Submodel Analysis for MQXFA End Shell H Pan

Submodel Analysis for MQXFA End Shell H. Pan LBNL, 01/12/2018

Outline • High stress in the end shell --- global model • Sub-model analysis – Sub-model verification – Peak stress with different cases – Stress Linearization 1/17/2022 2

Global Model Overview • Half-length coil: 2. 281 m (return end) • Octant model, using solid 186 elements • All components are in contact using contact elements • Contact 174 • Target 170 • Friction coefficient is 0. 2 • Inside coil winding --- bonded • Other interfaces --- frictional contact • Boundary conditions (MQXFAP 1) End shell is highlighted for this analysis • Azimuthal symmetry: at 0 and 45 degree of the assembly • Axial symmetry: Z=0, except for the rod • Rod is Shell tension: 580 με • Key interference: 640 μm 1/17/2022 3

High stress on the end shell 293 K, key shimming 1. 9 K MPa Von Mises Stress on the end shell of MQXFAP 1 7075 -T 6 aluminum properties * Ultimate stress (MPa) Yield Stress (MPa) 293 K 489 420 4. 2 K 674 550 KIC (MPa*√m) 20 (need test data) * Data is from: “Metallic Materials and Elements for Aerospace Vehicles Structure, Department of Defense Handbook, Version 5 J. --Aluminum 7075 Alloy” • High stress appears at the inner bottom corner of the alignment cutout. • Global model does not account for fillet and has relatively coarse mesh. • Stress singularity or concentration at this location needs to be analyzed by advance FE methods. 1/17/2022 4

High stress on the end shell (global model) mi --- initial mesh density, element size = 8 1000 1 x 900 2 x 800 4 x Peak stress get to plateau after cooldown 600 500 400 200 Peak stress increases with mesh density --- stress concentration or singularity 1000 700 300 mm 1200 Peak stress (MPa) 1100 800 600 400 293 K, key shimming 1. 9 K 200 Peak stress evolution with different mesh densities key Shimming 1. 9 K 130 T/m 140 T/m Global model results 0 0 1 2 3 Relative mesh density m/mi 1/17/2022 4 5 5

Sub-model---zoom the area of concern Local high stress is present because of 1. model singularity, such sharp corner; 2. Point load or restraints; 3. Cracks; 4. Change of geometry … Local refinement Sub-modelling • Sub-modeling is a way to “zoom in” on specific regions of a previously-analyzed model, create a fine mesh, include detail features, and obtain highly accurate results just for that region. • The process is based on St. Venant’s principle*. Therefore, it uses the displacement field (and temperature for cool-down) at the cut surfaces from the global model as a boundary condition. *Saint-Venant’s principle states that if two different load distributions are statically equivalent, the effects on a location a sufficient distance away will be the same. 1/17/2022 6

End shell sub-model The basic characteristics of sub-modeling: • The sub-model is a separate analysis from the global model. • The sub-model includes only the local region of interest. • The sub-model has its own geometry, so local features that may have been omitted in the global model can be included in the submodel. • Displacements at the cut surfaces from the global model were used as the cut boundary conditions for the sub-model The sub-model has its own mesh, so a much finer mesh and even different element types can be used to obtain more accurate results. Fillet and chamfer are modelled 1/17/2022 7

Sub-model verification Sub-model Global model 4 3 5 6 2 7 1 Unit: m 8 9 10 • Von Mises stress at 293 K on the cut edges. • The stress from sub-model is consistent with the results of the global model at the same location. Cut locations meet the St. Vanent’s principal. Stress (MPa) UX at 293 K, same mesh density 100 90 80 70 60 50 40 30 20 10 0 Global model 0 2 4 6 8 Locations on the cut edges Sub-model 10 12 1/17/2022 8

Sub-model cases Fillet radius (R): 1. 0. 5 mm; 2. 0. 75 mm --- specified in the drawing; 3. 1 mm. Mesh density multiples with respect to the global model: R 1. 2 x; 2. 4 x; 3. 10 x. 4. 20 x 1/17/2022 9

