST model included PEEK flanges (for ease of
manufacturing) and longitudinal ribs along the entire structure (to
combat the anticipated cantilever bending modes) located at the tops and
bottoms of the tubes (see Figure 1). The lowest modes all turned out to
be shell modes, however, emphasizing that the flange stiffness was much
more important than previously appreciated (even when the forward ends
of the PST are not constrained). Changing the flange material to
isotropic carbon fiber (without changing the shape, which is somewhat
non-physical, yet a good comparison) improved the vibrational stability
substantially, as shown in Results Table 1. Note that the forward ends,
when "fixed", are fixed only in radial dimensions (XY).

**Results Table 1. Effect of Carbon vs. PEEK Flanges.**

(Shell modes in italics, cantilever modes in bold)

+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ## Model      | #   |     |     |     | ##  |     |     |     |
|               | # F |     |     |     |  Fi |     |     |     |
|               | ree |     |     |     | xed |     |     |     |
|               |  Fo |     |     |     |  Fo |     |     |     |
|               | rwa |     |     |     | rwa |     |     |     |
|               | rds |     |     |     | rds |     |     |     |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ##            | **M | **M | **M | **M | **M | **M | **M | **M |
| # Description | ode | ode | ode | ode | ode | ode | ode | ode |
|               | 1** | 2** | 3** | 4** | 1** | 2** | 3** | 4** |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| PEEK Flanges  | *8  | *11 | *17 | *   | *12 | *13 | *   | *42 |
|               | .9* | .9* | .9* | 26* | .9* | .7* | 18* | .1* |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| Carbon        | *24 | *41 | *61 | *65 | *54 | *55 | *69 | *72 |
| Flanges       | .5* | .2* | .5* | .6* | .7* | .6* | .9* | .8* |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+

Since the vibrational improvement in going to carbon flanges is so
noticeable (at least a factor of 3, depending on forward fixation) it
was decided that the PST required this design element, and all further
analyses assume carbon flanges.

**6.2 Case 2 - Longitudinal Ribs**

The indication that shell stiffness was absolutely dominating PST
vibrational behavior led to immediate questioning of whether
longitudinal ribs were necessary. The ribs increase the overall PST
bending inertia by a factor of 2, but also increase the shell mass by
50%, see Table 2.

**Table 2. Longitudinal Rib Comparison for PST.**

  ------------------------------- ------------------ ---------------------
  **PST Rib Comparisons**         **Area (mm\^2)**   **Inertia (mm\^4)**

  PST Tube                        726                1.93E+07

  PST Ribs (6)                    360                1.92E+07

  Tube + Ribs                     1086               3.85E+07
  ------------------------------- ------------------ ---------------------

Analysis showed that, as suspected, the longitudinal ribs did almost
nothing to raise PST frequencies. In fact, removing the ribs *raised*
the lowest frequencies marginally, see Results Table 2. Although bending
modes were observed in the higher orders for the free-ended model, the
lowest modes remained shell modes, and so it was decided to remove ribs
from the PST design.

**Results Table 2. Effect of Longitudinal Ribs.**

(Shell modes in italics, cantilever modes in bold)

+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ## Model      | #   |     |     |     | ##  |     |     |     |
|               | # F |     |     |     |  Fi |     |     |     |
|               | ree |     |     |     | xed |     |     |     |
|               |  Fo |     |     |     |  Fo |     |     |     |
|               | rwa |     |     |     | rwa |     |     |     |
|               | rds |     |     |     | rds |     |     |     |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ##            | **M | **M | **M | **M | **M | **M | **M | **M |
| # Description | ode | ode | ode | ode | ode | ode | ode | ode |
|               | 1** | 2** | 3** | 4** | 1** | 2** | 3** | 4** |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| With Ribs     | *24 | *41 | *61 | *65 | *54 | *55 | *69 | *72 |
|               | .5* | .2* | .5* | .6* | .7* | .6* | .9* | .8* |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+
| Without Ribs  | *25 | *   | **  | **  | *58 | *59 | *67 | *68 |
|               | .8* | 44* | 55. | 62. | .5* | .2* | .9* | .5* |
|               |     |     | 9** | 4** |     |     |     |     |
+---------------+-----+-----+-----+-----+-----+-----+-----+-----+

