DESIGN AND MODELING OF INTERNALLY … AND MODELING OF INTERNALLY PRESSURIZED THICK-WALLED CYLINDER 2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17- …

DESIGN AND MODELING OF INTERNALLY PRESSURIZED THICK-WALLED CYLINDER

2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 1

Zhong Hu, Ph.D.

Associate ProfessorMechanical Engineering Department

South Dakota State University

Phone: (605) 688-4817, Fax: (605) 688-5878

E-mail: [email protected]

mailto:[email protected]�

OUTLINE


1. Introduction

2. Basic Concepts OF A Pressurized Thick-Walled Cylinder

3. Stress Analysis of A Single-Layer Pressurized Thick-Walled Cylinder

4. Stress Analysis of A Double-Layer Pressurized Thick-Walled Cylinder

5. Stress Analysis of A Composite-Wrapped Pressurized Thick-Walled Cylinder

6. Conclusions

7. Acknowledgements


Piping systems of chemical plant

Gun Barrel

Piping system of a nuclear power plant

1. INTRODUCTION


Cracked Barrel

1. INTRODUCTION – CONT.

Oil & Gas Pipeline Failure

Plastic Failure


1. INTRODUCTION – CONT.

2. BASIC CONCEPTS OF A PRESSURIZED THICK-WALLED CYLINDER


Closed cylinder with internal pressure, external pressure, and axial loads. (a) Closed cylinder. (b) Section e-e.

3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER


Stresses in thick-wall cylinder. (a) Thin annulus of thickness dz. (b) Cylindrical volume element of thickness dz.

Basic Assumptions:

(1). Static loads(2). Isotropic and homogenous material(3). Constant temperature(4). Elasto-plastic and small deformation(5). Ignoring axial load (stress)(6). Cross section keeping plane after deformation


Elastic Analysis:

Equilibrium Equation

Strain Compatibility Condition

Stress Components under Internal and External PressureStrain-Displacement Relations

Hooke’s Law (stress-strain relations)

3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.



Stress Components under Internal Pressure Only

Radial Displacement under Internal and External Pressure



Radial Stress, σr , Distribution in A Single Layer Thick-Wall Cylinder.

-2.5

-2

-1.5

-1

-0.5

0

0 20 40 60 80 100

σr/p

1

(r-a)/(b-a) ×100%

p2/p1=0p2/p1=0.5p2/p1=1p2/p1=1.5p2/p1=2

b/a = 2 -2.5

-2

-1.5

-1

-0.5

0

0 20 40 60 80 100

σr/p

1

(r-a)/(b-a) ×100%

p2/p1=0p2/p1=0.5p2/p1=1p2/p1=1.5

b/a = 1.5



Hoop Stress, σθ, Distribution in A Single Layer Thick-Wall Cylinder.

b/a = 2

b/a = 1.5

-4

-3

-2

-1

0

1

2

0 20 40 60 80 100

σθ/p

1

(r-a)/(b-a) × 100%

p2/p1=0p2/p1=0.5p2/p1=1p2/p1=1.5p2/p1=2

-5

-4

-3

-2

-1

0

1

2

3

0 20 40 60 80 100

σθ/p

1

(r-a)/(b-a) × 100%

p2/p1=0p2/p1=0.5p2/p1=1p2/p1=1.5p2/p1=2



Finite Element Model

3-D structural solid element

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.2 0.4 0.6 0.8 1

σr/p

1

(r-a)/(b-a)

Theoretical result

FEA result



Comparison of the analytical results with FEA results of an internally pressurized thick wall cylinder.

Radial Stress



Comparison of the analytical results with FEA results of an internally pressurized thick wall cylinder.

Hoop Stress

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

σθ/p

1

(r-a)/(b-a)

Theoretical result for hoop stressfea result



Comparison of the analytical results with FEA results of elastic strains in an internally pressurized thick wall cylinder.

