Buckling of a Cracked Cylindrical Shell Reinforced
With an Elastic Liner
A THESIS PRESENTED
BY
Yoontae Kim
TO
DEPARTMENT OF MECHANICAL AND INDUSTRIAL
ENGINEERING
IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
NORTHEASTERN UNIVERSITY
BOSTON, MASSACHUSETTS
August 2011
I
ABSTRACT
Shell structures have been widely used in engineering applications such as pipelines,
aerospace and marine structures, and cooling towers. Occurring suddenly and generally
inadvertently due to its nature, buckling is one of the main failure considerations in the
design of these structures. Presence of defects, such as cracks, corrosion pits, blow-out
holes, in shell structures may severely compromise their buckling behavior and
jeopardize the structural integrity.
In this study, a numerical investigation on the buckling behavior of a cracked
cylindrical shell reinforced with an elastic liner and subjected to combined axial
compression was carried out. The effect of supporting liner on the buckling behavior of
the cracked shell at different crack sizes and orientations were investigated. In the next
step, the buckling behavior of a cracked cylindrical shell with an elastic liner subjected to
the internal pressure and axial compression was studied. Different buckling modes of the
cracked shell, including global, transition and locales modes are identified for different
loading conditions.
The results showed that longitudinal crack has a more detrimental effect on the
buckling strength of the cylindrical shell in cylinders with no elastomeric liner or with
elastomeric liners with low relative stiffness. In addition, cylinders with elastomeric
liners of high relative stiffness circumferential crack have a more detrimental effect on
the buckling strength of the cylindrical shell.
II
The finite element analysis also showed that increasing the thickness of the
supporting layer or increasing its stiffness, can significantly increase the critical crack
size at each angle. The shells reinforced with elastic liners subjected to the internal
pressure and axial compression shown that the internal pressure does not affect the
overall buckling behavior of perfect cylindrical shells. For circumferential crack, the
internal pressure increases the buckling load of the cylindrical shell. In contrast, for
longitudinal crack, the internal pressure decreases the buckling load of the cylindrical
shell. We found that the critical buckling load of the cracked cylinder with various
thicknesses of the elastomeric liner can be expressed at each crack angle by a single
parameter, namely stretching stiffness ratio of liner and shell layers.
III
ACKNOWLEDGEMENTS
I wish to express my deep gratitude to Professor Ashkan Vaziri, my thesis supervisor,
for his guidance, constant encouragement, and his valuable advices for two years. I was
able to join his group after my first semester. It means that my research was started later
than other students. But, I could utilize my remaining semesters and make many results in
his group because he always shows me a very inspired way to approach research
problems and achieve those tasks. In addition, I really appreciate his thoughtful
treatments for my school life at Northeastern.
I hope to thank Professor Hamid Nayeb-Hashemi his sharp advices and valuable
suggestions for my better researches. Beside of my research group, I would like to thank
Professor Yung Joon Jung who gave a chance to me for learning of Nano-structures in his
research group. I just want to express my heartfelt gratitude to Professor Tai Joon Um for
all of his supporting in Soonchunhyang University, South Korea.
I also hope to give credit to my lab mates. Although they are younger than me, but
sometimes I felt that they were my teachers, friends and brothers. They helped me out a
lot in my research and shared useful knowledge with me. Truly thanks for the support
from everyone in Northeastern University. This acknowledgement would be incomplete,
if I miss to mention the support I received from my friends.
Finally, I would like to attribute this glory to my father, mother, brother and my wife
for their everlasting love. Without their help and support, I won’t be able to make this
happen. Now I would not be able to do my duty by my family, but I hope that my family
IV
and wife provide me a chance to reciprocate sometime soon after my Ph.D. study.
V
TABLE OF CONTENTS
ABSTRACT………………………………………………………………………..……Ⅰ
TABLE OF CONTENTS………………………………………………………………Ⅴ
LIST OF FIGURES…………………………………………………..…………….…. Ⅵ
CHAPTER 1 – INTRODUCTION
1. Introduction of Cylindrical Shell Structures.……………………….…………………..2
2. Loadings in Cylindrical Shell Structures……………………………………...………..3
3. Failure Modes of Cylindrical Shell Structures…………………………...……………..5
4. Literature Review……………………………………………………………...………..7
5. Objectives...…...……………………………………………………………………....10
6. References……………………………………………………………………..………12
CHAPTER 2 – METHODOLOGY
1. Buckling Theory………………………………………………………………………17
2. Finite Element Model…………………………………………………………………25
3. References……………………………………………………….……………………29
CHAPTER 3 – RESULTS AND DISCUSSION
1. Results and Discussions………………………………………………………….……31
1.1 A Cracked Cylindrical Shell Reinforced with an Elastic Liner under an Axial
Compression……………………………………………………………………………..31
1.2 A Cracked Cylindrical Shell Reinforced with an Elastic Liner under Internal Pressure
and an Axial Compression…..…………………………………………………………...40
4. References……………………………………………………………………….…….44
CHAPTER 4 – Conclusion
1. Conclusion…………………………………………………………………………….46
VI
LIST OF FIGURES
Figure 1. Applications of cylindrical shell structures (A) Aircraft and spacecraft (B)
Marine structures (C) Piping systems ................................................................................. 3
Figure 2. Failure modes of underground pipelines[17] ...................................................... 6
Figure 3. (A) The axi-symmetrical buckling mode shape of a cylindrical shell with an
elastic liner under axial compression. (B) The element cut out of the cylindrical shell by
two pairs of planes parallel to the xy and yz planes. ......................................................... 18
Figure 4. Example of Finite element models. (A) An elastic liner: Perfect cylinder (B) A
cracked cylindrical shell structure (C) A cracked cylindrical shell reinforced with an
elastic liner by Tie constraint in ABAQUS. ..................................................................... 26
Figure 5. Computational models of a cylindrical shell with (A) A circumferential crack
and (B) A longitudinal crack developed by employing a special meshing scheme at the
crack region, proposed by Estekanchi and Vafai [26]. ..................................................... 27
Figure 6. (A) Normalized buckling load of a lined cylindrical shell with a
circumferentially oriented crack (α = 0o) versus the crack length ratio, (a / R), for
different liner to shell thickness ratios, (tp / t). (B) Normalized buckling load of a lined
cylindrical shell with a circumferentially oriented crack (α = 90o) versus the crack length
ratio, (a / R), for different liner to shell thickness ratios, (tp / t). ...................................... 31
Figure 7. Example of the first buckling mode shapes. (A) Global Buckling (B) Local
Buckling (C) Transient ..................................................................................................... 32
Figure 8. (A) Normalized buckling load of a lined cylindrical shell with a circumferential
crack (α = 0o) versus liner to shell Young's modulus ratio, (Ep / E), for different liner to
shell thickness ratios, (tp / t). (B) Normalized buckling load of a lined cylindrical shell
with a circumferential crack (α = 90o) versus liner to shell Young's modulus ratio, (Ep / E),
for different liner to shell thickness ratios, (tp / t). ............................................................ 33
VII
Figure 9. Normalized buckling load of a lined cylindrical shell versus Ep / E for different
crack angles at tp / t = 0.1 and 2.5. .................................................................................... 34
Figure 10. (A) The first buckling mode shapes for different crack orientations (no liner)
(B) The first buckling mode shapes for different crack orientations (reinforced with an
elastic liner) ....................................................................................................................... 35
Figure 11. (A, B) The maps showing the dominance of global, transitional and local
buckling shapes in a lined cylindrical shell with a circumferential (top left) or
longitudinal (top right) crack based on the Young's modulus ratio, (Ep / E), and the
thickness ratio, (tp / t). ....................................................................................................... 36
Figure 12. Critical crack length ratio (ac / R) versus Young's modulus ratio, (Ep / E), for
different thickness ratios, (tp / t). ....................................................................................... 38