**6.3 Case 3 - Hoop Stiffeners and End Plugs (PP1)**

In addressing the problem of PST shell stiffness, it was obvious that
increased sections in the tube were important, particularly along the
forwards. The option of a cored structure for the PST shell was
considered, but it would more than double mass and cost. The simpler
approach was to incorporate circumferential ribs, or "hoops", located at
the midpoints of the forwards. In addition, it was realized that the PP1
panels (otherwise known as endplugs) would also add radial stiffness to
the PST, and thus these were included in the model (whereas they had
been omitted before). This load case was only examined for fixed forward
ends, however, so only these results are shown in Results Table 3 and
Figure 3.

**Results Table 3. Effect of Hoops and End Plugs.**

(Shell modes in italics, cantilever modes in bold)

+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ## Model       | #   |     |     |     | ##  |     |     |     |
|                | # F |     |     |     |  Fi |     |     |     |
|                | ree |     |     |     | xed |     |     |     |
|                |  Fo |     |     |     |  Fo |     |     |     |
|                | rwa |     |     |     | rwa |     |     |     |
|                | rds |     |     |     | rds |     |     |     |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| #              | **M | **M | **M | **M | **M | **M | **M | **M |
| ## Description | ode | ode | ode | ode | ode | ode | ode | ode |
|                | 1** | 2** | 3** | 4** | 1** | 2** | 3** | 4** |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| Without        | *25 | *   | **  | **  | *58 | *59 | *67 | *68 |
|                | .8* | 44* | 55. | 62. | .5* | .2* | .9* | .5* |
|                |     |     | 9** | 4** |     |     |     |     |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| With Hoops and | N/A | N/A | N/A | N/A | *   | *   | *   | *   |
| End Plugs      |     |     |     |     | 64* | 114 | 116 | 117 |
|                |     |     |     |     |     | .6* | .5* | .9* |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+

![](media/image3.png){width="2.7069444444444444in"
height="2.6770833333333335in"}

**Results Figure 3. Effect of Hoops and End Plugs.**

The most notable aspect of the results from this load case is that the
fundamental frequency is not increased very substantially (only about
10%). However, looking at the higher order modes (HOM's) it can be seen
that they are all increased by almost a factor of 2, to well above 100
Hz (the nominal design criteria). Additionally, it is evident that the
lowest mode is caused by a diaphragm type motion of the entire tube,
allowed by the lack of constraints across the PST diameter at the SCT
barrel. Constraining the PST across the diameter in addition to
increasing the barrel flange thicknesses would raise the fundamental
frequency, while causing no decrease in HOM's, thereby reaching a tube
with fundamental frequency nearer to, or above, 100 Hz.

**6.4 Case 4 -- Flange Shape**

The initial flange shape chosen for the PST was a simple ribbed annulus,
with the goal that it would be machined from a large block of PEEK. Even
after rejecting PEEK as the potential flange material, the shape was
retained in order to easily judge the effect of changing material
properties alone. Since that time, the requirements for the flange have
changed (given mounting and gas sealing issues) and a fabrication scheme
has been developed. The current flange design utilizes repeated carbon
fiber building blocks that are integrated into a bonded assembly, as
shown in Figure 3.

