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Optimization of Reinforcement Methods for Non-round Pressure Vessels

By

Shawn McMahon

A Presentation of a Thesis

In Partial Fulfillment of the

Requirements for the Degree of

Masters of Science

Major Subject: Mechanical Engineering

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Abstract

For a number of reasons the exhaust of a modern gas turbine engine is moving away from the conventional round pipe, and being replaced by one with an elliptical cross section. However, designing a low weight, non-round pressure vessel is more challenging than a typical round pressure vessel.

The problem posed is how to create the lightest weight round to elliptical pressure vessel. In order to accomplish this, analytical models were created and optimized based on a number of parameters. Two different optimization approaches were investigated.

The results showed that the first optimization method was simpler to build and optimize, but provided less than optimal weights. The second optimization method was much more complicated to build, was more sensitive to the controls of the optimization, but provided the lightest results.

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Optimization Methods

ANSYS was used as the finite element solver. The first optimization method used was shape optimization, also called

topological optimization. Simple ANSYS commands Pseudo-density manipulation Limited element selection and optimization controls

Simulated topological optimization. Design optimization of shell thickness Simulated manufacturability constraints

The second method used was design optimization. Requires parametric model to be built with APDL More difficult but more functionality Yields better results

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Pressure Vessel Description

Dimensions:

• 40” diameter

• 50” long

• 12” spool piece

12” spool pieceEdge fixed in all DOF

Material: Ti-6Al-V4

Constraints: Edge of spool fixed in all DOF

Load: 80 psi

A B A BModel # Ratio Major Minor Minor Major

1 1.0000 20.000 20.000 N/A N/A2 1.5000 24.495 16.330 1.22474 0.816503 2.0000 28.285 14.142 1.41423 0.707104 2.5000 31.623 12.649 1.58115 0.632455 3.0000 34.641 11.547 1.73205 0.577356 3.5000 37.416 10.691 1.87082 0.534537 4.0000 40.000 10.000 2.00000 0.50000

Scale FactorEllipse Dimensions

Table of Ellipse Parameters

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Quick Test of Topological Opt.

Dimensions:

• 10 inch long

• 1 inch high

Elements:

• Plane82

Goal:

• 75 percent reduction in volume

P

Results:

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Topological Optimization

Dimensions:

• Ellipse ratio 1.5

• 6 inches thick

Elements:

• Solid95

Goal:

• 50 percent reduction in volume

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Simulated Topological Opt.

Optimization:

• Vary segment thickness

• Simulated manufacturing constraints

Model Types:

• Axial segments

• Angular segments

Equal arc length segments

Equal angle segments

Angular Segments

Equal length segments

Axial Segments

Spool piece

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Simulated Topo. Opt. Results

Model 2 Model 3 Model 4 Model 5 Model 6 Model 7VTOT 746 1004 1237 1408 1537 1739DMAX 0.49973 0.51051 0.51035 0.51127 0.52229 0.50790T1 0.11047 0.10944 0.11520 0.12810 0.22251 0.19416T2 0.10797 0.10932 0.10955 0.10881 0.27498 0.23423T3 0.10758 0.10880 0.10988 0.10790 0.27376 0.29415T4 0.10869 0.11267 0.19896 0.25083 0.29698 0.32235T5 0.15409 0.25936 0.37540 0.44148 0.37598 0.50012T6 0.29878 0.44673 0.57943 0.65027 0.68499 0.86364T7 0.48878 0.66764 0.79971 0.86712 0.87785 0.94266T8 0.69788 0.91337 1.05967 1.12408 0.94874 0.75389T9 0.96575 1.12382 1.14631 1.03347 0.60998 0.51454T10 1.51187 2.04907 2.46349 2.86049 3.39258 4.01248

Results Table

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

1 2 3 4 5 6 7 8 9 10

Pressure Vessel Axial Segment

Sh

ell T

hic

knes

s (i

n)

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7

Shell Thickness per Axial Segment

Sample Results Screen Shot

Results:

• Aft most segment always thickest

• Segment 7 and/or 8 thicker in highly elliptical models

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Simulated Topo. Opt. Results