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0 0.2 0.4 0.6 0.8 1

ϵ

(r-a)/(b-a)*100%

fsalon-rfsalon tetafsalon teta feaes-r fea

Analytical εr

Analytical εθFEA εr

FEA εθ



From Wikipedia: Autofrettage is a metal fabrication technique in which a pressure vessel is subjected to enormous pressure, causing internal portions of the part to yield and resulting in internal compressive residual stresses. The goal of autofrettage is to increase durability of the final product. The technique is commonly used in manufacturing high-pressure pump cylinders, battleship and tank cannon barrels, and fuel injection systems for diesel engines. While some work hardening will occur, that is not the primary mechanism of strengthening.

When autofrettage is used for strengthening cannon barrels, the barrel is prebored to a slightly undersized inside diameter, and then a slightly oversized die is pushed through the barrel. The amount of initial underbore and size of the die are calculated to strain the material past its elastic limit into plastic deformation, sufficiently far that the final strained diameter is the final desired bore.

The technique has been applied to the expansion of tubular components down hole in oil and gas wells. The method has been patented by the Norwegian oil service company, READ, which uses it to connect concentric tubular components with sealing and strength properties outlined above.

Elasto-Plastic Analysis of Autofrettage

0.0E+00

5.0E+07

1.0E+08

1.5E+08

2.0E+08

2.5E+08

3.0E+08

3.5E+08

4.0E+08

0 0.02 0.04 0.06 0.08 0.1 0.12

Stre

ss σ

(Pa)

Strain ε



Stress-Strain Curve of AISI 304 Tensile Test

The stress-strain relationship of a strain-hardening material shown in the figure is assumed as:

Elasto-Plastic Analysis of Autofrettage



Elasto-Plastic AnalysisIt can be assumed that the elastic zone of vessel is a cylinder of inner radius ρ and outer radius bwhich is subjected to internal pressure p ρ

Where p ρ in plane-strain and plane-stress conditions is obtained as:

The elastic-limit pressure pe and the plastic-limit pressure py are:

The relation between internal pressure and the radius of the elastic-plastic boundary in plane-strain and plane-stress condition is determined as:



Comparison of the analytical results with FEA results of the elastic-plastic interface radius vs. the internal pressure.

0.E+00

1.E-01

2.E-01

3.E-01

4.E-01

5.E-01

6.E-01

0 20 40 60 80 100

Pi /

σy(N

/m2 )

Elastic-plastic interface radius ρ ((ρ-a)/(b-a)×100%)

Pi by analyticalPi by modeling



Comparison of the analytical results with FEA results of the strain and stress during pressuring process

Stresses at pi=86.9 MPa and ρ=53.33% Strain at pi=65.8 MPa and ρ=21.15%

-1.2

-0.7

-0.2

0.3

0.8

1.3

0 20 40 60 80 100σ/σy

(N/m

2 )

(r-a)/(b-a)*100 (%)

σrr by analyticalσrr by ansysσθθ by analyticalσθθ by ansys

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.6E-04

0 20 40 60 80 100vo

n M

isis

Pla

stic

Str

ain

Radial Position (r-a)/(b-a)*100 (%)



FEA results of residual stresses of autofrettaged cylinder

Residual stresses of autofrettaged cylinder (pi=7.39 MPa and ρ=20.0%

-3.E+07

-3.E+07

-2.E+07

-2.E+07

-1.E+07

-5.E+06

0.E+00

5.E+06

1.E+07

0 20 40 60 80 100

σ(Pa

)

(R-a)/(b-a)×100 (%)

σrby modeling

σr residual by Analytical

σθ by modelling

σθ residual by analytical


4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER

Initial geometric condition of the composite cylinders before assembly (Δ = ci – co).

During pressuring (p1≠0) after assembling, the radial displacement of the outer and inner layer are:

During Assembling, the radial displacement of the inner layer(p1=0):

During assembling, the radial displacement of the outer layer (p2=0):

Elastic Analysis:

Relationship between the difference of Δ/a and pi/E (b/a=1.5 and (ci+co)/2a=1.25, and υ=0.3).


4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.

Relation between Δ and u.

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.01 0.02 0.03 0.04 0.05

pi/E

Δ/a

a/b=1.5, (ci+co)/2a=1.25

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.01 0.02 0.03 0.04 0.05u/

a

Δ/a

ui at the interfaceuo at the interface



-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 20 40 60 80 100

Stre

ss/E

(r-a)/(b-a) ×100%

Ratio of Hoop Stress to Elastic ModulusRatio of Radial Stress to Elastic Modulus

The radial and hoop stress distributions by prestressed assembly (b/a=1.5 and (ci+co)/2a=1.25, and υ=0.3).