![](media/image4.png){width="3.3868055555555556in"
height="2.454861111111111in"}

**Figure 3. Proposed Carbon Fiber Flange Design.**

**Results Table 4. Effect of Flange Shape.**

(Shell modes in italics, cantilever modes in bold)

+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ## Model       | #   |     |     |     | ##  |     |     |     |
|                | # F |     |     |     |  Fi |     |     |     |
|                | ree |     |     |     | xed |     |     |     |
|                |  Fo |     |     |     |  Fo |     |     |     |
|                | rwa |     |     |     | rwa |     |     |     |
|                | rds |     |     |     | rds |     |     |     |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| #              | **M | **M | **M | **M | **M | **M | **M | **M |
| ## Description | ode | ode | ode | ode | ode | ode | ode | ode |
|                | 1** | 2** | 3** | 4** | 1** | 2** | 3** | 4** |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| Initial Flange | ##  | N/A | N/A | N/A | *   | *   | *   | *   |
| Shape          | ##  |     |     |     | 64* | 114 | 116 | 117 |
|                | N/A |     |     |     |     | .6* | .5* | .9* |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| Carbon Flange  | **  | **  | *55 | *62 | *88 | *   | *1  | *   |
| Concept        | 31. | 48. | .3* | .2* | .7* | 118 | 19* | 123 |
|                | 4** | 4** |     |     |     | .6* |     | .6* |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+

The assembled shape of this new flange design was modeled as a solid
with isotropic properties, which approximates the use of quasi-isotropic
laminates in the real assembly. This design has higher stiffness than
the original shape, and this stiffness can be seen in the vibrational
results (Results Figure 4) as the fundamental frequency is increased by
38%. As in load case 3, the free-ended version of the model with
original flange shape was not run. Also, the same diaphragm type mode
can be seen; although its frequency is higher, it is still not above the
desired 100 Hz threshold.

![](media/image5.png){width="4.127083333333333in"
height="2.5166666666666666in"}

**Results Figure 4. Effect of Flange Shape.**

**6.5 Case 5 -- Constraining Across the Barrel (SCT Diameter)**

The shape of the lowest mode from the previous analysis leads directly
to the hypothesis that constraining the PST across the barrel of the SCT
would eliminate the diaphragm type motion observed. Analysis results,
Results Table 5, show that this is indeed the case, raising the
fundamental frequency to well above 100 Hz, and moving all modes to high
frequency shell motion in the forwards, Results Figure 5.

**Results Table 5. Effect of Diametral Constraints at SCT Barrel.**

(Shell modes in italics, cantilever modes in bold)

+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ## Model       | #   |     |     |     | ##  |     |     |     |
|                | # F |     |     |     |  Fi |     |     |     |
|                | ree |     |     |     | xed |     |     |     |
|                |  Fo |     |     |     |  Fo |     |     |     |
|                | rwa |     |     |     | rwa |     |     |     |
|                | rds |     |     |     | rds |     |     |     |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| #              | **M | **M | **M | **M | **M | **M | **M | **M |
| ## Description | ode | ode | ode | ode | ode | ode | ode | ode |
|                | 1** | 2** | 3** | 4** | 1** | 2** | 3** | 4** |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| No Barrel      | 3   | 4   | 5   | 6   | *88 | *   | *1  | *   |
| Constraints    | 1.4 | 8.4 | 5.3 | 2.2 | .7* | 118 | 19* | 123 |
|                |     |     |     |     |     | .6* |     | .6* |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| Barrel Constr. | 5   | 5   | 5   | 6   | *   | *1  | *   | *   |
| Across SCT     | 1.3 | 3.8 | 5.3 | 2.2 | 118 | 19* | 123 | 123 |
|                |     |     |     |     | .8* |     | .6* | .7* |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+

Little mention has been made thus far about the effect of forward end
freedom, although results are tabulated for most load cases. In general,
it has always been perceived that allowing this degree of freedom is
unwise. From the results shown thus far, it can be seen that freeing the
forward ends lowers frequency by factors of 2 to 3. Introducing service
masses into the forwards will only exacerbate this difference, and is
thus thought to be unacceptable. Analysis results for the free-ended
analysis cases are therefore presented primarily for comparison
purposes.