Model 2 Model 3 Model 4 Model 5 Model 6 Model 7VTOT 1189 1786 2214 2592 2886 3175DMAX 0.53585 0.52757 0.52329 0.50587 0.51424 0.51439T1 1.25551 1.64626 1.82651 1.95039 2.03334 2.07348T2 1.16706 1.57205 1.83817 1.96738 1.97194 1.98805T3 1.05279 1.44335 1.64386 1.75217 1.83029 1.86388T4 0.90051 1.28259 1.46227 1.60907 1.67687 1.69969T5 0.71594 1.08000 1.25377 1.36976 1.43269 1.46014T6 0.48734 0.76950 0.92223 1.03357 1.09142 1.12506T7 0.10886 0.10889 0.10751 0.10986 0.12032 0.13234T8 0.51284 0.86880 1.14905 1.37951 1.50323 1.63999T9 0.65026 1.13154 1.49307 1.81857 2.06284 2.29106T10 0.70264 1.23961 1.64802 2.00344 2.26225 2.50213

Results Table

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1 2 3 4 5 6 7 8 9 10

Pressure Vessel Angular Segment (1 = TDC, 10 = Right, Aft Looking Fwd)

Sh

ell T

hic

knes

s (i

n)

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7

Shell Thickness per Axial Segment

Sample Results Screen Shot

Results:

• Segment 10 always the thickest

• Segment 7 always the thinnest

• Segment 7 is actually the middle segment

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Design Optimization

Model Types:

• Axially spaced ribs

• Addition of four circumferentially spaced ribs

Optimization Parameters:

• Number of ribs

• Distribution of ribs

• Position of first ribs

• Rib height

• Shell thickness

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6

Rib Number

Rib

Po

siti

on

fro

m F

orw

ard

Ed

ge

(in

)

Linear: D = 1 Cubic: D = 2 Quadratic: D = 3

Model Type A

Model Type B

Mesh Elements

Shell181

Beam188

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Design Optimization

Optimization Flowchart:

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Design Optimization Results

Model 2 Model 3 Model 4 Model 5 Model 6 Model 7VTOT 141 185 228 253 299 339DMAX 0.504 0.502 0.481 0.500 0.482 0.495R_total 6 6 6 7 8 7D_val 1.300 1.418 1.550 1.476 1.763 1.211shift 24.232 20.885 19.753 18.317 17.665 18.885S_thick 0.031 0.043 0.054 0.048 0.050 0.074R_height 3.036 4.030 4.927 5.157 5.517 5.971

Model A Results Table

Model 2 Model 3 Model 4 Model 5 Model 6 Model 7VTOT 171 215 258 302 361 407DMAX 0.503 0.496 0.495 0.496 0.499 0.494R_total 5 6 6 6 6 8D_val 1.414 1.695 1.966 1.718 1.796 2.075shift 14.439 19.773 19.229 20.452 22.373 20.723S_thick 0.032 0.035 0.043 0.053 0.074 0.062R_height 3.123 3.837 4.571 5.160 5.471 5.621

Model B Results Table

Total Volume vs. Model Number

Rib Number vs. Model Number

Rib Height vs. Model Number

Results:

• Higher total volume of model with circumferentially spaced ribs

• Short rib height of model with circumferentially spaced ribs

• Increase in rib number with increase ellipse ratio

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Hand Calculations

Info:

• Quick check of the results of the analysis

• Beam bending equations from Roark’s and Timoshenko

• ANSYS results and hand calc results match pretty well. Hand calcs predict slightly lower deflections than ANSYS.

b/ax/S 0.3 0.5 0.6 0.7 0.8 0.9

0.0 -0.172 -0.156 -0.140 -0.115 -0.085 -0.0450.1 -0.167 -0.152 -0.135 -0.112 -0.082 -0.0440.2 -0.150 -0.136 -0.120 -0.098 -0.070 -0.0380.4 -0.085 -0.073 -0.060 -0.046 -0.030 -0.0150.6 0.020 0.030 0.030 0.028 0.022 0.0150.7 0.086 0.090 0.082 0.068 0.050 0.0220.8 0.160 0.150 0.130 0.105 0.075 0.0380.9 0.240 0.198 0.167 0.130 0.090 0.0461.0 0.282 0.218 0.180 0.140 0.095 0.050

K Coefficient per Ellipse Ratio

Beam Nomenclature

Shell Nomenclature

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Q & A