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STABILITY ANALYSIS OF THIN CYLINDRICAL SHELLS USING DIRECT VARIATIONAL PRINCIPLE
ABSTRACT
This research work presents Stability Analysis of Thin Cylindrical Shells using Direct Variational Principle. Internally pressurized thin cylindrical shells stiffened with inclined stiffeners and rings-and-stringers were analyzed. The internally pressurized thin cylindrical shells were subjected to any of the following external pressures: uniform axial compression, bending and lateral pressure. The method of solution was carried out by the use of nonlinear large deflection theory and the effect of initial imperfections in the strain –displacement equations were considered. The direct variation method, called the Ritz method was used to find the governing equations for stability analysis of thin cylindrical shells subjected to any of the external pressures considered in this work. The numerical values of buckling stress parameter of all the stiffened thin cylindrical shells considered in this research were computed using the derived equations. It was found that the values of buckling stress parameter of cylinders stiffened with rings-and-stringers are averagely greater than the values of that stiffened with 450 inclined stiffeners with maximum percentage difference of 19.79914% and 15.20556% due to action of bending and lateral pressure respectively, but less than the values of those reinforced with 100, 200, 300, 500 and 600 inclined stiffeners with maximum percentage differences of 94.931% , 94.028%, 87.051%, 38.155% and 76.472% respectively for the different loading conditions considered. Hence, rings-andstringers are more effective than 450 inclined stiffeners, but less effective than 100, 200, 300, 500 and 600 inclined stiffeners respectively. The data obtained from the present study were also compared with that obtained from Timoshenko-Gere and Chai-Sung and found to be very close with maximum percentage difference of 0.9685% and 0.0648% due to action of lateral pressure and axial compression respectively. For easy computation of buckling stress parameter using the equations derived in this research work, a computer program was developed using MATLAB 7.5 version.
Keywords: Buckling, Cylindrical Shells, Deflection, Imperfections, Inclined Stiffeners, Nonlinear, Rings-and-Stringers, Stability.
TABLE OF CONTENTS
Page
Title page i
Certification ii
Dedication iii
Acknowledgements iv
Abstract v
Table of Contents vi
List of Tables x
List of Figures xii
Definition of Notations xiv
CHAPTER ONE: INTRODUCTION 1
1.1 Background of Study 1
1.2 Statement of Problem 5
1.3 Objective of Study 6
1.4 Justification of Study 6
1.5 Scope of Study 7
CHAPTER TWO: LITERATURE REVIEW 8
2.1 Stability 8
2.1.1 Types of Instability 10
2.1.1.1 Bifurcation Instability 10
2.1.1.2 Limit Point Instability 11
2.1.1.3 Finite Disturbance Instability 12
2.1.1.4 Snap-through Instability 13
2.2 Stability Models 13
2.3 Stability Equations Derivation 14
2.4 Thin Shell 15
2.5 Shell Buckling 16
2.6 Buckling in Cylindrical Shells 16
2.6.1 Types of Buckling for Perfect Cylindrical Shell 18
2.6.1.1 Types of Bifurcation Buckling 18
2.7 Nature of Buckling 23
2.7.1 Equilibrium Considerations of Buckling 23
2.7.2 States of Equilibrium and Buckling 23
2.8 Collapse Mode of Cylindrical Shell 24
2.9 Varational Formulation of Elastic Bodies 26
2.9.1 Strain Energy of Elastic Body 26
2.9.2 General Variational Principle 28
2.9.3 Minimum Potential Energy Criterion for Stability of Shells 30
2.9.4 Variational Principle Method 31
2.9.5 Advantages of Direct Variational Calculus 32
2.9.6 Disadvantages of Direct Variational Calculus 32
2.10 The Ritz Method 33
2.10.1 The Advantages of the Ritz Method 34
2.10.2 The Disadvantages of the Ritz Method 35
2.11 Review of Previous Work on Stability Analysis of Cylindrical shells 35
2.12 Summary of Research Works on Stability Analysis of
Cylindrical shells 41
CHAPTER THREE: METHODOLOGY 45
3.1 Derivation of stability Equation of Thin Cylindrical Shells 45
3.2 Assumptions for the Derivation of the Stability Equation 48
3.3 Method of Solution 49
3.4 Compatibility Equation of Thin Cylindrical Shells 50
3.5 Energy Expressions of the Cylindrical Shells 53
3.6 Stability Analysis of a Stiffened Cylindrical Shell under Axial
Compression 59