![](media/image6.png){width="4.246527777777778in"
height="2.779861111111111in"}

**Results Figure 5. Effect of Diametral Constraints at SCT Barrel.**

**6.6 Case 6 -- Effect of Service Mass in the Forwards**

Once the PST had been nominally optimized to fundamental frequencies
above 100 Hz, it was decided to incorporate service masses in the model.
Originally, it had been desired to model the entire service structure in
connection with the PST itself. However, this model is extremely large,
and the service mass structure has been constantly changing. For these
reasons, the service masses were incorporated into the shell material
densities (given in Table 1). Since this approximation adds mass, but no
stiffness, it yields the most conservative results. The service masses
are initially modeled at 13 kg per side (approximately the mass of
services on the side *opposite* the b-layer). Since this is *not*
conservative, models will soon be run with a higher, more accurate
service mass of 17 kg per side. It should be noted, however, that since
the service structure is currently under development, a final accounting
of all service masses will need to be made before an accurate assessment
of the effect on PST stiffness can be judged.

**Results Table 6. Effect of Service Mass in the Forwards.**

(Shell modes in italics, cantilever modes in bold)

+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| ## Model       | #   |     |     |     | ##  |     |     |     |
|                | # F |     |     |     |  Fi |     |     |     |
|                | ree |     |     |     | xed |     |     |     |
|                |  Fo |     |     |     |  Fo |     |     |     |
|                | rwa |     |     |     | rwa |     |     |     |
|                | rds |     |     |     | rds |     |     |     |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| #              | **M | **M | **M | **M | **M | **M | **M | **M |
| ## Description | ode | ode | ode | ode | ode | ode | ode | ode |
|                | 1** | 2** | 3** | 4** | 1** | 2** | 3** | 4** |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| No Service     | 5   | 5   | 5   | 6   | *   | *1  | *   | *   |
| Mass           | 1.3 | 3.8 | 5.3 | 2.2 | 118 | 19* | 123 | 123 |
|                |     |     |     |     | .8* |     | .6* | .7* |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+
| Service Mass   | **  | **  | **  | **  | *53 | *53 | *54 | *54 |
| (13 kg)        | 26. | 27. | 28. | 32. | .1* | .2* | .2* | .2* |
|                | 4** | 7** | 6** | 3** |     |     |     |     |
+----------------+-----+-----+-----+-----+-----+-----+-----+-----+

Initial results in Results Table 6 show that the addition of service
masses decreases the fundamental frequencies across the board by a
factor of almost 2 or more. As might be expected, all modes for the
free-ended case are beam modes (cantilever bending) while all modes for
the fixed case are of a shell nature. A plot of the deformed shape for
mode 1, fixed ends with service masses, yields a mode pattern almost
identical to that for the model without service mass (Results Figure 6).

![](media/image7.png){width="4.2027777777777775in"
height="2.701388888888889in"}

**Results Figure 6. Effect of Service Mass in the Forwards.**

**6.7 Case 7 -- Fiberglass Forward Shells**

It has recently come under attention that the PST structure, due to its
inherent bending stiffness, might impose unwanted loads on the SCT
during the unlikely displacement of the cryostat's end from its barrel
section. This motion would cause the end of the PST forward to move
relative to the SCT, imposing bending loads on the SCT interlinks. There
has been a proposal that this displacement may be as large as 2 mm in X
or Y (see model coordinate system). In order to mitigate the potential
loads that this fault condition would apply, the possibility of
fabricating the forward PST shells from fiberglass was investigated.
Using a layup similar to that proposed for carbon fiber, a glass
laminate would achieve a modulus approximately 20% that of carbon. The
fiberglass PST is analyzed to determine vibrational behavior, support
reactions under vertical and horizontal bending, and displacements due
to gravity sag.

