Numerical Simulation of Buckling of Numerical Simulation of Buckling of Thin Cylindrical ShellsThin Cylindrical Shells

Master of TechnologyMaster of Technology(M.Tech)(M.Tech)

in in

Structural EngineeringStructural Engineering

ByBy

Khaja misba uddinKhaja misba uddinoct 1oct 1stst, , 20132013

Department of Civil EngineeringDepartment of Civil EngineeringJNTUH College of Engineering, Kukatpally, HyderabadJNTUH College of Engineering, Kukatpally, Hyderabad

Outline of the Outline of the PresentationPresentation

• IntroductionIntroduction• ImportanceImportance• Literature ReviewLiterature Review• Lacuna in the fieldLacuna in the field• Present workPresent work• Numerical modeling and analysisNumerical modeling and analysis• ResultsResults• ConclusionsConclusions• Scope for future workScope for future work• ReferencesReferences• AcknowledgementAcknowledgement

IntroductionIntroduction

It is well known that thin walled It is well known that thin walled cylinders are proven efficient cylinders are proven efficient structures with a variety of structures with a variety of applications inapplications in

- construction- construction

- chemical - chemical

&&

- aero-space industry. - aero-space industry.

The The strengthstrength of these structures of these structures is is limited to their buckling strengthlimited to their buckling strength when subjected to axial compressive when subjected to axial compressive loads and external pressures.loads and external pressures.

FailureFailure of these thin cylindrical of these thin cylindrical shells shells under bucklingunder buckling loads is a loads is a matter of concernmatter of concern for engineers, for engineers, while predicting the reliability of while predicting the reliability of these structures. these structures.

Definition -Thin ShellsDefinition -Thin Shells

If the wall thickness of the shell is less that 10% If the wall thickness of the shell is less that 10% of the diameter of the shell then it can be treated of the diameter of the shell then it can be treated a thin shell.a thin shell.

When the shell is subjected to internal Pressure When the shell is subjected to internal Pressure the stresses developed are assumed to be uniform the stresses developed are assumed to be uniform throughout the wall thickness.throughout the wall thickness.

ImportanceImportance

Knowledge ofKnowledge of

- Stress distribution- Stress distribution

- Vibration Pattern- Vibration Pattern

- Buckling Behaviour- Buckling Behaviour

is very important in the design of is very important in the design of thin cylindrical shells.thin cylindrical shells.

Thin cylinder subjected Thin cylinder subjected to external pressure to external pressure (Vacuum inside the (Vacuum inside the

cylinder) cylinder)

Literature ReviewLiterature Review

There are several research papers and There are several research papers and technical reports published in this field.technical reports published in this field.

Euler’s research in 18Euler’s research in 18thth century gave century gave birth to classical theory of buckling.birth to classical theory of buckling.

Successful applications of a variety of Successful applications of a variety of the structures is well documented in the structures is well documented in Theory of Elastic Stability by Theory of Elastic Stability by Timoshenko and Gere.Timoshenko and Gere.

A very pertinent literature survey has A very pertinent literature survey has been conducted in this field.been conducted in this field.

Certain gaps are identified in the open Certain gaps are identified in the open published literature.published literature.

Some of these gaps are detailed Some of these gaps are detailed hereunder. hereunder.

According to the According to the classical theoryclassical theory of of buckling, for axially-loaded thin cylindrical buckling, for axially-loaded thin cylindrical shells, shells, buckling stressbuckling stress is directly is directly proportionalproportional to the wall thickness ‘ to the wall thickness ‘tt’’, other , other things being equal.things being equal.

Where, “Where, “σσ” is critical compressive stress, ” is critical compressive stress, “E” is Young’s modulus, ““E” is Young’s modulus, “υυ” is Poisson ” is Poisson ratio of iso-tropic material, “t” is the ratio of iso-tropic material, “t” is the uniform thickness and “R” is the radius of uniform thickness and “R” is the radius of the shell. the shell.

R

tEcr

)1(3 2

However, from the experimental However, from the experimental investigations of the several investigations of the several researchers researchers the empirical datathe empirical data show show clearly that the clearly that the buckling stressbuckling stress is is actually actually proportional toproportional to tt1.51.5, other , other things being equal.things being equal.