Peak Von Mises stresses 1300 Key shimming, 293 K 650 1200 600 1100 Peak stress (MPa) 700 550 500 450 Yield stress @ 293 K Fillet Radius 0. 5 mm 400 1000 Peak stress tends to converge with mesh density --- stress concentration (not singularity). 900 800 Fillet Radius 0. 5 mm 700 Fillet Radius 0. 75 mm 350 1. 9 K Fillet Radius 0. 75 mm 600 Yield stress @ 22 K Fillet Radius 1 mm 300 500 0 2 4 6 8 10 12 14 Relative mesh density m/mi 16 18 20 22 1300 130 T/m 1200 140 T/m 1200 1100 Peak stress (MPa) Fillet Radius 1 mm 1000 900 800 Fillet Radius 0. 5 mm 700 Yield stress @ 22 K 900 800 Fillet Radius 0. 5 mm 700 Fillet Radius 0. 75 mm 600 1000 Fillet Radius 0. 75 mm 600 Fillet Radius 1 mm Yield stress @ 22 K Fillet Radius 1 mm 500 0 2 4 6 8 10 12 14 Relative mesh density m/mi 16 18 20 22 0 2 4 6 8 10 12 14 Relative mesh density m/mi 16 18 1/17/2022 20 22 10

Peak stress from the sub-model Von Mises stress of an elastic-perfectly submodel Peak stress (MPa) 1300 1200 2 X 1100 4 X 1000 10 X 900 20 X 800 700 Yield at 1. 9 K 600 500 Yield at 293 K 400 300 Key shimming, 293 K key 1. 9 K 1300 Elasto-plastic model 1200 Peak stress is limited around the yielding points 1100 Peak stress (MPa) Size of the plastic deformation is about 1 mm x 2 mm 1000 Not much strain hardening --local yielding limits the peak stress. 1 x 2 x 4 x 900 800 700 600 Yield at 1. 9 K 500 Yield at 293 K 400 Plastic strain at 1. 9 K 130 T/m 140 T/m 300 key 1. 9 K 130 T/m 140 T/m 1/17/2022 11

Stress Linearization • According to the ASME code, stresses are separated in 3 classes that are associated with different types of failure [ASME sec VIII]: 1. Primary stress--- generally due to internal or external pressure or produced by sustained external forces and moments. Ø It is associated with gross plastic deformation. 2. Secondary stress --- Relenting loads (self-limiting) produce secondary stresses (self-limiting) Ø They can be relieved by local yielding minor distortions; 3. Peak stress --- Peak stresses are additive to primary and secondary stresses present at the point of the stress concentration. Ø It is associated with fatigue failure. • Stress linearization is a way to convert the six stress components into those primary, secondary, and peak stresses. 1/17/2022 12

Stress Linearization at the SCL • ANSYS provides the major three linearized stresses along a SCL: 1. 2. 3. Membrane stress Bending stress Total stress (membrane + bending) - is associated with incremental plastic deformation. Ø If the primary membrane is less than the limit, but the total Ø exceeds, local yield may happen. The yield length should be less than the detection limit (1 mm), refine the design on the localized area may be needed. SCL (Stress Classification Line) 1/17/2022 13

Discussion Path 3 (through stress concentration) Path 1 (45°) Path 2 (Mid-plane) Von Mises stress on paths (elastic model) 1. The process of advance analysis (including sub-modelling, stress linearization, and elasto-plastic model) has been developed. 2. According to Eric’s analysis, cracks are substantially driven by the membrane stress, the plots show the stress concentration is very high at the localized area, and get into the structure relatively deep. 3. Further LEFM with the small flaw size (2 mm or smaller) and other flaw propagate directions will be performed.

Extra slide Typical tensile stress-strain curve (full range) for 7075 -T 6 and T 651 aluminum alloy rolled or cold-finished bar at room temperature 1/17/2022 15