# Presentation: 880125

##  Magneto-Structural Analysis of the Fermilab TQC Nb3Sn High Gradient Quadrupole End Region

**CHATS-AS 2006**
**Lawrence Berkeley National Laboratory **
**September 5-7, 2006**

**Bob Wands**
**Fermilab**

**PPD/MD/Engineering Analysis Group**

**Fermilab**

## Outline

**Overview of TQC mechanical structure**
**Magnetic Model and results**
**Structural Model and results**
**Conclusions**

## The Fermilab TQC Quadrupole

- The Fermilab TQC magnet is a *T*echnological *Q*uadrupole based on the *C*ollar/yoke/skin mechanical structure used successfully in past dipoles and quadrupoles
- Aperture is 90 mm
- Target operating conditions are 14 kA at 1.9 K, with a resultant field gradient in excess of 200 T/m
- The magnet coils are two layer, with three current blocks, and consist of Nb3Sn Rutherford-type cable constructed by the wind-and-react method.
- Coils are confined by interleaving stainless steel collars, and the interaction of the collars, yoke, and skin through shrinkage of the skin closure weld.
- Current work is directed toward 1 m magnets to prove feasibility

## TQC Quadrupole Structure – Cross Section

## TQC Quadrupole Structure – End Region

**Aluminum Bronze Endparts**

**Inner Coil during Construction**

**Detail of End Region Current Blocks**

## Coil Structure

- Coil is Nb3Sn/Fiberglass/Epoxy composite with embedded wedges of aluminum bronze

- This figure shows a cross section of the final epoxy impregnated coil

- a slice through the straight section of a coil

- Block 2

- Block 1

- Block 3

## Magnet Assembly – Three Basic Steps

- 1. Collar

- 2. Add yoke, weld skin

- 3. Add endplate, apply axial preload

- typical collar piece

## A Major Challenge - Characterization of the Coil Stiffness for FEA

- Finished coils are well-known to exhibit non-linear stress-strain behavior
- Action shakes down to quasi-elastic, similar to strain hardening of metals
- Fermilab coils are not “massaged” or shaken down prior to assembly
- Best material model would be plastic – *convergence issues*
- Coil material model in this analysis is orthotropic and elastic based on experience with mechanical models and testing

- D.R. Chichili, et al, azimuthal loading of ten stack

## The Elastic Orthotropic Coil Material Model

- The natural material coordinate system for the straight section of such a model is cylindrical
- Young’s modulus varies somewhat in the three coordinate directions.
- Variation of thermal contraction is more pronounced
- In the end regions, material properties must remain correctly oriented as the conductors turn.

## Material Properties

| Component | Material | Young’s Modulus – GPa 
(293 K/1.9 K) |  |  | Thermal Contraction 
293 K to 1.9 K |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | Radial | Azimuthal | Axial | Radial | Azimuthal | Axial |
| Coils | Nb3Sn/fiberglass/ epoxy composite | 44/55 | 44/44 | 44/44 | 0.0026 | 0.0035 | 0.0023 |
| End Parts/Wedges | C642000 Aluminum Bronze | 110/120 | 110/120 | 110/120 | 0.0031 | 0.0031 | 0.0031 |
| Collars | Nitronic 40 Stainless Steel | 190/210 | 190/210 | 190/210 | 0.0025 | 0.0025 | 0.0025 |
| Yoke – central region | Extra low Carbon Steel | 213/224 | 213/224 | 213/224 | 0.0020 | 0.0020 | 0.0020 |
| Yoke – end region | 304 Stainless Steel | 190/210 | 190/210 | 190/210 | 0.0026 | 0.0026 | 0.0026 |
| Yoke Spacers | 316 Stainless Steel | 190/210 | 190/210 | 190/210 | 0.0029 | 0.0029 | 0.0029 |
| Skin/Endplate | 304 Stainless Steel | 190/210 | 190/210 | 190/210 | 0.0026 | 0.0026 | 0.0026 |

## Summary of Basic Analysis Approach

- Create a 1/8th symmetric magnetic ANSYS model, using SOURC36 finite elements to determine field at 14 kA and Lorentz forces
- Create a 1/4 symmetric structural ANSYS model, using first order SOLID45 elements. Define all interfaces between coils and endparts as bonded.
- Exercise structural model through assembly, cool down, and energizing load steps, tracking bond failure between coils and endpart.
- De-energize and reenergize to follow shaken-down behavior of failed bond regions

## The Magnetic Model

- The magnetic symmetry is 1/8th azimuthally.
- Structural symmetry is, strictly speaking, 1⁄2 due to the use of 180 degree iron laminations. However, the iron laminations are not strongly connected across their midplane, and interact with four stainless steel spacers (used to control the transfer of skin stress to the collars) symmetrically spaced about the azimuth. Therefore, approximate 1/4 symmetry is assumed.
- The difference in symmetry, the existence of non-magnetic (assembly and cool down) load steps, the need to include a universe of air, and complications arising when using the SOLID5 element to model the coils argued against the “coupled-field” approach.
- The magnetic model was run one time at 14 kA, and the resulting Lorentz forces scaled for lower currents.