-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

0 20 40 60 80 100Stre

ss/E

(r-a)/(b-a) ×100%

Radial stress ratio (p1=5pi)Radial stress ratio (p1=16pi)Hoop stress ratio (p1=5pi)Hoop stress ratio (p1=16pi)Radial stress ratio (pi=0)

The radial and hoop stress distributions by internal pressure p1 and the assembly pressure pi (b/a=1.5 and (ci+co)/2a=1.25, and υ=0.3).

-2

-1

0

1

2

3

4

5

6

7

8

0 20 40 60 80 100

Stre

ss/E

(r-a)/(b-a) ×100%

Radial stress ratio (p1=5pi)Radial stress ratio (p1=37pi)Hoop stress ratio (p1=5pi)Hoop stress ratio (p1=37pi)Radial stress ratio (pi=0)

The radial and hoop stress distributions by internal pressure p1 and the assembly pressure pi (b/a=1.2 and (ci+co)/2a=1.1, and υ=0.3).



Contact Model


4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER – CONT.

• Two layer cylinder tapered in dimension so that one can slide into another.

• Dimension- inner cylinder to outer cylinder in the model is 0.06-0.09 m

• Element Type- Solid45, Conta174,Targe170

• Total elements= 17920, ET1= 15360, ET2&3=1280Total nodes = 19040

Model is in transient condition with the first 10 sub steps used for assembly and the next 10 sub steps used to apply internal pressure



Modeling Results - Stresses

Hoop Stress

Radial Stress

Isotropic, elastic deformation with uniform pressure of 2.1e8 Pa

von Mises Stress



Modeling Results - Strains

Hoop Strain Radial Strain



• Relationship for dimensionless radii Δ/a & dimensionless pre-pressure after assembly pi/E.

• To generate more residual stress, more overlapping between the interface- more pressure

Relationship between pi/E and ∆/a



-0.006

-0.004

-0.002

-1E-17

0.002

0.004

0.006

0.008

0 20 40 60 80 100

Stre

ss/E

(r-a)/(b-a)*100(%)

Ratio of radial Stress to Elastic ModulusRatio of hoop Stress to Elastic ModulusHoop stress from FEARadial stress from FEA

Radial and hoop stress distributions by prestressed assembly


5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER


5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER - CONT.

Design in Laminated Composites



Governing EquationsThe elastic material property matrix [D]j for the layer j

Nonlinear Finite Strain Shell



Governing Equations


5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER

Different orientations were selected, such as (0/90/0/90), (0/90/45/135), (0/90/30/120/60/150)

Solid Model Meshed Model



Hoop Stress Radial Stress

Isotropic, elastic deformation with uniform pressure of 2.1e8 Pa

Von Mises Stress



• Failure criteria are curve fits of experimental data that attempt to predict failure under multi-axial stress.

• Failure criteria is defined as If =

• Failure is predicted when If ≥ 1• Considering failure for maximum

stress criterion.

FAILURE CRITERIA



COMPARISON

SINGLE LAYER THICK WALLCYLINDER

von Mises Stress0.671e9

DOUBLE LAYER THICK WALLCYLINDER 0. 618e9

COMPOSITE WRAPPED THICK WALL CYLINDER

0. 606e9

Best orientation for this model was (0-90-45-135)


Acknowledgement

This work was inspirited by the DoD projects in METLAB at South Dakota State University and supported by the Department of

Mechanical Engineering at SDSU. Calculation data contributed by Manjunath Gurumallappa and Sudhir Puttagunta is gratefully

acknowledged.


Contact:

Zhong Hu, Ph.D.

Associate ProfessorMechanical Engineering Department

South Dakota State University

Phone: (605) 688-4817, Fax: (605) 688-5878

E-mail: [email protected]

mailto:[email protected]�

DESIGN AND MODELING OF INTERNALLY … AND MODELING OF INTERNALLY PRESSURIZED THICK-WALLED CYLINDER 2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17- …

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