3.6.1 Expression of Total Potential for Stiffened Cylindrical Shell
subjected to Internal Pressure and Axial Compression 62
3.6.2 Minimization of Total Potential Energy of Internal
Pressurized Thin Cylindrical Shell subjected to Axial 69 Compression
3.6.3 Thin Cylindrical Shell under Axial Compressive force and
Reinforced with stringers-and-rings 74
3.6.4 Unstiffened Thin Cylindrical Shell under the action of Axial
Compression 77
3.7 Stability Analysis of a Stiffened Cylindrical Shell subjected
to Bending and Internal Pressure 80
3.7.1 Expression of Total Potential for Stiffened Cylindrical Shell subjected to Bending and Internal Pressure 83
3.7.2 Minimization of Total Potential Energy of Internally 87
Pressurized Thin Cylindrical Shell subjected to Bending
3.7.3 Thin Cylindrical Shell Reinforced with stringers-and-rings Subjected to Bending and Internal Pressure 91
3.7.4 Unstiffened Thin Cylindrical Shell subjected to Bending and
Internal Pressure 95
3.8 Stability Analysis of a Stiffened Cylindrical Shell subjected to Lateral Pressure and Internal Pressure 98
3.8.1 Expression of Total Potential for Stiffened Cylindrical Shell subjected to Lateral Pressure and Internal Pressure 100
3.8.2 Minimization of Total Potential Energy of Internally Pressurized Thin Cylindrical Shell subjected to Lateral 102
Pressure
3.8.3 Thin Cylindrical Shell Reinforced with stringers-and-rings Subjected to Lateral Pressure and Internal Pressure 106
3.8.4 Unstiffened Thin Cylindrical Shell subjected to Lateral
Pressure and Internal Pressure 110
3.9 Summary of The Equations Derived for Stability Analysis of
Internally Pressurized Thin Cylindrical Shells 114
CHAPTER FOUR: RESULTS AND DISCUSSION 115
4.1 Results of Internally Pressurized Thin Cylindrical Shell
subjected to Lateral Pressure 115
4.2 Results of Internally Pressurized Thin Cylindrical 118
Shell subjected to Uniform Bending
4.3 Comparison of Buckling stress parameter of Cylindrical
shell reinforced with Inclined Stiffeners and that
reinforced with Rings-and-stringer 121
4.4 Relationship between Buckling stress parameter and Deflection parameter, 𝜆1 128
4.5 Comparison of the Results of the Present Study and Results
of Other classical Theories 129
4.6 Computer Program for the Elastic Stability Analysis of
Internally Pressurized Thin Cylindrical shells 132
CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS 133
5.1 Conclusions 133
5.2 Recommendations 134
5.3 Contribution to Knowledge 134
REFERENCES 136
APPENDICES 147
CHAPTER ONE INTRODUCTION 1.1 BACKGROUND OF STUDY
A cylindrical shell is generated by moving a straight line along a curve while maintaining it parallel to its original position (Venstel and Krauthammer, 2001). Cylindrical shell can be either thick or thin. A thin cylindrical shell is one that maximum ratio of its thickness, h to the radius of curvature, R is less than or equal to , i.e.
max (1.1)
The cylindrical shell for which the inequality in equation (1.1) is violated is known as thick cylindrical shell (Calladine, 2007; Venstel and Krauthammer, 2001). Thin cylindrical shells are widely used in engineering application. They have wide applications in marine, mechanical and civil engineering and space structures which are highly susceptible to imperfections. They have efficient load carrying capacity with weight economy (Venstel and Krauthammer, 2001).
The design of cylindrical shell structures depends on a large number of factors, namely the economic aspects, material availability, response of each structure of the system to static and dynamic loads, temperature effects and so on. The designer is interested in arriving at an optimum design taking into considerations all these factors (Iyengar, 1988). The analysis of the structure is normally concerned with the determination of behaviour of the structure or the elements of the structure under the action of external loads. It explains the response of the structure when subjected to external loads and /or temperature changes. In other words, if the external loads are known, the deformation pattern and internal stress distribution in the structure can be determined. Also, the nature of equilibrium of the structure (stable or unstable equilibrium) shall be determined. The understanding of those responses of the structure is necessary for design of safe structure (Iyengar, 1988).
