Skip to main content
SpringerLink
Log in

The Influence of Boundary Conditions on the Buckling of Stiffened Cylindrical Shells

  • Conference paper
Buckling of Structures

Abstract

Theoretical and experimental studies on the influence of boundary conditions on the buckling of stiffened cylindrical shells and their vibrations are discussed. The effect of prebuckling deformations on the buckling loads and vibrations of stiffened shells is studied and compared with that in the case of unstiffened shells. The in-plane boundary conditions are found to be of particular importance for stiffened cylindrical shells and their effect differs significantly from that in unstiffened shells. The effect of axial restraints, which are found to be of prime importance in stringer-stiffened shells, are also studied.

By correlation with the vibration tests on the same shells a method is developed for definition of the actual boundary conditions of a stiffened shell non- destructively.

The effect of eccentricity of loading on stringer-stiffened shells is studied experimentally and correlated with vibration tests on the same shells.

Preliminary results of a non-destructive experimental method for prediction of buckling loads based on vibration testing of stiffened shells are also presented.

The research reported in this paper has been sponsored in part by the Air Force Office of Scientific Research, through the European Office of Aerospace Research, United States Air Force under Contract F44620-71-C-0116 and Grant 72-2394.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Abbreviations

A1, A2:

Cross section of stringer and ring, respectively

a :

Ring spacing (distance between centers of rings)

A, B :

Constants

b 1 :

Stringer spacing (distance between centers of stringers)

C1, C2, C3, C4:

Notation for clamped boundary conditions

c1 :

Width of stringer

d1 :

Height of stringer

E :

Young’s modulus

e1, e2 :

Stringer or ring eccentricity, respectively (distance from shell middle surface to stiffener centroid)

ē:

Eccentricity of loading (distance from shell middle surface to the point of application of load)

f :

Frequency

h :

Thickness of shell

I11, I22:

Moment of inertia of stringer or ring stiffener cross-section about its centroidal axis

k 1 :

Axial elastic restraint

k 4 :

Rotational elastic restraint

L :

Length of shell

M x :

Moment resultant in axial direction

m :

Number of half longitudinal waves

N x :

Axial membrane force resultant

N xy :

Shear membrane force resultant

n :

Circumferential wave number

P :

Compressive axial load

P CR :

Theoretical buckling load

P PRE :

Theoretical buckling load with nonlinear prebuckling defor­mations considered

P mem :

Theoretical buckling load with prebuckling deformation neglected

P EXP :

Experimental buckling load

R :

Radius to shell middle surface

SS1, SS2, SS3, SS4:

Notation for simply supported boundary conditions

u, v, w :

Displacements in axial, circumferential and radial directions respectively (radial direction positive inward)

x :

Axial coordinate

Z :

(1 - v 2)1/2 (L/R)2 (R/h), Batdorf shell parameter

v :

Poisson’s ratio

Q PRE :

PPRE/Pmem

Q :

PEXP/Pmem, “linearity”

η +1 :

Torsional stiffness parameter of stringer (see [3] or [22])

φ:

Circumferential coordinate

γ:

Free parameter for exponent of frequency

References

  1. Singer, J.; Rosen, A.: The Influence of Boundary Conditions on the Buckling of Stiffened Cylindrical Shells. TAE Report 213, Technion, Israel Institute of Technology, Dept. of Aeronautical Engineering, Haifa, Israel, June 1974.

    Google Scholar 

  2. van der Neut, A.: General Instability of Orthogonally Stiffened Cylindrical Shells. Collected Papers on Instability of Shell Structures —1962, NASA TN D-1510, pp. 309–319, Dec. 1962.

    Google Scholar 

  3. Singer, J.: Buckling of Integrally Stiffened Cylindrical Shells—A Review of Experiment and Theory. Contributions to the Theory of Aircraft Structures. Rotterdam: Delft University Press 1972, pp. 325–358.

    Google Scholar 

  4. Peterson, J. P.: Buckling of Stiffened Cylinders in Axial Compression and Bending-a Review of Test Data. NASA TN D-5561, December 1969.

    Google Scholar 

  5. Singer, J.; Arbocz, J.; Babcock, C. D.: Buckling of Imperfect Stiffened Cylindrical Shells Under Axial Compression. AIAA J. 9, No. 1, 68–75 (1971).

    Article  Google Scholar 

  6. Block, D. L.: Influence of Discrete Ring Stiffeners and Prebuckling Deformations on the Buckling of Eccentrically Stiffened Orthotropic Cylinders. NASA TN D-4283, January 1968.

