# 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.

## Keywords

Experimental Data Mathematical Modeling Theoretical Analysis Theoretical Result Test Specimen## Notations

*A*_{t}total area of panel section

*a*_{m}=1−

*M*_{n}/*M*_{y}(Appendix*E*)*C*_{or}, C_{or}, and*C*_{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}and*D*_{4}depth of tension or compression yield zones in shell plate (Fig. 4)

*d, d*_{1},*d*_{2},*d*_{3}and*d*_{4}dimensionless depth of tension or compression yield zones in shell plate,

*d*_{i}=D_{i}/T_{p}*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}and*f*_{4}axial or circumferential moment functions

*G*_{0},*G*_{1},*G*_{2}and*G*_{4}coefficients in eq. (2. 31)

*H*depth of stringer

*I*_{1},*I*_{2},*I*_{3},*I*_{4},*I*_{5}and*I*_{6}integrals in Appendix

*I*_{21}and*I*_{22}integrals of eqs. (2.17)

*k*=1+sec

^{2}β_{1}*L*length of cylindrical shell

*L*_{1}distance where the stringer is tripped

*M*_{0}and*N*_{0}defining fully plastic bending moment membrane force per unit length for rigid-plastic cylindrical shell

*M*_{D}moment caused by

*F*_{1}in shell plate at*X=L*_{1}in a deformed position*M*_{n}- moment of fully plastic membrane force about a horizontal axis through
*M*_{p}moment in shell plate at

*x=L*_{1}without considering deformation*M*_{Q}moment caused by the foundational spring force

*Q*_{r}*M*_{s}moment in stringer

*M*_{s}and*M*_{b}moment produced by σ

_{α}and σ_{β}per unit length of the eages parallel to the 0 and*X*-axis*M*_{O}plastic moment when

*n*_{α}=0*m*_{e}=

*M*_{e}/*M*_{y}*m*_{x}=

*M*_{e}/2*M*_{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 and

^{θ}*X*-axis*n*_{z}=

*N*_{z}/*N*_{0}*n*_{6}=

*N*_{θ}/*N*_{0}*P*external axial force

*Q*_{1}and*Q*_{2}two parts of the Q

_{θ}*Q*_{R}reaction on left and of panel

*Q*_{r11},*Q*_{r12},*Q*_{r}*Q*_{r14},*Q*_{r21},*Q*_{r23}*Q*_{r}segments of

*Q*_{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}and*R*_{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}and*V*_{4}coefficients in eq. (2. 29)

- ω
=δ/

*T*_{p}*X*=axial coordinate

*X*_{p}axial shortening of shell plate due to radial deflection

- X
*r* 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}/T_{0}- α and β
lower integral and difference between lower and upper integral

- α
_{1} angle of deflection of stiffened panel (Fig. 11)

- β
_{1} inclination of plastic hinge to direction of thrust (Fig. 7)

- Γ
=Γ

_{1}+Γ_{2}- Γ
_{1}and Γ_{2} axial distances of skew hinge lines (Fig. 8)

- γ
=Γ/T

_{2}- 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}maximum

*e*_{i}used in calculation of one specimen*e*_{y}compressive yield strain of the material (=σ

_{y}/E)- ζ
=θ/θ

_{1}*n*ξ/Γ

- θ
circumferential coordinate

- θ
_{1} circumferential angle of skew hinge lines

- λ
_{ψ},λ_{1},λ_{2}and λ_{4} coefficients in eq. (2.19)

- μ
Poisson's ratio

- ξ
local axial coordinate

- ξ
_{1} =ξ/Γ

_{1}- ξ
_{2} =ξ/Γ

_{2}- ρ
_{ψ},ρ_{1},ρ_{2}and ρ_{4} coefficients in eq. (2.23)

- σ
_{H}and σ_{1} axial stresses corresponding

*e*_{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]}- σ
_{x}and σ_{ψ} axial and circumferential stresses

- σ
_{xx}and σ_{1x}^{*} stress in the states of plane strain and stress, respectively; when circumferential strain e

_{gψ}=0- σ
_{y} uniaxial yield stress of the material

- τ
=1−Γ/2Γ

_{1}

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## References

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