In many cases, instability is not necessarily associated with the failure of the overall structure. For instance, in a structure like aircraft, if the skin wrinkles, or a stringer locally buckles, the entire fuselage or wing does not fail. However, if a portion of the fuselage between adjacent rings becomes unstable, the entire fuselage or wings fails
catastrophically. Thus, stability also plays an important part in designing a structure. A structure may have two kinds of failure, namely material failure and form failure. In material failure, the stresses in the structure exceed the specified safe limit, resulting in the formation of cracks which cause failure. In form failure, though the stresses may not exceed the safe value, the structure may not be able to maintain its original form. Here, the structure does not fail physically, but may deform to some other shape due to intolerable external disturbance. Furthermore, form failure depends on the geometry and loading of the structure. It occurs when the conditions of loading are such that compressive stresses get introduced. When magnitude of the load on the structure is such that the equilibrium changes from stable to neutral, the load is called the critical load. This phenomenon of change of equilibrium is called the buckling of the structure (Iyengar, 1988 and Houliara, 2008).
Buckling is often critical in thin-walled or light weight members such as slender columns, plates and cylindrical shells which are subjected to predominantly compressive action. Yet the demand for efficient, light weight structures often dictates the use of thin walled members. This demand is prevalent in the design of silos, liquid retaining structures, aerospace and hydrospace structures. The collapse of a structure like cylindrical shell structures, precipitated by buckling is often a more serious problem than fracture or yielding. Buckling sometime occurs suddenly without warning causing a catastrophic failure. Fracture or yielding, on the other hand, can also produce failure, but the elasticity of the material permits a redistribution of the stresses often allowing a progressive collapse rather than a sudden complete collapse characteristic of buckling.
Once buckling is initiated within the structure, there is little or no chance of recovery unless the load is suddenly reduced (Houliara, 2008).
A cylindrical shell structure is considered to fail by buckling when subjected to a compressive load; the structure undergoes a transition in deformation from that of the direction of compressive load application to a deformation that is predominantly perpendicular to the direction of load application. In fact, buckling phenomenon in cylindrical shell occurs when most of the strain energy which is stored as membrane energy has been converted to bending energy requiring large deformation resulting to catastrophic failure (Houliara, 2008 and Sosa, 2005). Hence, the design of thin cylindrical shells should be based on buckling criteria (Zhang and Han, 2007).
Typical failure theories based on material strength such as Tresca and Von Mises failure theories, have no method by which to address buckling and instability issue. Furthermore, the most significant material properties affecting the resistance of buckling failure are Young modulus of elasticity and Poisson ratio. The most significant geometrical parameter is the aspect ratio comprised of the diameter to length.
Cylindrical shells can be either stiffened or unstiffened. Cylindrical shells in engineering structures with large aspect ratios are typically stiffened against buckling by circumferential and longitudinal members known as ring and stringers stiffeners respectively. The use of the stiffeners improves the resistance of cylindrical shells to buckling (George, 1996; Arani et al, 2007).
Historically, buckling failures have occurred at compressive stress significantly less than the ultimate compressive stresses of the given shell material. Furthermore, buckling of cylindrical shells can occur when the structure is subjected to the individual or combined action of axial compression, external pressure and torsion.
Buckling behaviour of cylindrical shells (in particular, the critical buckling load) is not accurately predicted by linear elastic equations due to initial imperfections of the shell structure under the action of compressive loads. Such imperfections may be either geometrical imperfections(for example: out of straightness, initial ovality, dents, swells, circularity, cylindricity and geometrical eccentricities);structural imperfections(i.e. residual stresses and material inhomogenities) or loading imperfections (i.e. non uniform edge load distribution, unintended edge moments, load eccentricities and load alignments as well as imperfect boundary conditions).Also the constructional defects, such as small holes, cut-outs, rigid inclusions and delamination could be regarded as structural imperfections. Out of all these imperfections the geometrical imperfections are more dominant in determining the load carrying capacity of thin cylindrical shells (Zhang and Han, 2007).
Consequently, nonlinear large deflections which incorporated this imperfection are required to obtain accurate result. In contrast, classical theories employing nonlinear equation have been utilized extensively in the past to predict buckling behaviour of cylindrical shells. The details of the theories are discussed in the literature review of this work.
