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

41

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

46

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.