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

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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.

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Abbreviations

A t :

total area of panel section

a m :

=1−M n/M y (AppendixE)

C or, Cor, andC or :

coefficients

c :

=e/T 1

c u :

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

D, D 1,D 2,D 3 andD 4 :

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

d, d 1,d 2,d 3 andd 4 :

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

E :

Young's modulus

e :

eccentricity

F 1 :

total axial force in shell plate

F 1 :

total axial force instringer

f 1,f 2,f 3 andf 4 :

axial or circumferential moment functions

G 0,G 1,G 2 andG 4 :

coefficients in eq. (2. 31)

H :

depth of stringer

I 1,I 2,I 3,I 4,I 5 andI 6 :

integrals in Appendix

I 21 andI 22 :

integrals of eqs. (2.17)

k :

=1+sec2β1

L :

length of cylindrical shell

L 1 :

distance where the stringer is tripped

M 0 andN 0 :

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

M D :

moment caused byF 1 in shell plate atX=L 1 in a deformed position

M n :

moment of fully plastic membrane force about a horizontal axis through

figure 1
M p :

moment in shell plate atx=L 1 without considering deformation

M Q :

moment caused by the foundational spring forceQ r

M s :

moment in stringer

M s andM b :

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

M O :

plastic moment whenn α=0

m e :

=M e/M y

m x :

=M e/2M 0 Rθ 1

m x :

=M x/M c

m i :

=M θ/M 0

n :

number of stringers

N 2,N 0 :

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

n z :

=N z/N 0

n 6 :

=N θ/N 0

P :

external axial force

Q 1 andQ 2 :

two parts of the Qθ

Q R :

reaction on left and of panel

Q r11,Q r12,Q r Q r14,Q r21,Q r23 Q r :

segments ofQ r

Q θ :

the foundational spring force per unit length

R :

mean radius of shell

R 0 :

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

R a1h andR a14 :

coefficients

R 0 :

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

s :

half of the distance between the stringers

T p :

thickness of shell plate

T s :

thickness of stringer

V 0,V 1,V 2 andV 4 :

coefficients in eq. (2. 29)

ω:

=δ/T p

X :

=axial coordinate

X p :

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/T 0

Z 0 :

maximum deflection of shell plate

z 0 :

=Z 0/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:

=Γ1+Γ2

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)

e 1 :

value of calculating strain (times the yield strain)

e max :

maximume i used in calculation of one specimen

e y :

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

ζ:

=θ/θ1

n :

ξ/Γ

θ:

circumferential coordinate

θ1 :

circumferential angle of skew hinge lines

λψ,λ1,λ2 and λ4 :

coefficients in eq. (2.19)

μ:

Poisson's ratio

ξ:

local axial coordinate

ξ1 :

=ξ/Γ1

ξ2 :

=ξ/Γ2

ρψ,ρ1,ρ2 and ρ4 :

coefficients in eq. (2.23)

σH and σ1 :

axial stresses correspondinge 1 with and without consideration of material hardening, respectively

σ *1 and σ *x :

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 σ *1x :

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

σy :

uniaxial yield stress of the material

τ:

=1−Γ/2Γ1

References

  1. Murray, N.W., Buckling of stiffened panels loaded axially and in bending, The Structural Engineer, Vol. 51, No. 8, (1973), 285–301.

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  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).

  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.

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  4. Hodge, P.G., Jr., Plastic Analysis of Structures, McGraw-Hill Book Company, New York, (1959).

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  5. Matheson, J.A.L., Hyperstatic Structures, 2nd Edition, Bullesllonth (1979). (1959). 382.

  6. Horne, MR., Plastic Theory of Structures, 2nd Edition, Pergamon Press, (1979).

  7. Murray, N.W., Das aufnehmbare Moment in einem zur Richtung der Normalkraft schräg liegendeb plastischen Gelenk, die Bautechnik, Vol. 2, (1973), 57–58.

  8. Sokolnikoff, I.S., Mathematical Theory of Elasticity, McGraw-Hill, (1956).

  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).

  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.

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Authors and Affiliations

  1. Shanghai Research Institute of Ships and Shipping, Shanghai

    Shao Wen-jiao

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  1. Shao Wen-jiao
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Communicated by Zhong Wan-xie.

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Wen-jiao, S. Analysis of postbuckling strength of stringer stiffened cylindrical shells under axial compression. Appl Math Mech 5, 1831–1851 (1984). https://doi.org/10.1007/BF01904928

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  • DOI: https://doi.org/10.1007/BF01904928

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