5.1

5

R

t

Emean

Lacuna in the FieldLacuna in the Field It is well known that there is It is well known that there is wide scatter wide scatter

in the buckling-stress datain the buckling-stress data, ranging from , ranging from one one half to twicehalf to twice the mean value. the mean value.

Current theories of shell buckling Current theories of shell buckling attribute both the scatter and the low attribute both the scatter and the low buckling stress – in comparison with the buckling stress – in comparison with the classical – to “classical – to “imperfection-sensitive”, imperfection-sensitive”, non-linear structural behaviour.non-linear structural behaviour.

All those theories considered All those theories considered classical classical shell theoryshell theory as their as their ideal referenceideal reference and and treated as perfect.treated as perfect.

Present WorkPresent Work

The present work deals with the The present work deals with the investigation ofinvestigation of

1. The stress distribution – Static 1. The stress distribution – Static AnalysisAnalysis

2. Buckling Behaviour 2. Buckling Behaviour

using FEM when thin cylindrical shells using FEM when thin cylindrical shells are subjected to axial compressive are subjected to axial compressive loads and external pressures.loads and external pressures.

NUMERICAL MODELING NUMERICAL MODELING AND ANALYSISAND ANALYSIS

A complete ABAQUS analysis flow chart A complete ABAQUS analysis flow chart usually consists of three distinct stages: usually consists of three distinct stages:

1) Preprocessing,1) Preprocessing,

2) Simulation,2) Simulation,

3) Post-Processing.3) Post-Processing.

These three stages are linked together by These three stages are linked together by the corresponding files.the corresponding files.

FLOW CHARTFLOW CHART

MODELING AND MODELING AND MESHINGMESHING

MESHING MESHING (Four node shell element)(Four node shell element)

Applying loads and Applying loads and boundary conditionsboundary conditions

RESULTS AND RESULTS AND DISCUSSIONS DISCUSSIONS

Buckling pattern of thin Buckling pattern of thin walled (0.1mm) walled (0.1mm)

cylindrical shell subjected cylindrical shell subjected to axial compressionto axial compression

Stress distribution in thin cylinder Stress distribution in thin cylinder (0.3mm) (0.3mm)

subjected to axial compression.subjected to axial compression.

DEFORMATION PATTERN OF THIN WALLED SHORT DEFORMATION PATTERN OF THIN WALLED SHORT CYLINDRICAL SHELL SUBJECTED TO EXTERNAL CYLINDRICAL SHELL SUBJECTED TO EXTERNAL

PRESSURE PRESSURE

Buckling pattern of thin cylindrical Buckling pattern of thin cylindrical shell shell

subjected to external pressure subjected to external pressure

Deformed shape of the Deformed shape of the cylindrical shell under axial cylindrical shell under axial

compressioncompression

Deformed shape of the Deformed shape of the cylindrical shell cylindrical shell

Deformed shape of the Deformed shape of the cylindrical shell cylindrical shell

Deformed shape of the Deformed shape of the cylindrical shell cylindrical shell

Deformed shape of the Deformed shape of the cylindrical shell cylindrical shell

Shell subjected to axial Shell subjected to axial compressioncompression of 100 N and of 100 N and

varying the thickness of the varying the thickness of the shell the buckling load is shell the buckling load is

calculated.calculated.Length

inmm

Thicknessmm

Radiusmm

Eigen valuefor

Mode 1

BucklingLoad in

N

6000 0.1 500 31.592 3159.2

6000 0.3 500 237.24 23724

6000 0.5 500 618.9 61890

CONCLUSIONS CONCLUSIONS The following are the important conclusions The following are the important conclusions drawn from the present study on numerical drawn from the present study on numerical simulation of buckling of thin cylindrical shells. simulation of buckling of thin cylindrical shells.

During buckling, half sine waves will be formed During buckling, half sine waves will be formed along the generator and along the circumference.along the generator and along the circumference.

The number of half sine waves formed along the The number of half sine waves formed along the generator and along the circumference are generator and along the circumference are independent of the length of the cylindrical shell.independent of the length of the cylindrical shell.