Figure 13. (A-C) Normalized buckling loads versus normalized stretching stiffness of the
lined cylindrical shell with a crack for different liner to cylindrical shell thickness ratios,
(tp / t), at the crack orientations of 0°, 30° (top left) , 45°, 60° (top right), and 90°
(bottom). ............................................................................................................................ 39
Figure 14. (A-D) Normalized axial buckling [6]load, γ, of a lined cracked cylindrical
shell under uniform internal pressure versus the normalized loading parameter, λ, for
different liner to shell thickness ratios, (tp / t), at crack angles of α = 0o, 30o, 60o and 90o
........................................................................................................................................... 41
1
CHAPTER 1
INTRODUCTION
2
1. Introduction of Cylindrical Shell Structures
Thin-walled shells are widely used in many industrial components such as
pipelines, air- and space-crafts, marine structures, large dams, shell roofs, liquid-retaining
structures and cooling towers[1, 2]. The examples are shown in Fig 1. Occurring
suddenly and generally inadvertently due to its unique nature, buckling is regarded as one
of the critical failure considerations in the design of shell structures[3, 4]. Presence of
defects and particularly cracks in shell structures may severely compromise their
buckling behavior and undermine the structural stability[5-7]. One conventional approach
for restoring the load carrying capacity of the cracked structures is replacing the cracked
component. However, this approach might be costly and in some cases technically
difficult due to the integrity of the structure, as in pipelines. Therefore, in the recent
decade new technologies have been developed to locally repair the shell structure at the
damaged zone by a supportive adherent liner. Currently this technique is commercially
available in the in situ repair of pipelines (in this technique, a polymeric tube of smaller
diameter is inserted and inflated at the damage part of the pipeline whose length is
slightly larger than the length of the damaged area of the pipeline). However, there is still
a lack of understanding on the mechanical behavior of damaged structures reinforced by
polymeric liners. The study focused on the bifurcation buckling behavior of cracked
cylindrical shells combined elastic liners under multi loading. In this study we will focus
on the buckling behavior elastically lined cylindrical shells with the presence of a crack
as one of the most detrimental kind of defects. First, we will investigate the buckling of a
3
cracked cylindrical shell reinforced with an elastic liner under the action of uniform axial
compression. In the next step, both internal pressure and the uniform axial compression
are applied to the shell structures to understand the effect of the polymeric elastic layer
on the global and local buckling modes of the shell under biaxial loading. This multi-
loading condition also reflect the actual loading conditions common in structures such as
pipelines, aircraft, spacecraft and marine structures. The effect of crack length and
orientation, the elastic liner thickness and Young’s modulus are investigated.
Figure 1. Applications of cylindrical shell structures (A) Aircraft and spacecraft (B)
Marine structures (C) Piping systems
2. Loadings in cylindrical Shell Structures
We discussed briefly in identify the loads that are applied to cylindrical shell
structures to towards understanding the structural performance and reliability of the
cylindrical shell structure, such as piping system. Development of such knowledge
becomes even more critical for studying pipes with material deterioration and pipes
rehabilitated by various techniques, since the residual structural capacity of these pipes
4
are complex function of loading (e.g. internal pressure, frost loads) and materials. In this
study, we applied two kinds of the mechanical loadings to our computational models.
• Internal loads due to operational pressure: The internal operational pressure
applied to pipeline is a major source of hoop stress in the pipe. The contribution
of internal pressure in the pipe to longitudinal stress, although small, may
increase the risk of circumferential breaks when occurring simultaneously with
other mechanical loadings [8].
• Thermal loading: Thermal contraction or expansion of the constrained pipe due
to temperature variation of the water inside the pipe or the pipe surroundings
results in thermal stresses in the pipe, which alter the stress distribution and pipe
mechanical performance.
• Pipe-soil interaction: One of the key mechanical loadings applied to pipes is the
soil weight, which is transferred to the pipe via pipe-soil interaction. The pipe-
soil interaction could induce significant alteration in the level of stresses in the
pipes, both in the in-plane and longitudinal directions. Several empirical and
analytical models have been developed to estimate and study the effect of pipe-
soil interaction on the pipeline response [9-11]. The frost and traffic loads, which
are explained below, are also transferred to the pipe by pipe-soil interaction
based on the general principal discussed above.
• Traffic loads: Traffic load is a dynamic load that is applied to pipes during their
service due to passing of vehicles. The loads are affected to the pipeline by pipe-
5
soil interaction, which is governed by the soil and pipe material and pipe burial
depth [12, 13].
• Frost loads: Frost load is often corresponds to the increase in earth loads
(geostatic vertical stress) as a consequence of frost penetration, which results in
an increase in overall load exerted on underground pipes. It is well-established
that frost load is an important loading condition for underground pipes. In fact,
analysis of available data on the performance of water main indicates that water
main failures are greatly influenced by seasonal climate, i.e. a huge increase in
the number of breaks in the pipeline system is observed with the drop in seasonal
temperatures in cold regions [14, 15].
3. Failure modes of Cylindrical Shell Structures
The mechanisms that lead to pipe breakage are often very complex and not
completely understood. Pipe breakage types were classified by O’Day et al. [16] into
three categories: (1) Circumferential breaks, caused by longitudinal stresses; (2)
longitudinal breaks, caused by transverse stresses (hoop stress); and (3) split bell, caused
by transverse stresses on the pipe joint.
Makar et al. [17] introduced other possible failure modes. Figure 2 shows the
schematic pictures of these failure modes. A brief discussion of each of these failure
modes are also provided below. In this study, the cracked cylindrical shell structures of
the computational models are generated base on those failure modes.
6
Figure 2. Failure modes of underground pipelines[17]
• Circumferential cracking: Circumferential cracking in pipes occurs due to
excessive longitudinal stresses, which are generally due to: (1) thermal
contraction acting on a restrained pipe, (2) bending stresses due to soil
differential movement (especially clayey soils) or large voids in the bedding
near the pipe (resulting from leaks), (3) inadequate trench and bedding practices,
and (4) third party interference (e.g., accidental breaks, etc.)[16, 17].
• Longitudinal cracking: Longitudinal cracking is common in large diameter
pipes. Longitudinal breaks due to transverse stresses are typically the result of
one or more of the following factors: (1) hoop stress due to internal pressure, (2)
ring stress due to soil weight and pipe-soil interaction, (3) ring stress due to live
Circumferential cracking Longitudinal cracking
Bell splitting Corrosion pitting
Blow-out hole. Spiral cracking
7
loads caused by traffic, and (4) increase in ring loads when penetrating frost
causes the expansion of frozen moisture in the ground [16-18].
• Bell splitting: Bell splitting is most common in small diameter cast iron pipes
[16, 17]. The main reason for bell splitting is the sealing of the joints. The
difference of lead and sulphur in coefficient of thermal expansion may cause bell
splitting.
• Corrosion pitting and blow-out holes: A corrosion pit reduces the thickness and
mechanical resistance of the pipe wall. When the wall is thinned to a certain
point, the internal pressure blows out a hole. The size of the hole depends on the
distribution of corrosion and the pressure in the pipe. In some cases a blow-out
hole can be fairly large.
• Spiral cracking: Spiral cracking is a rather unique failure mode that sometimes
occurs in medium diameter pipes [17, 18]. The initially circumferential crack
propagates along the length of the pipe in a spiral fashion. Historically, this type
of failure is associated with pressure surges, but it can also be related to a
combination of bending force and internal pressure.