## The Magnetic Model – cont’d

- SOURC36 elements are used to provide currents to the model.
- They are overlaid on the inactive structural mesh to ease the calculation and application of the resulting Lorentz forces.
- Lorentz forces at 14 kA are extracted by an ANSYS macro, rotated into a cylindrical coordinate system, and output to an external file for reading and scaling during the structural solution

## The Magnetic Model – cont’d

- The ANSYS GSP (general scalar potential) solution routine was invoked

## Why not SOLID5’s with Voltage-induced Currents?

- In principle, the SOLID5 can be used as a current source, solving for a current density from an applied voltage and input resistivity, and calculating the resulting b-field and subsequent Lorentz forces.
- In practice, for large ratios of ro/ri, current distributions cannot be kept sufficiently uniform around curves to adequately describe the field and forces of this magnet.
- A “stranded” mesh of individual strings of SOLID5s could be used, with nodes coincident between strands but not merged, but bookkeeping could be an annoyance, plus the increase in dof to be carried into the structural solution.
- The element simply needs the ability to use a specified current density in a specified element coordinate direction as a body force loading.

## Results of Magnetic Model – comparison with 2-d straight section results

- The 3-d model contains a straight section where results can be compared to a 2-d model
- A highly-refined 2-d model using PLANE13 elements was created.
- The 3-d and 2-d field results agreed within 3% for maximum field in the straight section
- The 3-d and 2-d forces agreed within 2% in the straight section based on a reaction force summation

## Results of Magnetic Model - Lorentz Forces in the End Region

**Schematic Depiction of End Region Lorentz Forces**

- From the 3-d magnetic model, the total *axial* Lorentz force acting on the quadrupole is 360 kN.
- This is distributed as:
- Block 1: 11%
- Block 2: 16%
- Block 3: 73%
- Forces act to straighten and stretch the blocks at the turn
- Note: For reference, distribution of total current in the blocks is 18%, 35%, and 47% for Blocks 1,2, and 3, respectively

- Symmetry plane

## The Structural Finite Element Model

- The 3-d structural model is expected to predict displacements and primary stresses with some accuracy; peak stresses (concentrations) are beyond the consideration of this analysis
- Given the primary stress requirement, the approximate material models, the need to model 1⁄4 of the cross section, and the goal of running the magnet through multiple energizing cycles, linear elements (ANSYS SOLID45 eight node hexahedra) are chosen.
- All surfaces between bonded parts are modeled with TARGE170 and CONTA173 elements.
- The analysis will monitor the tensile stress in the bonds between the coil blocks and endparts, and remove from the solution those contact elements whose tensile stress exceeds 30 MPa. This value is based on tests of slices of actual coils.

## Quantities of Interest from the Structural Model

**Previous analyses have shown that gaps will tend to develop between Block 1 and Block 3 and their respective endparts at the symmetry plane of a coil when the interfaces between those parts are modeled as frictionless/separating surfaces**
**Axial preload (reaction with the endplate) can delay the opening of these gaps, and influence their final size.**
**The structural model attempts to quantify the effects of three different axial preloads on bond failure initiation, ultimate gap size, and coil stress at the symmetry plane**

**Gaps at the symmetry plane between Block 1 and Block 3 and their respective endparts**

## Components of the Structural Model

## Aligning the Material Property Axes of the Coil Elements

- For the SOLID45 element, material property directions are controlled by the element coordinate system.
- This system defaults to the global cartesian system, unless specified with element keyopt or esys
- As-generated elements did not have the desired orientation
- A user macro was written to delete and redefine the element nodal incidence to keep element Z in the direction of the current

**Z**

**X**

**Y**

## Coil Material Axis Alignment in End Region – Thermal Contraction Effects

- The strongest effect of the orthotropic material model is in the thermal contraction behavior at the ends.