In this work, direct variational principle which was incorporated with imperfections in shell structures was employed in the stability analysis of thin cylindrical shells under different loading conditions. This method is easy, both conceptually and mathematically. It is extremely powerful to obtain reusable analytical solutions, and it provides a valuable preparation for the understanding of finite element method, which have rapidly become the most dominant numerical method in the structural analysis. Some examples of direct variational method are the Ritz method, Galerkin method etc. In this work, the Ritz method which applies the principle of minimum potential energy shall be used in the stability analysis of cylindrical shells. Ritz method is one of the best methods for deriving Eigen-values. This method is a powerful tool for the analysis of structures such that the assumed displacement functions of the structure may satisfy any combination of boundary conditions. Also, with this method, the power of polynomial equations assumed for the displacement functions can be adjusted according to the boundary conditions. The results obtained using this method was compared with those obtained from other classical theories methods.
1.2 STATEMENT OF PROBLEM
Thin cylindrical shells are generally highly efficient structures. Their structural analysis constitutes a classical problem of mechanics with numerous applications in civil, aerospace, mechanical and marine structures. In particular, nonlinear response and the loss of structural stability of thin cylindrical shells as well as their postbuckling behaviour is a topic of fundamental and applied research. In the past, this problem has caused significant controversy due to the unreasonably high analytical predictions of buckling loads, compared with low buckling loads obtained from relevant experiments. This controversy remained a major issue of concern and dispute among structural engineers and researchers.
The major cause of this discrepancy has been discovered to be the initial imperfections of the shells, due to their manufacturing difficulties. These imperfections affect the load carrying capacity of these shells. Thus, reliable prediction of buckling strength of these structures is important, because the buckling failure is catastrophic in nature. Due to these initial imperfections on the shells, the critical buckling pressure of the shells cannot be accurately predicted by using either linear elastic equations or small deflection theory under the action of external pressures.
The present study is motivated by the need for an accurate prediction of the response of internally pressurized thin walled cylinders subjected to uniform axial compression, bending and lateral pressure using advanced numerical tools. The numerical method, direct variational principle that would be used in this work shall incorporate nonlinear large deflection theory and initial imperfections of the shell in its analysis. The results obtained using this method shall be compared with that obtained using other classical methods of analysis that considered the influence of initial imperfection on the buckling load of the shells.
1.3 OBJECTIVES OF THE STUDY
The main objective of this work is Stability Analysis of Thin Cylindrical Shells using Direct Variational Principle. The specific objectives are:
- To formulate differential equations for computing critical buckling stress of thin cylindrical shells subjected to different loadings.
- To solve the differential equations developed in (i) using direct variational principle and determine the critical buckling stress of the thin cylindrical shells, both unstiffened and stiffened subjected to different loadings.
- To compare the results obtained from the present study with those obtained from other classical methods.
- To compare the collapse pressure of cylindrical shells reinforced with inclined stiffeners with that reinforced with stringers -and -rings subjected to the same loading conditions.
- To develop a computer program that will facilitate the solution obtained using Ritz method.
1.4 JUSTIFICATION OF THE STUDY
The following shall be benefited from this research work:
- The knowledge derived from this work will assist engineers in determination of critical buckling stresses of thin circular cylindrical shells under different loadings using Ritz method.
- Stability analysis of thin cylindrical shell structures can be done easily by engineers.
- The knowledge derived from this research will be very essential for the safe design of agricultural storage bins like silos and liquid retaining structures.
- The rate of accidents caused by improper design due to inadequate check of stability of thin cylindrical shell structures like submarines, aerospace and liquid retaining structures shall be minimized.
- Engineers would be able to select suitable reinforcement (i.e. inclined stiffeners or stringers and rings) for thin-wall cylindrical shell structures.
1.5 SCOPE OF THE WORK
Due to their favorable stiffness-to-mass ratio, thin cylindrical shells are encountered in a wide variety of applications. It is well known that under compressive loading, such structures may lose their stability, that is, they buckle.
Consequently, in this research work, isotropic stiffened thin cylindrical shells were considered for stability analysis using direct variational principle method called the Ritz method. The cylinders that were considered for the analysis were internally pressurized cylinders that were subjected to any of the following loadings:
- Uniform axial compression
- Uniform bending
- Uniform lateral pressure
However, this work was covered in five chapters. Chapter one dealt with introduction, which discussed the background of study, statement of problem, objective of study, justification of the study and scope of the study.
In chapter two, the literature of the past works related to the subject matter was reviewed. The differential equations for the stability analysis of thin-walled circular cylindrical shells were developed and solved in chapter three.
The numerical examples of the equations obtained in chapter three were given in chapter four. Also in chapter four, the results obtained from the developed equation in the present study were compared with the results obtained from other classical theory methods. Computer programme for analysis of thin cylindrical shells using Ritz method was also developed in chapter four. This research was concluded in chapter five with recommendations.