**6.7.1 Vibrational Behavior**

Initially, the carbon model was analyzed with only the forward shells
reduced in stiffness. This change, however, resulted in extremely low
frequencies, due to large shell modes in the less stiff forwards. In
order to combat this deficiency, additional hoop stiffeners were added
at the forwards' quarter points, bringing the number of hoops per
forward section to three, instead of one, see Figure 4. This addition
improved the frequency significantly, as can be seen in Results Table
7.1. In fact, decreasing the bending stiffness of the PST forwards by
almost a factor of five results in frequencies that are reduced by only
20%. This indicates an efficient design, and while the overall
frequencies are low (barely more than 40 Hz) this side effect must be
accepted if loads on the SCT are to be minimized. (Note that service
masses are included in all models, but they are the less conservative 13
kg mass, not the more accurate mass of 17 kg. Also, no free-ended
analysis cases are shown, since the approach of decreasing forward shell
stiffness eliminates any impetus to leave the forward ends of the PST
unconstrained.)

![](media/image8.png){width="4.593055555555556in"
height="2.8819444444444446in"}

**Figure 4. Fiberglass PST with Extra Hoop Stiffeners in Forwards.**

**Results Table 7.1. Effect of Fiberglass Forwards.**

(Shell modes in italics, cantilever modes in bold)

+--------------------------+----------+---------+---------+----------+
| ## Model                 | #        |         |         |          |
|                          | # Fixed  |         |         |          |
|                          | Forwards |         |         |          |
+--------------------------+----------+---------+---------+----------+
| ### Description          | **Mode   | **Mode  | **Mode  | **Mode   |
|                          | 1**      | 2**     | 3**     | 4**      |
+--------------------------+----------+---------+---------+----------+
| Carbon Shells -- 1 Hoop  | *53.1*   | *53.2*  | *54.2*  | *54.2*   |
+--------------------------+----------+---------+---------+----------+
| Glass Shells -- 1 Hoop   | *27.4*   | *27.4*  | *27.8*  | *27.9*   |
+--------------------------+----------+---------+---------+----------+
| Glass Shells -- 3 Hoops  | *43.7*   | *43.7*  | *43.8*  | *43.8*   |
+--------------------------+----------+---------+---------+----------+

**6.7.2 Support Reactions**

The major concern with PST stiffness is the effect it has on the SCT
interlinks (the PST supports) when it undergoes bending. In order to
analyze this effect without modeling the SCT, the mount pads on the PST
flanges were assumed to be rigidly fixed, according to the fixation
scheme explained earlier. The PST forward ends were then displaced by 2
mm in either the vertical or horizontal (and on either one side or both
sides). Independent analysis by the SCT group shows that this
approximation is not perfect, but it provides a good basis for
comparison. Since the PST is mounted only in the horizontal plane,
bending of the tube in the vertical is different than bending of the
tube in the horizontal, and these two load cases are analyzed
independently, with the reactions on the most highly loaded supports
shown.

**6.7.2.1 Vertical Bending**

> Since the PST mounts are on the horizontal diameter, vertical bending
> is far more compliant than bending in the orthogonal direction. For
> this reason, the loads in vertical bending are fairly low, as shown in
> Results Table 7.2.1. Both the glass and carbon PST designs are shown
> for comparison, and displacement is made on *both* ends of the PST. It
> can be seen that the glass forwards reduce forces and moments by
> almost a factor of three.
>
> Results Table 7.2.1. Reaction Loads Under Vertical Bending.

  --------------------- -------- -------- -------- -------- -------- --------
  **Model Material**    **FX     **FY     **FZ     **MX     **MY     **MZ
                        (N)**    (N)**    (N)**    (Nm)**   (Nm)**   (Nm)**

  Carbon                \-       151      0        253      \-       \-

  Glass                 \-       53       0        89       \-       \-
  --------------------- -------- -------- -------- -------- -------- --------