    Google Scholar 

  7. Almroth, B. O.; Bushneil, D.: Computer Analysis of Various Shells of Revolution. AIAA J. 6, No. 10, 1848–1855 (1968).

    Article  Google Scholar 

  8. Card, M. F.; Jones, R. M.: Experimental and Theoretical Results for Buckling of Eccentrically Stiffened Cylinders. NASA TN D-3639, October 1966.

    Google Scholar 

  9. Weiler, T.; Singer, J.; Batterman, S. C.: Influence of Eccentricity of Loading on Buckling of Stringer-Stiffened Cylindrical Shells. Thin-Shell Structures, Theory, Experiment and Design, Fung, Y. C. and Sechler, E. E. (ed.). Englewood-Cliffs, N.J.: Prentice-Hall 1974, pp. 305–324.

    Google Scholar 

  10. Bushneil, D.: Stress, Stability, and Vibration of Complex Shells of Revolution: Analysis and User’s Manual for BOSOR3. Lockheed Missiles & Space Co. Report N-5J-69–1, Sept. 1969.

    Google Scholar 

  11. Weiler, T.: Further Studies on the Effect of In-Plane Boundary Conditions on the Buckling of Stiffened Cylindrical Shells. TAE Report No. 120, Technion—Israel Institute of Technology, Dept. of Aeronautical Engineering, Haifa (Israel) January 1974.

    Google Scholar 

  12. Yamaki, N.; Kodama, S.: Buckling of Circular Cylindrical Shells under Compression. Rep. Inst. High Speed Mech. (Tohoku University) 28, 99–123 (1970); 24, 111–142 (1971); 25, 99–141 (1972); 27, 1–30 (1973).

    Google Scholar 

  13. Rosen, A.; Singer, J.: Vibrations and Buckling of Eccentrically Loaded Stiffened Cylindrical Shells. TAE Report 205, Technion—Israel Institute of Technology, Dept. of Aeronautical Engineering, Haifa, Israel, June 1974.

    Google Scholar 

  14. Weiler, T.; Baruch, M.; Singer, J.: Influence of In-Plane Boundary Conditions on Buckling Under Axial Compression of Ring Stiffened CylindricalShells. Proceedings of the Fifth Israel Annual Conference of Mechanical Engineering, Israel J. of Techn. 9, No. 4, 397–410 (1971).

    Google Scholar 

  15. Sobel, L. H.: Effects of Boundary Conditions on the Stability of Cylinders Subject to Lateral and Axial Pressures. AIAA J. 2, No. 8, 1437–1440 (1964).

    Article  MATH  Google Scholar 

  16. Singer, J.: The Effect of Axial Constraint on the Instability of Thin Circula Cylindrical Shells under External Pressure. J. of Appl. Mech. 27, No. 4, 737–739 (1960).

    Article  Google Scholar 

  17. Forsberg, K.: Influence of Boundary Conditions on the Modal Eesponse of Thin Cylindrical Shells. AIAA J. 2, No. 12, 2150–2157 (1964).

    Article  Google Scholar 

  18. Eesnick, B. S.; Dujundji, J.: Effects of Orthotropicity, Boundary Conditions and Eccentricity on the Vibrations of Cylindrical Shells. AFOSR Scientific Rep. AFOSR 66–2821, ASRL TR 134–2, M.I.T. Aeroelastic and Structures Research Laboratory, Nov. 1966.

    Google Scholar 

  19. Penzes, L. E.: Effect of Boundary Conditions on Flexural Vibrations of Thin Orthogonally Stiffened Cylindrical Shells. J. of Acous. Soc. Am. 42, No. 4, 901–903 (1967).

    Article  Google Scholar 

  20. Sewall, J. L.; Naumann, E. C.: An Experimental and Analytical Vibration Study of Thin Cylindrical Shells With and Without Longitudinal Stiffeners. NASA TN D-4705, September 1968.

    Google Scholar 

  21. Patnaik, S.; Sankaran, G. V.: Vibrations of Initially Stressed Stiffened Circular Cylinders and Panels. J. of Sound and Vibration 31, No. 3, 369–382 (1973).

    Article  Google Scholar 

  22. Rosen, A.; Singer, J.: Vibrations of Axially Loaded Stiffened Cylindrical Shells: Part I-Theoretical Analysis. TAE Report No. 162, Technion- Israel Institute of Technology, Dept. of Aeronautical Engineering, Haifa, Israel, February 1974.

    Google Scholar 

  23. Rosen, A.; Singer, J.: Vibrations of Axially Loaded Stiffened Cylindrical Shells: Part II-Experimental Analysis. TAE Report No. 163, Technion- Israel Institute of Technology, Dept. of Aeronautical Engineering, Haifa, Israel, August 1973.