4. Literature Review
Shell structures have been widely used in engineering applications such as pipelines,
aerospace and marine structures, and cooling towers, so the pipelines structures requires
deep studies on the cylindrical shell which is widely used in the world. Occurring
8
suddenly and generally inadvertently due to its nature, buckling and vibration are the
main failure considerations in the design of these structures, so in-depth researches were
on this filed[19, 20]. The dynamic stability[21] or parametric resonance[22] of a
cylindrical shell under periodic loads have also been introduced. Presence of defects,
such as cracks, may severely jeopardize integrity and effect the vibration, buckling and
dynamic stability of the structures[23]. Then some researchers used finite element
method[24] to analyze the vibration behavior of cylindrical shells. Padovan[25] first used
the finite element method obtaining the frequency and buckling eigenvalue of pre-
stressed rotating anisotropic shells of revolution.
The numerical approach for solving of the problem seems to be the most
promising method, as the analytical formulation of the problem is formidably
complicated. Even in numerical analysis, the large number of interacting parameters and
the complicated shell buckling behavior makes it quite difficult to ascertain generally
applicable conclusions [6, 26]. Computational models of cracked cylindrical shells with
various crack lengths and orientations are developed by employing the meshing scheme
proposed by Estekanchi and Vafai [26] for cracked thin plates and shells. The suitability
of this meshing scheme for the present study is examined by comparing the numerical
results obtained using the proposed meshing scheme with those obtained from finite
element models with eight-node quadratic and six-node triangular elements with crack tip
singularity[27].
9
Buckling is another main consideration of cylindrical shell structures and many studies
have focused on it. Von Karman and Tsien[28, 29] derived a buckling load expression
and analyzed the post-buckling equilibrium path of an axially compressed thin
homogeneous cylindrical shell. Shen and Chen[30] investigated the buckling and post-
buckling of cylindrical shells under combined loading of external pressure and axial
compression. In the post-buckling analysis, deformation of cracked plates and shells
could cause a considerable amplification of the stress intensity around the crack tip [31].
On the other hand, increasing the load can cause propagation of the local buckling
leading to the catastrophic failure of the structure [32]. Reviews of the research
conducted in the context of buckling behavior of defected plates and shells are presented
in references[26, 33, 34].
Li and Batra[35] used Flugge’s theory and a semi-inverse method to studied the
buckling of a simply supported three-layer circular cylindrical shell under axial
compressive load, and they found that the thin shell buckles in an axisymmetric mode. El
Naschie[6] considered the buckling problem of a cracked shell for the first time.
Vaziri[36] found that the composite cylindrical shells is particularly sensitive to the
presence of cracks, which may develop during manufacturing or service life of composite
cylindrical shells, the crack could severely affect the buckling behavior of structures not
only by reducing their load carrying capacity but also by introducing local buckling at the
crack region. Vaziri and Estekanchi[27] considered the buckling of crack cylindrical shell
10
subjected to combined internal pressure and axial compression, they studied the effect of
crack type, size and orientation on buckling behavior of that structures.
The effect of crack orientation and internal pressure on the buckling behavior of
the cracked shells without elastic liners was studied in [Vaziri, add some more references
please]. It was shown that the internal pressure, in general, increases the buckling load
associated with the global buckling mode of the cylindrical shells. In contrast, the effect
of internal pressure on buckling loads associated with the local buckling modes of the
cylindrical shell depends mainly on the crack orientation[27, 37].
5. Objectives
The objective of this project is to examine the structural capacity of the
cylindrical shell reinforced with elastic liners and to provide a more in-depth
understanding of the validity and practicality of the relining process materials and
techniques and their limitations. To clarify the motivation for this study, let me tell you
that supporting the cracked cylindrical shells with an elastomeric/polymeric liner is an
established way of restoring the mechanical behavior of cracked pipelines and shells.
In this study, we investigated the effect of supporting polymeric liner on the
buckling behavior of the cracked shell at different crack sizes and orientations under the
application of axial force and internal pressure. Firstly, we introduces a theoretical
relationship for buckling of a perfect cylindrical composite shell which is used to
normalize the buckling load obtained from computational models of cracked, liner
11
reinforced cylindrical shells. Then, we discussed the computational models of cracked
shells for the parametric linear eigenvalue analysis on the buckling behavior of the
cracked cylinder over a wide range of crack size and orientation and reinforcement
properties. This part of our study complements previous investigations on the buckling
behavior of combined cylindrical shells combined elastic liners under the uniform axial
compression. Finally, we studied the effect of internal pressure on the buckling behavior
of cracks reinforced structure.
12
6. References
[1] M. Farshad, “Design and Analysis of Shell Structures.,” Kluwer Academic, 1992.
[2] M. W. Hilburger, “Buckling and Failure of Compression-Loaded Composite
Laminated Shells with Cutouts, “ 2007.
[3] S. Sallam, G. J. Simitses, “Delamination Buckling of Cylindrical Shells under Axial
Compression.,” Composite Structures, vol. 7, pp. 83-101, 1987.
[4] G. J. Lord, “Computation of Localized Post Buckling in Long Axially Compressed
Cylindrical Shells.,” Mathematical, Physical and Engineering Sciences vol. 355, pp.
2137-2150, 1997.
[5] J. W. Hutchinson, D. B. Muggeridge, R. C. Tennyson, “The Effect of a Local
Axisymmetric Imperfection on the Buckling Behaviour of a Circular Cylindrical
Shell under Axial Compression.,” AIAA Journal, vol. 9, pp. 48-52, 1971.
[6] El Naschie, “A Branching Solution for the Local Buckling of a Circumferentially
Cracked Cylindrical Shell.,” International Journal of Mechanical Sciences, vol. 16,
pp. 689-697, 1974.
[7] A. Barut, et al., “Buckling of a Thin, Tension-Loaded, Composite Plate with an
Inclined Crack.,” Engineering Fracture Mechanics, vol. 58, pp. 233-248, 1997.
[8] http://www.undergroundsolutions.com/duraliner-operating-pressures.php.
[9] M. G. Spangler, “The Structural Design of Flexible Pipe Culverts.,” Iowa
Engineering Experimental Station, vol. 17, pp. 235-239, 1941.
[10] R. Watkins, “Some Characteristics of the Modulus of Passive Resistance of Soil: a
Study in Similitude.,” Highway Research Board vol. 37, pp. 576-583, 1958.
[11] B. Rajani, S. Kuraoka, “Pipe-soil Interaction Analysis of Jointed Water Mains.,”
Canadian Geotechnical Journal, vol. 33, pp. 393-404, 1996.
13
[12] M. M. Patel, A.P. Janssen, R. R. Tardif, M. Herring, U. D. Parashar, “Traffic Load
Modelling and Factors Influencing the Accuracy of Predicted Extremes.,” BMC
Pediatr, vol. 7, pp. 270-278, 2007.
[13] M. Zhan, “Load Transfer Analysis of Buried Pipe in Different Backfills.,” Journal of
Transportation Engineering, vol. 123, pp. 447-453, 1997.
[14] R. E. Morris, “Principal Causes and Remedies of Water Main Breaks.,” Journal of
the American Water Works Association, vol. 59, pp. 782-798, 1967.
[15] A. S. Ciottoni, “Computerized Data Management in Determining Causes of Water
Main Breaks: The Philadelphia Case Study.,” Proceedings of the International
Symposium on Urban Hydrology, 1983.
[16] D. O’day, “Organizing and Analyzing Leak and Break Data for Making Main
Replacement Decisions.,” Journal of American Water Works Association, vol. 74, pp.
588-594, 1982.
[17] R. D. J. Makar, S. McDonald, “Failure Modes and Mechanisms in Gray Cast Iron
Pipe, Underground Infrastructure Research; Municipal, Industrial and Environmental
Applications.,” Kitchener, vol. 10, pp. 1-10, 2001.
[18] Rajani, et al., “Comprehensive Review of Structural Deterioration of Water Mains:
Physically Based Models.,” Oxford, ROYAUME-UNI: Elsevier, vol. 3, 2001.
[19] R. N. Arnold, “Flexural Vibrations of the Walls of Thin Cylindrical Shells Having
Freely Supported Ends.,” Proceedings of the Royal Society, vol. A197, pp. 238-256,
1949.
[20] H. J. Budiansky, “Buckling of Circular Cylindrical Shells under Axial Compression.,”
Delft University Press, pp. 239-260, 1972.
[21] V. V. Bolotin, “The Dynamic Stability of Elastic Systems.,” American Journal of
Physics vol. 33, p. 752, 1965.
14
[22] S. K. Sahu, “Parametric Resonance Characteristics of Laminated Composite Doubly
Curved Shells Subjected to Non-Uniform Loading.,” Journal of Reinforced Plastics
and Composites, vol. 20, pp. 1556-1576, 2001.
[23] M. Javidruzi, et al., “Vibration, Buckling and Dynamic Stability of Cracked
Cylindrical Shells.,” Thin-Walled Structures, vol. 42, pp. 79-99, 2004.
[24] G. M. L. Gladwell, D. K. Vijay, “Natural Frequencies of Free Finite-Length Circular
Cylinders.,” Journal of Sound and Vibration, vol. 42, pp. 387-397, 1975.
[25] J. Padovan, “Traveling Waves Vibrations and Buckling of Rotating Anisotropic
Shells of Revolution by Finite Elements.,” International Journal of Solids and
Structures, vol. 11, pp. 1367-1380, 1975.
[26] H. E. Estekanchi, A. Vafai “On the Buckling of Cylindrical Shells with Through
Cracks Under Axial Load.,” Thin-Walled Structures, vol. 35, pp. 255-274, 1999.
[27] A. Vaziri, H. E. Estekanchi, “Buckling of Cracked Cylindrical Thin Shells under
Combined Internal Rressure and Axial Compression.,” Thin-Walled Structures, vol.
44, pp. 141-151, 2006.
[28] Y. Chen, et al., “Vibrations Of High Speed Rotating Shells with Calculations For
Cylindrical Shells.,” Journal of Sound and Vibration, vol. 160, pp. 137-160, 1993.
[29] P. Mandal and C. R. Calladine, “Buckling of Thin Cylindrical Shells under Axial
Compression.,” International Journal of Solids and Structures, vol. 37, pp. 4509-
4525, 2000.
[30] H. S. Shen, T. Y. Chen, “Buckling and Postbuckling Behaviour of Cylindrical Shells
under Combined External Pressure and Axial Compression.,” Thin-Walled Structures,
vol. 12, pp. 321-334, 1991.
[31] E. Riks, et al., “The Buckling Behavior of a Central Crack in a Plate under Tension.,”
Engineering Fracture Mechanics, vol. 43, pp. 529-548, 1992.
15
[32] E. Chater, J. W. Hutchinson, “On the Propagation of Bulges and Buckles.,” Journal
of Applied Mechanics, vol. 51, pp. 269-277, 1984.
[33] A. Khamlichi, et al., “Buckling of Elastic Cylindrical Shells Considering the Effect
of Localized Axisymmetric Imperfections.,” Thin-Walled Structures, vol. 42, pp.
1035-1047, 2004.
[34] M. W. Hilburger, J. H. Starnes, “Effects of Imperfections of the Buckling Response
of Composite Shells.,” Thin-Walled Structures, vol. 42, pp. 369-397, 2004.
[35] S. R. Li, R. C. Batra, “Buckling of Axially Compressed Thin Cylindrical Shells with
Functionally Graded Middle Layer.,” Thin-Walled Structures, vol. 44, pp. 1039-1047,
2006.
[36] A. Vaziri, “On the Buckling of Cracked Composite Cylindrical Shells under Axial
Compression.,” Composite Structures, vol. 80, pp. 152-158, 2007.
[37] A. Vaziri, et al., “Buckling of the Composite Cracked Cylindrical Shells Subjected to
Axial Load.,” ASME Conference Proceedings, vol. 2003, pp. 87-93, 2003.
[38] R. D. Cook, “Concepts and applications of finite element analysis.,” Wiley, 1981.
16
CHAPTER 2
METHODOLOGY
17
1. Buckling Theory
The first step in analytically estimating the buckling load of a cylindrical shell is
to find the neutral axis of the composite shell. Consider the composite shell which
deforms according to the Euler-Bernoulli beam deformation requirements (i.e. planes
perpendicular to the neutral axis of beam remain planar). Setting the net axial force per
unit of the shell equal to zero,
�𝐸𝑦𝑑𝑦.
𝑦= � 𝐸𝑝(𝑧 − 𝑡̅)𝑑𝑧
𝑡𝑝
0+ � 𝐸(𝑧 − 𝑡̅)𝑑𝑧 = 0
𝑡
𝑡𝑝
The location of the neutral axis of the cylindrical shell is obtained, with the 𝑡̅ and
�̅� parameters, shown in Fig. 3(B), are obtained as follows:
𝑡̅ =𝑡𝑝2 × 𝐸𝑝𝑡𝑝 + (𝑡𝑝 + 𝑡
2) × 𝐸𝑡𝐸𝑡 + 𝐸𝑝𝑡𝑝
�̅� =𝑡2 × 𝐸𝑡 + (−
𝑡𝑝2 ) × 𝐸𝑝𝑡𝑝
𝐸𝑡 + 𝐸𝑝𝑡𝑝
The bending moment per unit length of the edges parallel to the y axis is denoted
by 𝑀𝑥, and the moment per unit length of the edges parallel to the x axis is 𝑀𝑦. These
moments are considered positive when they produce compression at the upper surface of
the plate and tension at the lower. These thicknesses of the cracked shell and the elastic
liner are denoted by 𝑡, 𝑡𝑝, respectively. In general, the amount of thicknesses is assumed
too small in comparison with the other dimensions. Let us consider an element cut out of
the cylindrical shell by two pairs of planes as shown in Fig. 3(b). Since the thickness of
the shell is very small, the lateral sides of the element can be considered as rectangles.
Moreover, assuming that during bending of the element lateral sides of this element
18
remain plane and rotate about the neutral axes n-n so as to remain normal to the
deflection surface, it can be concluded that the middle plane of the element does not
undergo any deformation during this bending and is therefore a neutral surface. In
addition, there is no strain in the middle plane of a plate during bending is usually
sufficiently accurate as long as the deflections of the plate are small in comparison with
its thickness t.
Figure 3. (A) The axi-symmetrical buckling mode shape of a cylindrical shell with an
elastic liner under axial compression. (B) The element cut out of the cylindrical shell by
two pairs of planes parallel to the xy and yz planes.
Let 1 𝜌𝑥⁄ and 1 𝜌𝑦⁄ denote the curvatures of this neutral surface in sections
parallel to the zx and yz planes, respectively, with positive curvature corresponding to
bending which is convex down. Then the unit elongations in the x and y directions of an
l
19
elemental lamina at distance 𝑧 from the neutral surface can be found as in the case of a
beam. Therefore, we obtain
𝜀𝑥 =𝑧𝜌𝑥
𝜀𝑦 =𝑧𝜌𝑦
From Hook’s law we get these relations
𝜀𝑥 =1𝐸�𝜎𝑥 − 𝑣𝜎𝑦� 𝜀𝑦 =
1𝐸�𝜎𝑦 − 𝑣𝜎𝑥�
where v is Poission’s ratio, and therefore the corresponding stresses in the lamia is
𝜎𝑥 =𝐸𝑧
1 − 𝑣2�
1𝜌𝑥
+ 𝑣1𝜌𝑦� 𝜎𝑦 =
𝐸𝑧1 − 𝑣2
�1𝜌𝑦
+ 𝑣1𝜌𝑥�
The stresses are proportional to the distance z of the lamina abcd from the neutral
surface and depend upon the magnitude of the curvatures of the bent plate. Moreover,
bending moment per unit length of the edge can be obtained below,
𝑀𝑥 = �𝜎𝑥𝑧𝑑𝑧 =.