**6.7.2.2 Horizontal Bending**

> Up to this point, it has not been explicitly stated whether the PST is
> fixed to the SCT in Z (the beam direction) at only one point. However,
> all previous analyses have been completely insensitive to this
> constraint. The horizontal bending analysis, however, is very
> sensitive to this constraint, as it can apply a large Z load to the
> SCT during horizontal displacement. The PST has been analyzed with
> this constraint by the SCT group, and there is concern about the
> resultant loads. In order to make a comparison here, the PST is
> analyzed with both 1 and 2 Z constraints at the SCT interlinks (both
> on side C of the detector). The loads generated here are larger than
> those generated by the PST analysts, since the models here do not
> include SCT compliance. The absolute values of these results,
> therefore, should be viewed with caution. The relative magnitudes,
> however, should be indicative of the performance advantage of
> fiberglass forward tubes. In particular, note that the Z force is
> reduced by a factor of 4 when glass forwards are introduced into the
> model, Results Table 7.2.2.
>
> Results Table 7.2.2. Reaction Loads Under Horizontal Bending.

+---------------+--------+--------+--------+--------+--------+--------+
| ##### Both Si |        |        |        |        |        |        |
| des Bent -- 1 |        |        |        |        |        |        |
|  Z Constraint |        |        |        |        |        |        |
+---------------+--------+--------+--------+--------+--------+--------+
| **Model       | **FX   | **FY   | **FZ   | **MX   | **MY   | **MZ   |
| Material**    | (N)**  | (N)**  | (N)**  | (Nm)** | (Nm)** | (Nm)** |
+---------------+--------+--------+--------+--------+--------+--------+
| Carbon        | 146    | \-     | \-     | \-     | 233    | \-     |
+---------------+--------+--------+--------+--------+--------+--------+
| Glass         | 52     | \-     | \-     | \-     | 83     | \-     |
+---------------+--------+--------+--------+--------+--------+--------+
| #             |        |        |        |        |        |        |
| #### Both Sid |        |        |        |        |        |        |
| es Bent -- 2  |        |        |        |        |        |        |
| Z Constraints |        |        |        |        |        |        |
+---------------+--------+--------+--------+--------+--------+--------+
| **Model       | **FX   | **FY   | **FZ   | **MX   | **MY   | **MZ   |
| Material**    | (N)**  | (N)**  | (N)**  | (Nm)** | (Nm)** | (Nm)** |
+---------------+--------+--------+--------+--------+--------+--------+
| Carbon        | 297    | \-     | 1870   | \-     | 536    | \-     |
+---------------+--------+--------+--------+--------+--------+--------+
| Glass         | 108    | \-     | 468    | \-     | 110    | \-     |
+---------------+--------+--------+--------+--------+--------+--------+
| ##### One S   |        |        |        |        |        |        |
| ide Bent -- 1 |        |        |        |        |        |        |
|  Z Constraint |        |        |        |        |        |        |
+---------------+--------+--------+--------+--------+--------+--------+
| **Model       | **FX   | **FY   | **FZ   | **MX   | **MY   | **MZ   |
| Material**    | (N)**  | (N)**  | (N)**  | (Nm)** | (Nm)** | (Nm)** |
+---------------+--------+--------+--------+--------+--------+--------+
| Carbon        | 390    | \-     | \-     | \-     | 623    | \-     |
+---------------+--------+--------+--------+--------+--------+--------+
| Glass         | 137    | \-     | \-     | \-     | 218    | \-     |
+---------------+--------+--------+--------+--------+--------+--------+

**6.7.3 Gravity Sag**

Though less important than the above analysis cases, gravity sag is an
important criteria to consider in the PST design, primarily because it
determines whether the PST remains inside its envelope during removal
and insertion of the SCT forwards (when the forward ends of the PST are
free). Since the pixel detector may at some point remain inside the PST
during SCT forward access, it is also important to model the gravity sag
with service masses included. In these analyses only, the service masses
are modeled as 17 kg per side, which is the more accurate estimation.
The results show that the gravity sag for the fiberglass model is less
than a factor of 3 greater than that of the carbon one, due primarily to
the mount constraints and stiffness of the barrel tube (which is carbon
in both models), see Results Figure 7.3. The resultant loads from
gravity sag (the same for both models) are also given in Results Table
7.3, and are seen to be reasonable.