    Google Scholar 

  24. Singer, J.: The Influence of Stiff ener Geometry and Spacing on the Buckling of Axially Compressed Cylindrical and Conical Shells. Theory of Thin Shells, Proceedings of 2nd lUTAM Symposium on Theory of Thin Shells, Copenhagen, September 1967. Berlin, Heidelberg, New York: Springer 1969, pp. 234–263.

    Google Scholar 

  25. Weiler, T.; Singer, J.: Experimental Studies on the Buckling of 7075-T6 Aluminum Alloy Integrally Stringer-Stiffened Shells. TAE Report No. 135, Technion Research and Development Foundation, Haifa, Israel, November 1971.

    Google Scholar 

  26. Rosen, A.; Singer, J.: Further Experimental Studies of Vibrations of Axially Loaded Stiffened Cylindrical Shells. TAE Report 210, Technion-Israel Institute of Technology, Dept. of Aeronautical Engineering, Haifa, Israel, May 1975.

    Google Scholar 

  27. Rosen, A.; Singer, J.: Vibrations of Axially Loaded Stiffened Cylindrical Shells with Elastic Restraints. TAE Report 208, Technion—Israel Institute of Technology, Dept. of Aeronautical Engineering, Haifa, Israel, January 1975.

    Google Scholar 

  28. Almroth, B. O.: Influence of Imperfections and Edge Restraint on the Buckling of Axially Compressed Cylinders. NASA CR-432, April 1966. Also presented at the AIAA/ASME 7th Structures and Materials Conference, Cocoa Beach, Florida, April 18–20, 1966.

    Google Scholar 

  29. Cohen, G. A.: Buckling of Axially Compressed Cylindrical Shells with Ring- Stiffened Edges. AIAA J. 4, No. 10, 1859–1862 (1966).

    Article  Google Scholar 

  30. Lurie, H.: Effective End Restraints of Columns by Frequency Measurements. J. of the Aeronautical Sciences 18, No. 8, 566–567 (1951).

    Google Scholar 

  31. Klein, B.: Determinations of Effective End Fixity of Columns with Unequal Rotational End Restraints by Means of Vibration Test Data. J. of the Royal Aeronautical Society 61, No. 554, 131–132 (1957).

    Google Scholar 

  32. Jacobson, M. J.; Wenner, M. L.: Predicting Buckling Loads from Vibration Data. Experimental Mechanics 8, No. 10, 35N-38N (1968).

    Article  Google Scholar 

  33. Burgreen, D.: End Fixity Effect on Vibration and Stability. J. Engng. Mech. Div. ASCE 86, 13–28 (1960).

    Google Scholar 

  34. Segall, A.: Nondestructive Dynamic Methods for the Determination of the Critical Loads of Elastic Columns. M. Sc. (Aeronautical Engineering) Thesis, Technion —Israel Institute of Technology, Haifa, Israel, June 1973 (in Hebrew).

    Google Scholar 

  35. Tobias, A. S.: A Theory of Imperfections for the Vibrations of Elastic Bodies of Revolution. Engng. 172, 409–410 (1951).

    Google Scholar 

  36. Lurie, H.: Lateral Vibrations as Related to Structural Stability. J. of Appl. Mech. ASME 19, 195–204 (1952).

    MATH  Google Scholar 

  37. Johnson, E. E.; Goldhammer, B. F.: The Determination of the Critical Load of a Column or Stiffened Panel in Compression by the Vibration Method. Proc. of the Society for Experimental Stress Analysis 11, No. 1, 221–232 (1953).

    Google Scholar 

  38. Okubo, S.; Whittier, J. S.: A Note on Buckling and Vibrations of an Externally Pressurized Shallow Spherical Shell. J. of Appl. Mech. ASME 34:, 1032–1034 (1967).

    Article  Google Scholar 

  39. Becker, H.: Nondestructive Testing for Structural Stability. J. of Ship Research 13, 272–275 (1969).

    Google Scholar 

  40. Craig, J. I.; Duggan, M. F.: Nondestructive Shell-Stability Estimation by a Combined-loading Technique. Experimental Mechanics 13, No. 9, 381–388 (1975).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technion-Israel Institute of Technology, Haifa, Israel

    J. Singer & A. Rosen

Authors
  1. J. Singer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Rosen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Harvard University, Cambridge, Massachusetts, USA

    Bernard Budiansky

Rights and permissions

Reprints and permissions

Copyright information

© 1976 Springer-Verlag, Berlin/Heidelberg

About this paper

Cite this paper

Singer, J., Rosen, A. (1976). The Influence of Boundary Conditions on the Buckling of Stiffened Cylindrical Shells. In: Budiansky, B. (eds) Buckling of Structures. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50992-6_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-50992-6_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-50994-0

  • Online ISBN: 978-3-642-50992-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation