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Applied Mathematics and Mechanics

, Volume 5, Issue 6, pp 1831–1851 | Cite as

Analysis of postbuckling strength of stringer stiffened cylindrical shells under axial compression

  • Shao Wen-jiao
Article
  • 31 Downloads

Abstract

An analysis of the postbuckling strength of stringer stiffened cylindrical shells, subject to axial compression is described. The method used in this paper is based on plastic analysis extending Murray's method which was used to analyse postbuckling behaviour of stiffened plates loaded axially and in bending. The mechanism tripping of stringer and crumpling of shell plate is described based on how the test specimens are deformed after buckling.

Finally the theoretical analyses are compared with the experimental results of steel specimens. The theoretical results coincide quite well with the experimental data. It should therefore be possible to use the method described here to analyse postbuckling strength of stringer stiffened cylindrical shells and to estimate energy absorption capabilities in relation to collision studies.

Keywords

Experimental Data Mathematical Modeling Theoretical Analysis Theoretical Result Test Specimen 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notations

At

total area of panel section

am

=1−Mn/My (AppendixE)

Cor, Cor, andCor

coefficients

c

=e/T1

cu

\( = 4\left( {w + c + \frac{{M_y }}{{\sigma _y A,T_\rho }} + \frac{1}{2}} \right)\)

D, D1,D2,D3 andD4

depth of tension or compression yield zones in shell plate (Fig. 4)

d, d1,d2,d3 andd4

dimensionless depth of tension or compression yield zones in shell plate,di=Di/Tp

E

Young's modulus

e

eccentricity

F1

total axial force in shell plate

F1

total axial force instringer

f1,f2,f3 andf4

axial or circumferential moment functions

G0,G1,G2 andG4

coefficients in eq. (2. 31)

H

depth of stringer

I1,I2,I3,I4,I5 andI6

integrals in Appendix

I21 andI22

integrals of eqs. (2.17)

k

=1+sec2β1

L

length of cylindrical shell

L1

distance where the stringer is tripped

M0 andN0

defining fully plastic bending moment membrane force per unit length for rigid-plastic cylindrical shell

MD

moment caused byF1 in shell plate atX=L1 in a deformed position

Mn
moment of fully plastic membrane force about a horizontal axis through
Mp

moment in shell plate atx=L1 without considering deformation

MQ

moment caused by the foundational spring forceQr

Ms

moment in stringer

Ms andMb

moment produced by σα and σβ per unit length of the eages parallel to the 0 andX-axis

MO

plastic moment whennα=0

me

=Me/My

mx

=Me/2M01

mx

=Mx/Mc

mi

=Mθ/M0

n

number of stringers

N2,N0

axial and circumferential stress resultant per unit length of the edges parallel to the andθX-axis

nz

=Nz/N0

n6

=Nθ/N0

P

external axial force

Q1 andQ2

two parts of the Qθ

QR

reaction on left and of panel

Qr11,Qr12,QrQr14,Qr21,Qr23Qr

segments ofQr

Qθ

the foundational spring force per unit length

R

mean radius of shell

R0

the ratio of stringer area to area of shell plate of panel

Ra1h andRa14

coefficients

R0

\( = \left( {\frac{{1 - \mu }}{{1 - 2\mu }}} \right)^2 \)

s

half of the distance between the stringers

Tp

thickness of shell plate

Ts

thickness of stringer

V0,V1,V2 andV4

coefficients in eq. (2. 29)

ω

=δ/Tp

X

=axial coordinate

Xp

axial shortening of shell plate due to radial deflection

Xr

axial shortening at the end edge of the stringer due to lateral deflection Δ

Z

derlection of shell plate for θ,⩽θ⩽θπ/n

z

=Z/T0

Z0

maximum deflection of shell plate

z0

=Z0/T0

α and β

lower integral and difference between lower and upper integral

α1

angle of deflection of stiffened panel (Fig. 11)

β1

inclination of plastic hinge to direction of thrust (Fig. 7)

Γ

12

Γ1 and Γ2

axial distances of skew hinge lines (Fig. 8)

γ

=Γ/T2

Δ

local maximum deflection of stringer

δ

radial deflection of stiffened panel (Fig. 11)

e1

value of calculating strain (times the yield strain)

emax

maximumei used in calculation of one specimen

ey

compressive yield strain of the material (=σy/E)

ζ

=θ/θ1

n

ξ/Γ

θ

circumferential coordinate

θ1

circumferential angle of skew hinge lines

λψ12 and λ4

coefficients in eq. (2.19)

μ

Poisson's ratio

ξ

local axial coordinate

ξ1

=ξ/Γ1

ξ2

=ξ/Γ2

ρψ12 and ρ4

coefficients in eq. (2.23)

σH and σ1

axial stresses correspondinge1 with and without consideration of material hardening, respectively

σ1* and σx*

local buckling stress of an equivalent simply supported panel and overall buckling stress of the shell shell[9]

σx and σψ

axial and circumferential stresses

σxx and σ1x*

stress in the states of plane strain and stress, respectively; when circumferential strain e=0

σy

uniaxial yield stress of the material

τ

=1−Γ/2Γ1

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References

  1. (1).
    Murray, N.W., Buckling of stiffened panels loaded axially and in bending, The Structural Engineer, Vol. 51, No. 8, (1973), 285–301.Google Scholar
  2. (2).
    Carlsen, C.A., W.J. Shao and S. Fredheim. Experimental and Theretical Analysis of Post Buckling Strength of Flat Bar Stiffeners Subject to Tripping, Det norske Veritas, Report No. 80-149, (1980).Google Scholar
  3. (3).
    Hodge, P.G., Jr., Yield conditions for rotationally symmetric shells under axisymmetric loading, J. of Applied Mechanics, Vol. 27, Trans. ASME, Vol. 82, Series E, June (1960), 323–331.Google Scholar
  4. (4).
    Hodge, P.G., Jr., Plastic Analysis of Structures, McGraw-Hill Book Company, New York, (1959).Google Scholar
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    Matheson, J.A.L., Hyperstatic Structures, 2nd Edition, Bullesllonth (1979). (1959). 382.Google Scholar
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    Horne, MR., Plastic Theory of Structures, 2nd Edition, Pergamon Press, (1979).Google Scholar
  7. (7).
    Murray, N.W., Das aufnehmbare Moment in einem zur Richtung der Normalkraft schräg liegendeb plastischen Gelenk, die Bautechnik, Vol. 2, (1973), 57–58.Google Scholar
  8. (8).
    Sokolnikoff, I.S., Mathematical Theory of Elasticity, McGraw-Hill, (1956).Google Scholar
  9. (9).
    Sridharan, S. and A.C. Walker, Experimental Investigation of the Buckling Behaviour of Stiffened Cylindrical Shells, Department of Energy Report No. OT-R7835, Department of Civil and Municipal Engineering, University College, London, Feb. (1980).Google Scholar
  10. (10).
    Walker, A.C. and P. Davies, The Collapse of Stiffened Cylinders Steel Plated Structures, An International Conference, Ed. Dowling et al. Crosby Lockwood Staples, (1976), 791–901.Google Scholar

Copyright information

© HUST Press 1984

Authors and Affiliations

  • Shao Wen-jiao
    • 1
  1. 1.Shanghai Research Institute of Ships and ShippingShanghai

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