The number of half sine waves formed along the The number of half sine waves formed along the generator and along the circumference are generator and along the circumference are dependent on the radius and thickness of the dependent on the radius and thickness of the shell.shell.

SCOPE FOR FUTURE WORK SCOPE FOR FUTURE WORK As an extension to the present work:As an extension to the present work:

A number of variations of the problem A number of variations of the problem specifications can be tried with the available specifications can be tried with the available soft wares in estimating the critical buckling soft wares in estimating the critical buckling loads. loads.

There is always a need for the experimental There is always a need for the experimental data to be generated to validate the data to be generated to validate the numerical results. numerical results.

In these endeavors an experimental test In these endeavors an experimental test facility can be built with data acquisition and facility can be built with data acquisition and recording systems. recording systems.

There is a wide scope to further There is a wide scope to further investigate the effect of geometric investigate the effect of geometric nonlinearity and material nonlinearity and material nonlinearity on the critical buckling nonlinearity on the critical buckling loads. loads.

Buckling of thin shells, made of Buckling of thin shells, made of functionally graded materials, functionally graded materials, subjected to internal/ external loads.subjected to internal/ external loads.

Vibration behaviour and impact Vibration behaviour and impact resistance of thin cylindrical shells. resistance of thin cylindrical shells.

ReferencesReferences1.1. Euler, L., 1744. Methodus inveniend lincas Euler, L., 1744. Methodus inveniend lincas

curves maxim: minimive proprietate gaudentes curves maxim: minimive proprietate gaudentes (Appendix, de curvis elastics). (Appendix, de curvis elastics). Marcum Marcum Michaelem Bousquet, Lausanne and Geneva.Michaelem Bousquet, Lausanne and Geneva.

2.2. Timoshenko SP, Gerri JM. Theory of elastic Timoshenko SP, Gerri JM. Theory of elastic stability, 2nd ed. New York: McGraw-Hill, stability, 2nd ed. New York: McGraw-Hill, 19611961

3.3. Von Karman, T., Dunn, L.G., Tsien, H., 1940. Von Karman, T., Dunn, L.G., Tsien, H., 1940. The influence of curvature on the buckling The influence of curvature on the buckling characteristics of structures. Journal of the characteristics of structures. Journal of the Aeronautical Sciences 7, 276-289..Aeronautical Sciences 7, 276-289..

4.4. Koiter WT. On the stability of elastic Koiter WT. On the stability of elastic equilibrium (in Dutch with English summary). equilibrium (in Dutch with English summary). Ph.D. thesis, Delft, H.J. Paris, Amsterdam, Ph.D. thesis, Delft, H.J. Paris, Amsterdam, 1945. Air Force Dynamics Laboratory, 1945. Air Force Dynamics Laboratory, Technical Report, AFFDL-TR-70-25, Ohio, Technical Report, AFFDL-TR-70-25, Ohio, February 1970(English translation)February 1970(English translation)

5.5. Arbocz., J., 1974. The effect of initial Arbocz., J., 1974. The effect of initial imperfections on shell stability. In: Fung, Y.C., imperfections on shell stability. In: Fung, Y.C., Sechler, E.E. (Eds), thin shell structures.Sechler, E.E. (Eds), thin shell structures.

6.6. Abramovich H, Singer J, Weller T. The influence Abramovich H, Singer J, Weller T. The influence of initial imperfections on the buckling of of initial imperfections on the buckling of stiffened cylindrical shells under combined stiffened cylindrical shells under combined loading. In: Jullian JF, editor. Buckling of shell loading. In: Jullian JF, editor. Buckling of shell structures on land, in the sea, and on the air. structures on land, in the sea, and on the air. London: Elsevier Applied Science, 1991.p. 205-London: Elsevier Applied Science, 1991.p. 205-45.45.

7.7. Yamaki, N., 1984. Elastic Stability of Circular Yamaki, N., 1984. Elastic Stability of Circular Cylindrical Shells. North – Holland, Amsterdam.Cylindrical Shells. North – Holland, Amsterdam.

