1. BUCKLING ANALYSIS
OF SANDWICH PIPES
UNDER EXTERNAL PRESSURE
Presented by Rouzbeh Hashemian
Supervisor: Dr. Magdi Mohareb
Fall 2014

2. Outline of the Presentation
• Introduction
• Literature
• Formulation
• Verification
• Results
• Summary and Conclusions
2

3. Typical Sandwich Pipe
External Pipe- Steel (Es)
Core Layer
Soft Material (Ec)
Es/Ec= 200-8000
Internal Pipe- Steel (Es)
3

4. Objective
• Critical External Pressure
Pcr
Pint
4

5. Study
Brush and
Almroth
(1975)
Sato and Patel
(2007)
Arjomandi
and Taheri
(2011)
Abaqus
(CPE8R)
Present Study
5
Most Relevant Studies on Buckling of Sandwich
Pipes Under external pressure
shear
deformation
Steel
pipes
Core
Destabilizing terms
due to
follower
effects
due to pre-buckling
stresses undergoing
nonlinear strains
Contribution
of Internal
Pipe
Captured
Work
Conjugate
Triplets
Multiple
conditions at
pipe-core
interfaces

6. Assumptions
D L>>D
• Problem modelled as 2D-Plane strain system
• Material is considered Linearly Elastic
• External hydrostatic pressure modeled as a Follower Force
• Full bonding is assumed between the layers
A
A'
𝜃
𝜃′ ≠ 𝜃
σ
ε
Constant
6

7. ε 𝜃, σ 𝜃 ε 𝑟, σ 𝑟
𝛾 𝑟𝜃, 𝜏 𝑟𝜃
Strains and Stresses
7
1 0
1 0
(1 )(1 2 ) 1 2
0 0
2
r r
r r
E
 
 
 
 
 
 
  
 
 
            
      
    
 

8. r
𝜃
A
A'
u (r,𝜃) -v (r,𝜃)
Displacement Fields
8
r
𝜃
A
A'
u (r,𝜃) -v (r,𝜃)
   
   
   
2 2
2 2 2 2 2
1 2
11
1
2 2 2
1
r
r
u v
r u v uv vu u v
r u
u
r u u
r u v vv v vu u





  
 
 

     
     

 
 


 

9. Un-Deformed Configuration
Pint
Pextλ
Pre-Buckling Configuration under
Internal Pressure
Pre-Buckling Configuration under
Internal and Reference External Pressure
Onset of BucklingBuckled Configuration
9

10. 2
2
0
21
2
du dv
v u v u
d d
urP d


 
 
   

 

  
 

Total Potential Energy
 
1
2
r r r r
V
dV          
U W  
10
U W

11. 11
Field Equations Boundary Equations
Pre-buckling Closed
form Solution
0p  
 1 2 0b  
Neutral Stability
Conditions
Boundary
Conditions
Finite Difference
Solution
Section
3.6
Section
3.7
Chapter 4

12. 12
Field Equations Boundary Equations
Pre-buckling Closed
form Solution
0p  
 1 2 0b  
Finite Element Discretized form of
Section
5.2
Section
5.3
Assumed Shaped
Functions
       0GK K u 
 Obtain , u

13. Convergence characteristics
1. Finite Element (FEA)-converges from above
2.0
2.4
2.8
3.2
3.6
4.0
0 10 20 30 40 50 60 70 80
Normalized
Critical
Pressure
(Pcr/E)×10-5
Number of elements in FE solution or intervals in FD solution
13
2. Finite Difference –converges from below D/t=40
D
t

14. Mesh Comparison
14
Present Study ABAQUS (CPE8R)
2008 DOFs62 DOFs

15. Verification
15
1
10
100
1000
10000
0.5 0.6 0.7 0.8 0.9
Critical Pressure
/Critical Pressure
of External Pipe
Rint/Rext(Rext/text)=50
(Rext/tint)=25
υc=0.4
Es/Ec=100
Es/Ec=10
Es/Ec=1000
Es/Ec=10000

16. 14
18
22
26
30
34
5 15 25 35
Critical
Pressure
(MPa)
Internal pipe thickness (mm)
Effect of Steel Pipes Thicknesses
16
14
18
22
26
30
34
5 15 25 35
External pipe thickness text (mm)
Es/Ec = 100
Rint = 535 mm
Rext = 813 mm
tint =6 mm
tint =30 mm
text =30 mm
text =6 mm
Buckling CapacityInternal Pipe Thickness Buckling CapacityExternal Pipe Thickness
Critical Pressure
(MPa)

17. Effect of Core Material
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0 100 200 300 400 500
Normalized
Critical
Pressure
(Pcr/Es)
Young Elasticity Modulus Ratio (Es/Ec)
17
Core Stiffness Buckling Capacity
Poisson’s Ratio Buckling Capacity

18. Effect of Core Thickness
18
0
10
20
30
40
50
60 70 80 90 100 110
Normalized
Critical
Pressure
(Pcr/Es)
Radius/Thickness for External Pipe
Rint/tint=61.5
tint/text=0.92
Es/Ec=200
Es/Ec=1000

19. Effect of Internal Pressure
0
20
40
60
80
0 10 20 30 40
Critical
External
Pressure
(MPa)
Internal Pressure (MPa)
19
Internal Pressure Buckling Capacity
,R 01 0,R 01
,R 02 0,R 02
0.886
0.875
cr cr
cr cr
t
int
in
P P P
P P P  
  

20. Summary
1.Features:
– Energy conjugate stress-strain-constitutive parameters
– Polar coordinates
– Uses the orthogonally properties to reduce the computational effort
2. Effects Captured:
– Shear deformation
– Destabilizing terms due to follower effects
– Destabilizing terms due to pre-buckling stresses undergoing nonlinear
strains
20

21. Conclusions
• Buckling Capacity depends upon
21
Core
Young Modulus
Poisson ratio
Thickness
Internal Pipe Thickness
External Pipe Thickness
Internal Pressure
Es/Ec
υc
Rint/Rext
Rint/tint
Rext/text
Pint
Steel Pipes
High
Moderate
Low
High
High
Moderate

22. Acknowledgment
• Dr. Magdi Mohareb
• Dr. Shawn Kenny
• Dr. Beatriz Martin-Perez
• Sepideh Zaghian
• Family and Friends
22

23. Thank You!