𝑦�
𝐸𝑧2
1 − 𝑣2�
1𝜌𝑥
+ 𝑣1𝜌𝑦�𝑑𝑧
.
𝑦
= �𝐸𝑧2
1 − 𝑣2�
1𝜌𝑥
+ 𝑣1𝜌𝑦�𝑑𝑧
𝑡𝑝
0+ �
𝐸𝑝𝑧2
1 − 𝑣2�
1𝜌𝑥
+ 𝑣1𝜌𝑦�𝑑𝑧
𝑡
𝑡𝑝
=1
1 − 𝑣2�
1𝜌𝑥
+ 𝑣1𝜌𝑦�𝐸𝐼0 +
11 − 𝑣2
�1𝜌𝑥
+ 𝑣1𝜌𝑦�𝐸𝑝𝐼𝑝0
Where this quantity of D is called the flexural rigidity of the cylindrical shell combined
the elastic liner.
20
𝐷 =1
1 − 𝑣2(𝐸𝐼0 + 𝐸𝑝𝐼𝑝0)
We mentioned the thickness of the shell structure is very small above, the lateral
sides of the element can be considered as rectangles; hence the resultant forces will act in
the middle surface of the shell. Using the same notations as in the case of plates, we
obtain
𝑁𝑥 = � 𝜎𝑥𝑑𝑧 =𝐸
1 − 𝑣2�𝜀 + 𝑣𝜀𝑝�𝑑𝑧
𝑡/2
−𝑡/2= (𝐸𝑡 + 𝐸𝑝𝑡𝑝)
𝜀 + 𝑣𝜀𝑝1 − 𝑣2
𝑁𝑦 = � 𝜎𝑦𝑑𝑧 =𝐸
1 − 𝑣2�𝑣𝜀 + 𝜀𝑝�𝑑𝑧
𝑡/2
−𝑡/2= (𝐸𝑡 + 𝐸𝑝𝑡𝑝)
𝑣𝜀 + 𝜀𝑝1 − 𝑣2
And also
𝑀𝑥 = � 𝜎𝑥𝑧𝑑𝑧 = −𝐷�𝑥𝑥 + 𝑣𝑥𝑦�𝑑𝑧𝑡/2
−𝑡/2
𝑀𝑦 = � 𝜎𝑦𝑧𝑑𝑧 = −𝐷�𝑥𝑦 + 𝑣𝑥𝑥�𝑑𝑧𝑡/2
−𝑡/2
Where D has the same meaning as mentioned above and denotes the flexural rigidity of
the shell structures.
The relation between the shearing stress τxy and the twisting of the element
ABCD can be established exactly in the same method as in the case of an element cut out
from a plate; in this way we obtain
𝜏𝑥𝑦 = −2𝐺𝑧𝑥𝑥𝑦 𝑀𝑥𝑦 = 𝐷(1 − 𝑣)𝑥𝑥𝑦
21
where, 𝑥𝑥𝑦 takes the place of 𝜕2𝑥 𝜕𝑥𝜕𝑦⁄ in the case of a plate and represents the twist of
the element ABCD during bending of the shell, so that 𝑥𝑥𝑦𝑑𝑥 is the rotation of the edge
BC relative to Oz with respect to the x-axis. If, in addition to twist, there is a shearing
strain γ in the middle surface of the shell, we obtain
𝜏𝑥𝑦 = (𝛾 − 2𝑧𝑥𝑥𝑦)𝐺
𝑁𝑥𝑦 = � 𝜏𝑥𝑦𝑑𝑧 =+𝑡/2
−𝑡/2
𝛾𝑡𝐸2(1 + 𝑣)
𝑀𝑥𝑦 = −� 𝜏𝑥𝑦𝑧𝑑𝑧 =+𝑡 2⁄
−𝑡 2⁄𝐷(1 − 𝑣)𝑥𝑥𝑦
Thus assuming that during bending of a shell the linear elements normal to the
middle surface remain straight and become normal to the deformed middle surface, we
can express the resultant forces 𝑁𝑥, 𝑁𝑦, and 𝑁𝑥𝑦 and the moments 𝑀𝑥, 𝑀𝑦, and 𝑀𝑥𝑦 in
terms of six quantities: the three components of strain 𝜀1, 𝜀2, and γ of the middle surface
of the shell and the three quantities 𝑥𝑥, 𝑥𝑦, and 𝑥𝑥𝑦 representing the changes of curvature
and the twist of the middle surface.
The strain energy of a deformed shell consists of two parts: (1) the strain energy
due to bending and (2) the strain energy due to stretching of the middle surface. For the
first part of this energy we can use two formulas below
𝑈 =12𝐷��(
𝜕2𝑤𝜕𝑥2
)2 + (𝜕2𝑤𝜕𝑦2
)2 + 2𝑣𝜕2𝑤𝜕𝑥2
∙𝜕2𝑤𝜕𝑦2
+ 2(1 − 𝑣)(𝜕2𝑤𝜕𝑥𝜕𝑦
)2� 𝑑𝑥𝑑𝑦
22
𝑈 =12𝐷���
𝜕2𝑤𝜕𝑥2
+𝜕2𝑤𝜕𝑦2
�2.
− 2(1 − 𝑣) �𝜕2𝑤𝜕𝑥2
∙𝜕2𝑤𝜕𝑦2
+ �𝜕2𝑤𝜕𝑥𝜕𝑦
�2
�� 𝑑𝑥𝑑𝑦
Substituting in it the changes of curvature 𝑥𝑥 , 𝑥𝑦 , and 𝑥𝑥𝑦 , instead of curvatures
𝜕2𝑤 𝜕𝑥2⁄ , 𝜕2𝑤 𝜕𝑦2⁄ , and 𝜕2𝑤 𝜕𝑥⁄ 𝜕𝑦, we obtain
U1 = 12
D∫[ (𝑥𝑥 + 𝑥𝑦)2 − 2(1 − 𝑣)(𝑥𝑥𝑥𝑦 − 𝑥𝑥𝑦2 )]𝑑𝐴 (1)
where the integration should be extended over the entire surface of the shell. That part of
the energy due to stretching of the middle surface is
U2 = Et2(1−𝑣2)∫∫(𝜀 + 𝜀𝑝)2 − 2(1 − 𝑣)(𝜀𝜀𝑝 −
14𝛾.2)]𝑑𝐴 (2)
The total energy of deformation is gained by adding together expressions (1) and (2). We
assume for radial displacements during buckling the expression
𝑤 = −𝐴𝑠𝑖𝑛 𝑚𝜋𝑥𝑙
(a)
Where, h is the length of the cylinder. The strains ε1 and ε2 in the longitudinal and
circumferential directions after buckling will be found from the conditions that the axial
compressive forces during buckling remain constant. Using for the axial strain before
buckling the notation
ε0 = −𝑁𝑐𝑟𝐸𝑡
(b)
Where t is the thickness of the shell, we obtained
23
𝜀2 = −𝑣𝜀0 −𝑤𝑎
= −𝑣𝜀0 + 𝐴𝑎𝑠𝑖𝑛 𝑚𝜋𝑥
𝑙 (c)
We know that from the references
ε1 = ε0 − 𝑣 𝐴𝑎𝑠𝑖𝑛 𝑚𝜋𝑥
𝑙 (d)
The change of curvature in the axial plane is
𝑥𝑥 = 𝜕2𝑤𝜕𝑥2
= 𝐴𝑚2𝜋2
𝑙2𝑠𝑖𝑛 𝑚𝜋𝑥
𝑙 (e)
Substituting expressions (c), (d), and (e) in Eqs. (1) and (2) for strain energy and noting
that, owing to symmetry of deformation,
𝛾 = 𝑥𝑦 = 𝑥𝑥𝑦 = 0
We gained for the increase of strain energy during buckling the following expression:
∆U = −2π�𝐸𝑡 + 𝐸𝑝𝑡𝑝�𝑣𝜀0 ∫ 𝐴 sin𝑚𝜋𝑥𝑙𝑑𝑥 + πA2�𝐸𝑡+𝐸𝑝𝑡𝑝�𝑙
2𝑎+ A2 π4𝑚4
2𝑙4𝑙0 𝜋𝑎𝑙𝐷 (f)
The work done by compressive force during buckling is
ΔT = 2πNcr(v∫ Al0 𝑠𝑖𝑛 𝑚𝜋𝑥𝑙𝑑𝑥 + 𝑎
4𝐴2 𝑚
2π2
𝑙) (g)
Where the first term in the parentheses is due to the change ε1 − ε0 of the axial strain and
the second term is due to bending of the generators given by Eq.(a). Equating expressions
(f) and (g), we obtain
𝜎𝑐𝑟 = 𝑁𝑐𝑟𝑡
= 𝐷 �𝑚2π2
ℎ𝑙2+ 𝐸𝑡+𝐸𝑝𝑡𝑝
ℎ𝑎2𝐷∙ 𝑙2
𝑚2π2� (h)
24
Assuming that there are many waves formed along the length of the cylinder
during buckling and considering 𝜎𝑐𝑟 as a continuous function of 𝑚𝜋 𝑙⁄ , we find that the
minimum value of expression (h) is
𝜎𝑐𝑟 =2𝑎𝑡��𝐸𝑡 + 𝐸𝑝𝑡𝑝�𝐷 =
2𝑎𝑡��𝐸𝑡 + 𝐸𝑝𝑡𝑝� ∙ (𝐸𝐼0 + 𝐸𝑝𝐼𝑝0)
1 − 𝑣2
𝑁𝑐𝑟 =2𝑎𝑡��𝐸𝑡 + 𝐸𝑝𝑡𝑝� ∙ �
𝐸𝑡312 +
𝐸𝑝𝑡𝑝312 �
1 − 𝑣2× 2𝜋𝑡𝑎
=1
�3(1 − 𝑣2)∙ ��𝐸𝑡 + 𝐸𝑝𝑡𝑝� × (𝐸𝑡3 + 𝐸𝑝𝑡𝑝3)
And occurs at
𝑚𝜋𝑙
= �𝐸𝑡 + 𝐸𝑝𝑡𝑝𝑎2𝐷
4
The results presented in next part are based on bifurcation buckling analysis of
cracked cylindrical shells under compression and internal pressure. We applied the basic
theories to our computational models which are reinforced by inserting an elastic liner
into the cracked cylindrical shell. A normalized loading parameter, 𝜆, is defined as the
ratio of the induced constant hoop stress due to the internal pressure in an uncracked