8.8. Heyman J. Equilibrium of shell structures. Heyman J. Equilibrium of shell structures. Oxford: Clarendon Press, 1977Oxford: Clarendon Press, 1977

9.9. Lancaster, E.R., Calladine, C.R., Palmer S.C., Lancaster, E.R., Calladine, C.R., Palmer S.C., 1998. Experimental observations on the 1998. Experimental observations on the buckling of a thin cylindrical shell subjected to buckling of a thin cylindrical shell subjected to axial compression. International Journal of axial compression. International Journal of Mechanical Sciences.Mechanical Sciences.

10.10. T.D. Park and S.Kyriakides, “On the collapse of T.D. Park and S.Kyriakides, “On the collapse of dented cylinders under external pressure”, dented cylinders under external pressure”, Int.J.Mech.Sci. v.38, No.5, pp. 557-558 (1996)Int.J.Mech.Sci. v.38, No.5, pp. 557-558 (1996)

11.11. Y. Bai, R.T.Igland, T.Moan, “Tube Collapse Y. Bai, R.T.Igland, T.Moan, “Tube Collapse Under Combined External Pressure, Tension Under Combined External Pressure, Tension and Bending”, Marine Structures, v.10, 389-410 and Bending”, Marine Structures, v.10, 389-410 (1997)(1997)

12.12. G.Forasassi, R.Lo Frano., “Buckling of imperfect G.Forasassi, R.Lo Frano., “Buckling of imperfect thin cylindrical shells under lateral pressure”, thin cylindrical shells under lateral pressure”, Journal of Achievements in Materials and Journal of Achievements in Materials and Manufacturing Engineering, pp287-290, Vol.18, Manufacturing Engineering, pp287-290, Vol.18, Issue 1-2, September – October 2006.Issue 1-2, September – October 2006.

13.13. R.Lo Frano and G.Forasassi,., “Buckling of R.Lo Frano and G.Forasassi,., “Buckling of imperfect thin cylindrical shells under lateral imperfect thin cylindrical shells under lateral pressure”, Journal of Achievements in Materials pressure”, Journal of Achievements in Materials and Manufacturing Engineering, pp1-8, Vol.20, and Manufacturing Engineering, pp1-8, Vol.20, 2008. 2008.

ACKNOWLEDGEMENTACKNOWLEDGEMENT I profusely thankI profusely thank Dr. N. V. Ramana RaoDr. N. V. Ramana Rao, ,

PRINCIPAL & PROFESSOR, JNTUH PRINCIPAL & PROFESSOR, JNTUH College of Engineering, Kukatpally, College of Engineering, Kukatpally, Hyderabad, for his valuable guidance and Hyderabad, for his valuable guidance and constant inspiration at every stage of this constant inspiration at every stage of this dissertation. I sincerely thank for dissertation. I sincerely thank for supporting my work.supporting my work.

I am indebted to I am indebted to Dr. M.V. Seshagiri Dr. M.V. Seshagiri RaoRao, , the Head of the Department of Civil the Head of the Department of Civil Engineering for his valuable suggestions Engineering for his valuable suggestions and co-operation during the entire period and co-operation during the entire period of work, without whom, this project could of work, without whom, this project could not have been completed.not have been completed.

I would like to express my gratitude to the I would like to express my gratitude to the then Head of the Department of Civil then Head of the Department of Civil Engineering Engineering Dr. G. K. ViswanadhDr. G. K. Viswanadh,, for his for his support in a number of ways to M.Tech support in a number of ways to M.Tech Program in the department during the Program in the department during the entire period of my course.entire period of my course.

I sincerely express my thanks to I sincerely express my thanks to all the all the faculty membersfaculty members of Civil Engineering of Civil Engineering Department who all helped me at different Department who all helped me at different stages of my M.Tech course work and stages of my M.Tech course work and project work.project work.

I would like to sincerely thank I would like to sincerely thank all all my classmates my classmates who made my who made my stay in JNTUH College of stay in JNTUH College of Engineering, really a memorable Engineering, really a memorable one.one.

V.L.S. V.L.S. BANUBANU

Thank YouThank You