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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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: Zhong.Hu@sdstate.edu
OUTLINE
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 2
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 3
Piping systems of chemical plant
Gun Barrel
Piping system of a nuclear power plant
1. INTRODUCTION
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 4
Cracked Barrel
1. INTRODUCTION – CONT.
Oil & Gas Pipeline Failure
Plastic Failure
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 5
1. INTRODUCTION – CONT.
2. BASIC CONCEPTS OF A PRESSURIZED THICK-WALLED CYLINDER
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 6
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 7
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 8
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.
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 9
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 10
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 11
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 12
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 13
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
Comparison of the analytical results with FEA results of an internally pressurized thick wall cylinder.
Radial Stress
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 14
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 15
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 16
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 17
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 18
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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:
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 19
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 20
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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 (%)
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 21
3. STRESS ANALYSIS OF A SINGLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 22
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).
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 23
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 24
4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
-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).
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 25
4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
-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).
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 26
4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
Contact Model
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 27
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 28
4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
Modeling Results - Stresses
Hoop Stress
Radial Stress
Isotropic, elastic deformation with uniform pressure of 2.1e8 Pa
von Mises Stress
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 29
4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
Modeling Results - Strains
Hoop Strain Radial Strain
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 30
4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
• 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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 31
4. STRESS ANALYSIS OF A DOUBLE-LAYER PRESSURIZED THICK-WALLED CYLINDER - CONT.
-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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 32
5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 33
5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER - CONT.
Design in Laminated Composites
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 34
5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER - CONT.
Governing EquationsThe elastic material property matrix [D]j for the layer j
Nonlinear Finite Strain Shell
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 35
5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER - CONT.
Governing Equations
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 36
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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 37
5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER - CONT.
Hoop Stress Radial Stress
Isotropic, elastic deformation with uniform pressure of 2.1e8 Pa
Von Mises Stress
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 38
5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER - CONT.
• 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
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 39
5. STRESS ANALYSIS OF A COMPOSITE-WRAPPED PRESSURIZED THICK-WALLED CYLINDER - CONT.
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)
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 40
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.
2010-05-10 Presentation at 2010 NDIA Conference, Dallas, Texas, May 17-20, 2010 41
Contact:
Zhong Hu, Ph.D.
Associate ProfessorMechanical Engineering Department
South Dakota State University
Phone: (605) 688-4817, Fax: (605) 688-5878
E-mail: Zhong.Hu@sdstate.edu
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