cylindrical shell,
𝜆 =2𝜋𝑅2𝑃𝐹
25
2. Finite Elements Models
In the computational modeling of the lined cracked shells, we assumed that
cracked cylindrical shells and elastic liners have isotropic, linear elastic material behavior
to fulfill eigenvalue buckling analysis. The cylindrical shell is corresponding to the 3000
series aluminum alloy with the Young’s modulus, E = 69 GPa, and Poisson’s ratio, ν =
0.35. The computational model of cracked cylindrical shells has L = 1.0 m, R = 0.2 m
and t = 1.2 mm, where L, R and t denote overall length, radius and thickness of the
cylindrical shell, respectively.
For elastic liners, we assumed that these shell structures do not have any defect on
their surface, perfect cylindrical shell structures, which have same Poission’s ratio,
overall length, radius that of the cracked cylindrical shell. In addition, various material
properties such as Young’s modulus and thickness are applied to the elastic liners in the
computational modeling. Those computational models which were used in ABAQUS are
shown in Fig. 4(A)-(C).
The constraint conditions are decided through the real-life application. Few
companies are using the special technology to repair a cracked pipe by inserting a snug
PVC pipe into inside of the cracked pipe. Based on this example, we assumed that the
cylindrical shell and the elastic liner are perfectly combined by tie constraint conditions.
The combined computational model is shown in Fig. 4(C). Therefore, the one end of the
cracked cylindrical shell and the elastic liner were fixed by tie constraint condition and
26
was able to move only in the axial direction to the other end to apply an axial
compression conditions.
Figure 4. Example of Finite element models. (A) An elastic liner: Perfect cylinder (B) A
cracked cylindrical shell structure (C) A cracked cylindrical shell reinforced with an
elastic liner by Tie constraint in ABAQUS.
The meshing scheme suggested by Estekanchi and Vafaie [26] was used for
modeling of the crack zone and the example of the finite element mesh developed based
on this meshing scheme is shown in Fig 5(A) and (B). In this meshing method, the
element size is incrementally reduced at the crack zone from the element size employed
in meshing the cylinder to the crack tip. Moreover, the singularity of stresses in the crack
tip zone is well captured and the amount of computational complexity is decreased
considerably[38] .
27
Figure 5. Computational models of a cylindrical shell with (A) a circumferential crack
and (B) a longitudinal crack developed by employing a special meshing scheme at the
crack region, proposed by Estekanchi and Vafai [26].
We have developed a Matlab code that help automatic creation of the finite
element model of cracked cylindrical shells with different crack lengths, a, and
orientations, α (where from the circumferential direction to the longitudinal direction
corresponding to α = 0°–90°) shortly. The eight-node doubly curved thick shell elements
with reduced integration were used for meshing of the computational models to make a
densely meshed zone. For the meshing of crack zone, the zooming step of 6 and the
zooming ratio of 1/2 were applied, meaning the element length is reduced by 50% of
posterior element length in each step. This meshing scheme was starting from the
uncracked zone approaching the crack tip. As the result, the crack tip of the element
length is 1/64 of the element length of the uncracked zone. Fig. 5 demonstrates the
28
concept of the meshing procedure used for meshing of the crack zone in this study,
however, the actual mesh quality used in the computational calculations was much better
than the meshing shown in Fig. 5(A) and (B).
The buckling mode shapes and buckling loads of the cracked cylindrical shell
combined with the elastic liner were obtained by the linear eigenvalue analysis. By using
this analysis method, we are able to get the bifurcation point of isotropic and linear elastic
material structures theoretically. In addition, the analysis time is comparatively short and
the result of buckling modes can be used for nonlinear buckling analysis as an initial
imperfection. The calculated critical buckling load of the cracked shell combined with the
elastic liner, Fcrack, was normalized by the theoretical buckling load of the perfect
cylinder reinforced with the elastic liner which has similar geometry subject to an axial
compression, Ftheory
γ =𝐹𝑐𝑟𝑎𝑐𝑘𝑁𝐹𝑡ℎ𝑒𝑜𝑟𝑦
and γ denotes the normalized buckling load of elastic liner-reinforced cylindrical shells
with crack.
29
3. References
[27] A. Vaziri and H. E. Estekanchi, “Buckling of cracked cylindrical thin shells under
combined internal pressure and axial compression.,” Thin-Walled Structures, vol. 44,
pp. 141-151, 2006.
[37] A. Vaziri, et al., “Buckling of the Composite Cracked Cylindrical Shells Subjected to
Axial Load.,” ASME Conference Proceedings, vol. 2003, pp. 87-93, 2003.
[38] R. D. Cook, “Concepts and applications of finite element analysis..,” Wiley, 1981.