Results Table 7.3. Reaction Loads Under Gravity Sag.

  --------------------- -------- -------- -------- -------- -------- --------
  **Model Material**    **FX     **FY     **FZ     **MX     **MY     **MZ
                        (N)**    (N)**    (N)**    (Nm)**   (Nm)**   (Nm)**

  Both                  \-       110      \-       192      \-       29
  --------------------- -------- -------- -------- -------- -------- --------

![](media/image9.png){width="3.7375in" height="2.8118055555555554in"}

![](media/image10.png){width="3.7375in" height="2.8118055555555554in"}

**Results Figure 7.3. Displacement Under Gravity Sag (Carbon at Top,
Glass at Bottom) - shown in meters.**

7.0 Conclusions and Recommendations

The PST design has progressed through a series of iterations in order to
arrive at a design that performs adequately well, while allowing maximum
flexibility and minimum impact on surrounding detectors. The last design
shown above is considered to be a working baseline, and may change in
response to concerns that have not been addressed yet. In particular,
the effects of possible moments on the SCT interlinks must be addressed
(although this effect is mitigated by the support blocks, which will
lessen these moments somewhat). Also, the end plugs must be modeled as
real structures (rather than the flat plate approximations shown here)
and the flanges will change as fabrication designs are worked out.
Including service structures, which are stiffer than the distributed
mass added to the shells, may also increase frequency somewhat. It will
be necessary to work with SCT analysts to answer these questions and any
others that may arise.

# Appendix 1

**Comparison of PST Model Results (Neal H.) to those generated by SCT
Analyst (Joel C.)**

+------------+-------------------------+---------+---------+---------+
| ##### SCT  |                         |         |         |         |
| Analysis C |                         |         |         |         |
| omparisons |                         |         |         |         |
+------------+-------------------------+---------+---------+---------+
| **Bend     | **Load Case**           | **FX    | **FY    | **FZ    |
| D          |                         | (N)**   | (N)**   | (N)**   |
| irection** |                         |         |         |         |
+------------+-------------------------+---------+---------+---------+
| Y          | dy1 = dy2 = 2 mm SCT    |         | 141     |         |
+------------+-------------------------+---------+---------+---------+
| Y          | dy1 = dy2 = 2 mm Carbon |         | 151     |         |
+------------+-------------------------+---------+---------+---------+
| Y          | dy1 = dy2 = 2 mm Glass  |         | 53      |         |
+------------+-------------------------+---------+---------+---------+
| Y          | dy1 = dy2 = 2 mm        |         | **49**  |         |
|            | **Projected\***         |         |         |         |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = dx2 = 2mm SCT\^   | 325     |         | 588     |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = dx2 = 2mm         | 297     |         | 1870    |
|            | Carbon\^                |         |         |         |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = dx2 = 2mm Glass\^ | 108     |         | 468     |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = dx2 = 2mm         | **118** |         | **147** |
|            | **Projected\*\^**       |         |         |         |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = 2 mm SCT          | 462     |         |         |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = 2 mm Carbon       | 390     |         |         |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = 2 mm Glass        | 137     |         |         |
+------------+-------------------------+---------+---------+---------+
| X          | dx1 = 2 mm              | **162** |         |         |
|            | **Projected\***         |         |         |         |
+------------+-------------------------+---------+---------+---------+

**\*Projected values are to be expected when SCT compliance is
considered, as it was in SCT models (By Joel C.). These are computed by
taking the ratio of SCT result to Carbon result and multiplying by Glass
result (all shown above).**

**NOTE: These models have two Z constraints (for bending in X), so Z
Forces are COMPLETELY ELIMINATED when only 1 Z constraint is used, as is
currently proposed.**