elastic-plastic behavior of an ideal cylinder subject to mechanical and thermal loads
TRANSCRIPT
Elastic-Plastic Behavior of an Ideal Cylinder Subject to
Mechanical and Thermal Loads
Objective
Examine the stresses, strains, and displacements of ideal cylinders subject to axisymmetric mechanical (pressure) and thermal loads, to include:– Thin-walled and thick-walled cylinders– Plane-strain and plane-stress end conditions– Elastic and elastic-plastic behavior– Analytical and finite-element methods
Elastic Response
• Cylinders under pressure• Thin-shell theory, r/t < 10
p rt
r p
rpE
1 rt
p
E
r
t
z pE
1r
t
urp rE
r
t
Hoop Stress - Plane-Stress
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Distance Through Thickness
No
rmal
ized
Str
ess
r/t = 4.0
r/t = 5.7
r/t = 9.0
r/t = 12.3
r/t = 19.0
r/t = 39.0
r/t = 79.0
Elastic Response
• Cylinders under pressure• Thick-shell theory, Lamé’s Equations
rp a2
b2
a21
b2
r2
p a2
b2
a21
b2
r2
rp a2
E b2
a2
1 b2
r21
p a2
E b2
a2
1 b2
r21
z2 p a
2
E b2
a2
urp a2
E b2
a2
1 b2
r21
r
Hoop Stress - Plane-Stress
15000.00
17000.00
19000.00
21000.00
23000.00
25000.00
27000.00
29000.00
7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10
Radius (in)
Ho
op
Str
es
s (
ps
i)
Exact
Elastic Response
• Cylinders under thermal load
rE ti
2 lnb
a
lnb
r
a2
b2
a21
b2
r2
lnb
a
E ti
2 lnb
a
1 ln
b
r
a2
b2
a21
b2
r2
lnb
a
r ti
2 lnb
a
1 ln b
r
a2
b2
a2
1 1 b
2
r2
lnb
a
ti
2 lnb
a
1 1 ln b
r
a2
b2
a2
1 1 b
2
r2
lnb
a
z ti
2 lnb
a
2 1 ln
b
r
2 a
2
b2
a2lnb
a
ur
ti
2 lnb
a
1 1 ln b
r
a2
b2
a2
1 1 b
2
r2
lnb
a
r
Hoop Stress - Plane-Stress
-25000.00
-20000.00
-15000.00
-10000.00
-5000.00
0.00
5000.00
10000.00
15000.00
20000.00
25000.00
7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00
Radius (in)
Ho
op
Str
es
s (
ps
i)
Exact
Elastic Response
• Cylinders under combined load– Superposition
rp a2
b2
a21
b2
r2
E ti
2 lnb
a
lnb
r
a2
b2
a21
b2
r2
lnb
a
Hoop Stress - Plane-Stress
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
7 7.25 7.5 7.75 8 8.25 8.5 8.75 9 9.25 9.5 9.75 10
Radius (in)
Ho
op
Str
es
s (
ps
i)
Exact
ABAQUS
Finite-Element ModelABAQUS used to model 1/16th cylinder.Parameterized input files allow rapidconvergence studies and model variations.
Progess Report
• Elastic response finished– Excellent correlation between analytical and
finite-element solutions
• Elastic-plastic analysis underway
Schedule• Proposal draft (Deliverable) – (COMPLETE)• First progress report (Deliverable) – (COMPLETE)• Elastic response (COMPLETE)
– Both analytical and finite-element solutions– Stresses, strains, and radial displacements– Thin-walled vs. thick-walled cylinders– Pressure loading– Steady-state thermal loading– Combined pressure/thermal loading
• Second progress report (Deliverable) – 11/4• Elastic-plastic response – (IN PROGRESS)
– Both analytical and finite-element solutions– Stresses, strains, and radial displacements– Pressure loading– Steady-state thermal loading– Combined pressure/thermal loading
• Final draft (Deliverable)– 11/25• Final report (Deliverable) – 12/9
References[1] Young, W.C., 1989, Roark’s Formulas for Stress & Strain, 6th Edition, McGraw-Hill, New York, NY.
[2] Avalone, E.A. & Baumeister (III), T, 1987, Marks’ Standard Handbook for Mechanical Engineers, 9th Edition, McGraw-Hill, New York, NY.
[3] Case, J, 1999, Strength of Materials and Structures, 4th Edition, John Wiley & Sons Inc., New York, NY.
[4] Timoshenko, S., 1956, Strength of Material Part II, Advanced Theory and Problems, 3rd Edition, D. Van Nostrand Company Inc., Princeton, NJ.
[5] ABAQUS, v6.7-2, DSS Simulia, Providence, RI.
[6] Hojjarti, M.H. & Hassani, A., 2006, “Theoretical and finite-element modeling of autofrettage process in strain-hardening thick-walled cylinders,” International Journal of Pressure Vessels and Piping, 84 (2007) 310-319.