30
CHAPTER 3
RESULTS AND DISCUSSION
31
1. Results and Discussions
1.1 A Cracked Cylindrical Shell Reinforced with an Elastic Liner under an Axial
Compression
Fig. 6(A) and (B) show the normalized buckling load of elastic liner-reinforced
cylindrical shells with a circumferential crack (α = 0o) and a longitudinal crack(α = 90o),
respectively, for different elastic liner thickness to the cracked cylindrical shell thickness
ratio, tp / t. In this case, we assumed that elastic liners, in general, have lower Young’s
modulus than the cracked cylindrical shell structures.
Figure 6. (A) Normalized buckling load of a lined cylindrical shell with a
circumferentially oriented crack (α = 0o) versus the crack length ratio, (a / R), for
different liner to shell thickness ratios, (tp / t). (B) Normalized buckling load of a lined
cylindrical shell with a circumferentially oriented crack (α = 90o) versus the crack length
ratio, (a / R), for different liner to shell thickness ratios, (tp / t).
32
We checked the first buckling mode shapes via the computational analysis. As
the result, we could find three kinds of buckling mode shape. Those buckling mode
shapes are shown in Fig. 7(A)-(C). The global buckling is generated from the perfect
cylindrical shell structure in general. In contrast, both local and transient buckling modes
are generated from cracked cylindrical shells.
Figure 7. Example of the first buckling mode shapes. (A) Global Buckling (B) Local Buckling (C) Transient
The short cracked cylindrical shell reinforced with elastic liner shows the short
crack lengths could not change the buckling behavior of this reinforced shell structures;
the first buckling mode shape was occurred global buckling at γ ≅ 1 under specific
Young’s modulus ratio, Ep / Ε = 0.01. In contrast, the larger cracked cylindrical shells
combined with elastic liner appeared the local buckling at the crack zone under an axial
compression loading. In addition, the amount of the first buckling load are sharply
decreased, compare to that of the perfect cylindrical shell. Moreover, these results show
thinner structure which is coupled the cylindrical shell and the elastic liner is more
33
sensitive to the presence of a crack length. In general, the longitudinal cracked shells
have a wider range of the normalized buckling loads than that of the circumferential
cracked shells. Normalized buckling loads are well classified around a / R = 0.2 at the
longitudinal and circumferential cracked shells. Therefore, the further researches were
focused on a / R = 0.2 cracked models.
Figure 8. (A) Normalized buckling load of a lined cylindrical shell with a circumferential
crack (α = 0o) versus liner to shell Young's modulus ratio, (Ep / E), for different liner to
shell thickness ratios, (tp / t). (B) Normalized buckling load of a lined cylindrical shell
with a circumferential crack (α = 90o) versus liner to shell Young's modulus ratio, (Ep / E),
for different liner to shell thickness ratios, (tp / t).
Fig. 8 (A) and (B) show that the normalized buckling load of a cylindrical shell
with a circumferential crack and a longitudinal crack reinforced with elastic liners versus
different Young’s modulus ratios, Ep / E, for the five different elastic liner thickness
ratios, tp / t. Similar to the results presented above, the longitudinal cracked cylindrical
34
shells combined with the elastic liner have a wider range of the normalized buckling
loads than that of the circumferential cracked cylindrical shells combined with the elastic
liner, and the normalized buckling loads of longitudinal cracked multi shells smaller than
that of axial cracked cylindrical multi shells at the specific Young’s modulus ratio, Ep / E
= 0.05. For the more than Ep / E = 0.05, both the longitudinal cracked multi shells and
the axial cracked multi shells have similar normalized buckling loads. It means that the
cracked angle can be affected the normalized buckling loads under the lower Young’s
modulus ratios (Ep / E). In general, by increasing of Ep / E and tp / t, the local buckling
mode precedes the first global buckling mode. The buckling mode shape was occurred
global at γ ≅ 1, especially, tp / t = 2.5 models usually show the global buckling mode
shapes.
Figure 9. Normalized buckling load of a lined cylindrical shell versus Ep / E for different
crack angles at tp / t = 0.1 and 2.5.
35
The effect of crack orientations on the buckling behavior of cracked cylindrical
shells combined with elastic liners under an axial compression is studied by emphasizing
on four crack orientation of 0o, 30o, 60o, and 90o, measured from the circumferential line
of the cylindrical shell and the result is shown in the Fig. 9. As the results, the normalized
buckling load can be changed by the crack angle, α the maximum normalized buckling
load was observed at α = 30o at the less than Ep / E = 0.05. On the other hand, the
minimum normalized buckling load was shown at α = 90o as mentioned earlier.
Figure 10. (A) The first buckling mode shapes for different crack orientations (No liner)
(B) The first buckling mode shapes for different crack orientations (Reinforced with an
elastic liner)
The first buckling mode shapes of cracked cylindrical shells reinforced with
elastic liners and cracked cylinders without liners of the different crack orientations are
investigated. The differences are shown in Fig. 10. Thumbnail crack are generated at all
36
of crack angles of both liner and without liner models. Moreover, we could get similar
buckling mode shapes from the both models. When the crack angles are 0o, 30o, the local
buckling mode shapes are closed. However, the local buckling shapes of the cracked
cylindrical shells are opening shapes at α = 60o, 90o but, these holes are blocked up by
elastic liners which is inserted into the cylindrical shells.
In this paper, we inquired minutely into the first buckling mode shape of the
cracked shell combined with the elastic liner under varying some parameters such as a
crack angle, an elastic liner thickness, and Young’s modulus ratio. Dependence of global,
transitional and local buckling mode shapes of an elastic liner-reinforced cylindrical shell
with a circumferential crack and a longitudinal crack is shown in Fig. 11 (A) and (B),
respectively.
Figure 11. (A, B) The maps showing the dominance of global, transitional and local
buckling shapes in a lined cylindrical shell with a circumferential (top left) or
longitudinal (top right) crack based on the Young's modulus ratio, (Ep / E), and the
thickness ratio, (tp / t).
37
By increasing of the crack orientation from 0o to 90o, the area of local buckling
mode shapes was reduced remarkably and the area of global buckling mode shapes was
expanded slightly. As the result, the transitional buckling mode shape area was expended
remarkably and the slopes of two boundary lines among three buckling mode shapes
were changed finally.
In general, the buckling shapes of uncracked cylindrical shells combined with an
elastic liner have global buckling mode shapes. On the other hand, for the cracked shells
combined with an elastic liner, by increasing Young’s modulus ratio or the elastic liner
thickness ratio, the local buckling mode precedes the first global buckling mode. Even
though the elastic liner-reinforced cracked cylindrical shells, they can be transferred to
the global mode directly without a generating local buckling mode by changing both
material properties of the elastic liner such as a thickness and a Young’s modulus. For
instance, for the circumferential cracked shells, the local buckling mode was observed at
the Ep / E = 0.5 and tp / t = 0.5 in Fig. 11(A). By increasing the elastic liner thickness ratio,
tp/t, from 0.5 to 1.75, the local buckling mode can be changed to the global buckling
mode as same as the uncracked cylindrical shells combined with an elastic liner. For the
longitudinal cracked models, by increasing the elastic liner thickness ratio, tp / t, from 0.5
to 1.1, the global buckling mode can be reached. These results show the longitudinal
cracked shells are usually required smaller elastic liner thicknesses to reach the global
buckling modes. In Fig. 12, crack length ratios versus Young's modulus ratios for
different plastic liners thickness ratios of tp / t = 0.25, 1 and 2.5 are shown. This graph can
be utilized to repair cracked shells by inserting an elastic liner. According to the change
38
in buckling load and shape as the size of the crack is increased, one can define a critical
crack length as the certain size of the crack where the local buckling precedes the global
buckling of the cylindrical shell. You can see the huge increase in the critical crack length
as the thickness and relative density of the crack increase. For some reason, Fig. 12 can
be used for the optimization design to repair cracked shells by inserting elastic liners into
cracked shells to increase strength of it.
Figure 12. Critical crack length ratio (ac / R) versus Young's modulus ratio, (Ep / E), for
different thickness ratios, (tp / t).