**straight section azimuthal contraction**
**straight section axial contraction**

## Modeling the Interleaving Collar Laminates

- Two coexisting sets of collar pack elements are used, with in plane moduli reduced by 50%, and axial modulus set at 4% of the full value.
- The two sets of collar pack elements communicate only through the keys and keyways (keys not shown)

**(Left collar pack separated for clarity)**

## Applying the Assembly Loads

**For collaring load step, *****collars are allowed to slide*****, and coils are preloaded by key interference or coil midplane displacement**

**For axial preload load step, interference is applied here**

**Note: During the first load step (collaring), the collars are allowed to slide with respect to the coil to prevent generating fictitious shear stresses**
**In subsequent load steps, the sliding collars are killed, and a set of coexisting locking collar elements activated**

## The Structural Runs

- The structural model was exercised through the following load steps:
- Apply collars around coils – midplane displacement
- Add yoke and skin – simulate skin weld shrinkage with azimuthal displacement
- Add endplate and adjust axial preload – initial interference
- Cool down – track bond stresses between coil and endparts and eliminate elements with tensile stress exceeding 30 MPa
- Energize to 14 kA – continue tracking and eliminating “failed” bond elements
- De-energize and reenergize to observe “shake-down” effect on final gap sizes.
- This procedure was followed for axial preloads of 0, 3500 N/quadrant
- (the design preload), and 35000 N/quadrant

## Results of Structural Model – Comparison with 2-d Model Azimuthal Stress in Straight Section

**Load Step 1.**
**Assembly**

**2-d Model**

**3-d Model**

## Results of Structural Model – Comparison with 2-d Model Azimuthal Stresses in Straight Section – cont’d

**Load Step 2.**
**Assembly + Cooldown**

**2-d Model**

**3-d Model**

## Results of Structural Model – Comparison with 2-d Model Azimuthal Stress – cont’d

**Load Step 3.**
**Assembly+Cooldown+Energize**

**2-d Model**

**3-d Model**

## Results of the Structural Model – Coil-to-Endpart Bonds and Coil Stresses

- The following quantities were extracted from the solutions:
- The maximum bond stresses between Block 1 and Block 3 and their respective endparts at the symmetry plane during assembly and cool down. These provide the initial condition of the bonds just prior to the first energization
- The resulting width of the gaps caused by bond failure in those regions, and the variation of that width under a second energizing cycle
- The total cross sectional area of failed bond in those regions
- The variation of Block 1 coil stress at the symmetry plane over the load history
**Note: Block 2 of the inner coil shows no failure of bonding in any incarnation of the simulation, and will be omitted from this discussion henceforth**

## Maximum Bond Stress During Assembly and Cool down for Block 1

**Collars sliding**

**Collars locked**

## Maximum Bond Stress During Assembly and Cool down for Block 3

**Collars sliding**

**Collars locked**

## Comparison of Maximum Bond Stress in Block 1 and Block 3 for Design Axial Preload

**Collars sliding**

**Collars locked**

## Width of Gap between Block 1 and its Endpart

## Width of Gap between Block 3 and its Endpart

## Growth of the Failed Bond Area for Block 1

## Growth of the Failed Bond Area for Block 3

**Note: results suggest that mesh is too coarse to adequately track failed area at high current**

## The Failed Bond Area for Design Axial Preload for Blocks 1 and 3

- Block 1 failed bond area ~135 mm2

- Block 3 failed bond area ~380 mm2

**Note: Half area shown, full area listed**

## Block 1 Stresses at Symmetry Plane – Location for evaluation

- X-direction stresses are linearized over this path and the membrane value extracted.
- This should be indicative of the primary membrane plus bending stress about the y axis of section

## Block 1 Stresses at Symmetry Plane

## Conclusions

- The bonds between the coils and endparts at the symmetry plane of the end region almost certainly fail, and although axial preload can delay the effect, at the currently achievable levels of the TQC design it can’t prevent it.
- For the case of design axial preload, a gap of up to 50 microns might exist between Block 3 and its endpart when energized.
- Even when de-energized, and cold, this gap may still be nearly 40 microns because bond failure allows the coil to respond to its natural axial thermal contraction and the differential azimuthal contractions of its neighbors
- The Block 1 azimuthal stress at the symmetry plane is most likely greater after the first energizing/de-energizing cycle, due to bond failure and loss of the tensile load path to the endpart.

## For the Future

- A post-mortem examination of the TQC magnet to detect bond failures in the regions considered in this analysis
- A plastic material model for the coil