Fig. 13 (A)-(C) show the normalized buckling loads versus the stretching modulus
for the five different elastic liner thicknesses. The broken line and dot-chain lines
represent trends of the normalized buckling load of the reinforced crack shell with
different crack orientations from the circumferential to longitudinal direction.
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Figure 13. (A-C) Normalized buckling loads versus normalized stretching stiffness of the
lined cylindrical shell with a crack for different liner to cylindrical shell thickness ratios,
(tp / t), at the crack orientations of 0°, 30° (top left) , 45°, 60° (top right), and 90°
(bottom).
In the Fig. 13(A), the broken line and the one dot-chain line represent the crack
orientation 0 o and 30o, respectively. The normalized buckling load was observed at α =
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30o at the start point and the minimum normalized buckling load was observed at the
crack angle, α = 60o under the low stretching modulus in Fig. 13(B). In addition, the
slope of the trend lines is continuously increased by increasing the crack angle from the
circumferential to the longitudinal direction. In the Fig. 13(C), the buckling loads of the
shell with longitudinal crack combined elastic liner were not normalized well at the lower
stretch modulus zone, around Eptp/Et = 0.05. Then, the buckling loads are well
normalized under the higher stretch modulus zone. When the normalized buckling load is
plotted versus the relative stretching stiffness of liner and shell layers, all the buckling loads for
different thickness and stiffness of the polymeric layer are collapsed into one line. In other words
the buckling forces could be expressing by the single parameter of stretching stiffness ratio of
layers instead of thickness ratio and material stiffness ratio. However, at 90 degrees the buckling
load cannot be expressed only by the relative stretching stiffness of elastomeric and shell layers.
1.2 A Cracked Cylindrical Shell Reinforced with an Elastic Liner under Internal
Pressure and an Axial Compression
The computational model is used to study the first buckling modes of cracked
shells with different crack orientations combined elastic liners under internal pressure and
an axial compression. A normalized loading parameter, λ
𝜆 =2𝜋𝑅2𝑃𝐹
where P denotes the internal pressure applied to the inside surface of the elastic liner. The
effect of the internal pressure on the normalized buckling load of cylindrical shells
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combined with an elastic liner with an axial crack is depicted in Fig. 14 for various crack
orientations.
Figure 14. (A-D) Normalized axial buckling [6]load, γ, of a lined cracked cylindrical
shell under uniform internal pressure versus the normalized loading parameter, λ, for
different liner to shell thickness ratios, (tp / t), at crack angles of α = 0o, 30o, 60o and 90o
Fig. 14(A) shows the dependence of the normalized buckling loads of the
circumferential cracked cylindrical shells reinforced with an elastic liner on the
normalized loading parameter λ, for four different elastic thickness, tp / t= 0, 0.25, 1, and
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2.5. The buckling modes of the elastic liner-reinforced cylindrical shells with a through
circumferential crack show local buckling modes at the low internal pressure zone.
For the circumferential cracked shells combined with an elastic liner, the buckling
load associated with the first buckling mode sharply increased from the low internal
pressure to high internal pressure region for all of elastic thickness ratios. The results can
be explained that the internal pressure effects to the reinforced shell structures as a
resistance. Then, the effect of crack angle on the buckling behavior of cracked cylindrical
shells under internal pressure and axial compression is studied by emphasizing on two
crack angles of 30o and 60o. The first normalized buckling load of reinforced crack shells
with crack angle of 30o and 60o versus the normalized loading parameter are depicted in
Fig. 14(B) and (C), respectively. When the crack angle is 30o, the normalized buckling
load was increased at the low pressure zone, it can be seen that at low internal pressure,
the internal pressure tends to increase the normalized buckling load for the normalized
loading parameter lower than approximately 2. In the high pressure region, the internal
pressure has a negative effect on the normalized buckling load for all of elastic liners. As
the crack angle increased from 30o to 60o, the normalized buckling load increased at the
low pressure, λ <1, thereafter, the normalized buckling load was sharply decreased
continuously to the high pressure region in Fig. 14(C).
Fig. 14(D) shows the normalized buckling loads of longitudinal cracked
cylindrical shells versus the normalized loading parameter, λ, for various elastic thickness
ratios. To explain the effect of internal pressure on the local buckling of the reinforced
cylindrical shells, Vaziri and Estekanchi[27] were introduced two mechanisms: (1) the
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local disturbance of the stress field, in combination with the induced local compressive
stress at the crack edges, which tends to decrease the local buckling loads (the dominant
effect for axially cracked shells); (2) the stabilizing effect of the internal pressure, which
tends to suppress the lower buckling mode of cylindrical shells (the dominant effect for
circumferentially cracked shells). The relative influence of these two mechanisms on the
local buckling behavior of cracked cylindrical shells combined with an elastic liner
depends on the crack orientation and the internal pressure. For a cracked cylindrical shell
with a crack oriented at 30o under relatively low pressure, λ < 2, the former mechanism
associated with the internal pressure dominates the buckling behavior of the multi-shells.
By increasing the internal pressure, the buckling load associated with the first local
buckling mode of the cylindrical shell reduces considerably, Fig. 14(B). For a crack
oriented at 60o from the circumferential line, the second mechanism dominates the local
buckling of the cylindrical shells and the internal pressure reduces the buckling load
associated with the first buckling mode of the cracked cylindrical shells.
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References
[27] A. Vaziri and H. E. Estekanchi, “Buckling of cracked cylindrical thin shells under
combined internal pressure and axial compression.,” Thin-Walled Structures, vol. 44, pp.
141-151, 2006.
45
CHAPTER 4
CONCLUSION
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1. Conclusion
In this study, buckling analyses are performed to investigate the bifurcation
buckling behavior of perfect and cracked cylindrical shells combined with elastic liners
under a uniform axial compression. It is shown that the longitudinal cracked shells
reinforced with an elastic liner have a wider range of the normalized buckling loads than
that of the circumferential cracked shells corrected combined elastic liners. Moreover, the
circumferential cracks have a more detrimental effect on the mechanical strength of the
cylindrical shell: first, they decrease the buckling load of the cylinder and second: the
cylindrical shell reaches its collapse strength as the local buckling mode due to the
presence of circumferential crack occurs.
And then, eigenvalue analysis is performed to explore the linear buckling analysis
of a cracked cylindrical shell reinforced with an elastic liner under the internal pressure
and uniform axial compression. Those multi-loading boundary conditions are considered
to the elastic liner-reinforced cylindrical shells at the same time.
The results showed that longitudinal crack has a more detrimental effect on the
buckling strength of the cylindrical shell in cylinders with no elastomeric liner or with
elastomeric liners with low relative stiffness. In addition, cylinders with elastomeric
liners of high relative stiffness circumferential crack have a more detrimental effect on
the buckling strength of the cylindrical shell.
The finite element analysis also showed that increasing the thickness of the
supporting layer or increasing its stiffness, can significantly increase the critical crack
size at each angle. The shells reinforced with elastic liners subjected to the internal
47
pressure and axial compression shown that the internal pressure does not affect the
overall buckling behavior of perfect cylindrical shells.
For circumferential crack, the internal pressure increases the buckling load of the
cylindrical shell. In contrast, for longitudinal crack, the internal pressure decreases the
buckling load of the cylindrical shell. The study also showed that the critical buckling
load of the cracked cylinder with various thicknesses of the elastomeric liner can be
expressed at each crack angle by a single parameter, namely stretching stiffness ratio of
liner and shell layers. The internal pressure may stabilized the against local buckling by
suppression the lower internal pressure zone or may provoke the local buckling of the
reinforced cylindrical shells due to stress concentration under higher internal pressure
zone. In addition, this is mainly depends on the elastic liner thickness and the crack
orientation.