DNV-RP-C202: Buckling Strength of Shells · PDF fileDNV-RP-C202 Buckling Strength of Shells ... This RP treats the buckling stability of shell structures based on the load and resistance

RECOMMENDED PRACTICE

The electronic

DNV-RP-C202

Buckling Strength of ShellsJANUARY 2013

DET NORSKE VERITAS AS

pdf version of this document found through http://www.dnv.com is the officially binding version

FOREWORD

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© Det Norske Veritas AS January 2013

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Recommended Practice DNV-RP-C202, January 2013Changes – Page 3

CHANGES

General

This document supersedes DNV-RP-C202, October 2010.

Text affected by the main changes in this edition is highlighted in red colour. However, if the changes involvea whole chapter, section or sub-section, normally only the title will be in red colour.

Main Changes

• General— This edition of the document contains only minor corrections and clarifications.

• Sec.1 Introduction— 1.3: “Design applying numerical methods” has been added.

• Sec.3 Buckling Resistance of Cylindrical Shells— Table 3-2: Note 1 and 2 have been updated. — 3.5: Ring Stiffened Shell Limitation in equation 3.5.8 has been corrected.

In addition to the above stated main changes, editorial corrections may have been made.

Editorial Corrections


Recommended Practice DNV-RP-C202, January 2013Contents – Page 4

CONTENTS

1. Introduction............................................................................................................................................ 51.1 Buckling strength of shells........................................................................................................................51.2 Working Stress Design .............................................................................................................................51.3 Design applying numerical methods.........................................................................................................61.4 Symbols and Definitions...........................................................................................................................61.5 Buckling modes ........................................................................................................................................82. Stresses in Closed Cylinders ............................................................................................................... 102.1 General....................................................................................................................................................102.2 Stresses....................................................................................................................................................103. Buckling Resistance of Cylindrical Shells.......................................................................................... 133.1 Stability requirement...............................................................................................................................133.2 Characteristic buckling strength of shells ...............................................................................................133.3 Elastic buckling strength of unstiffened curved panels ..........................................................................143.4 Elastic buckling strength of unstiffened circular cylinders.....................................................................143.5 Ring stiffened shells................................................................................................................................153.6 Longitudinally stiffened shells................................................................................................................193.7 Orthogonally stiffened shells ..................................................................................................................213.8 Column buckling.....................................................................................................................................213.9 Torsional buckling ..................................................................................................................................233.10 Local buckling of longitudinal stiffeners and ring stiffeners..................................................................244. Unstiffened Conical Shells................................................................................................................... 264.1 Introduction.............................................................................................................................................264.2 Stresses in conical shells.........................................................................................................................264.3 Shell buckling .........................................................................................................................................27


Recommended Practice DNV-RP-C202, January 2013Sec.1 Introduction – Page 5

1 Introduction

1.1 Buckling strength of shells

This RP treats the buckling stability of shell structures based on the load and resistance factor design format(LRFD). Sec.2 gives the stress in closed cylinders. Sec.3 treats the buckling of circular cylindrical steel shells,see Figure 1-1. The shell cylinder may be stiffened by longitudinal stiffeners and/or ring frames.

Figure 1-1Stiffened cylindrical shell

It is assumed that the edges are effectively supported by ring frames, bulkheads or end closures.

Stiffened circular cylindrical shells have to be dimensioned against several buckling failure modes. Therelevant modes are defined in 1.5. To exclude local buckling of longitudinal stiffeners and rings, explicitrequirements are given in 3.10.

In Table 3-1 reference is made to recommended methods for buckling analysis with respect to differentbuckling modes. The methods are to be considered as semi-empirical. The reason for basing the design on semi-empirical methods is that the agreement between theoretical and experimental buckling loads for some caseshas been found to be non-existent. This discrepancy is due to the effect of geometric imperfections and residualstresses in fabricated structures. Actual geometric imperfections and residual stresses do not in general appearas explicit parameters in the expressions for buckling resistance. This means that the methods for bucklinganalysis are based on an assumed level of imperfections. This level is reflected by the tolerance requirementsgiven in DNV OS-C401; Fabrication and Testing of Offshore Structures.

The recommended methods for buckling analyses may be substituted by more refined analyses or model teststaking into account the real boundary conditions, the pre-buckling edge disturbances, the actual geometricimperfections, the non-linear material behaviour, and the residual welding stresses.

Sec.4 treats the buckling of unstiffened conical shells.

1.2 Working Stress Design

This Recommended Practice is written in the load and resistance factor design format (LRFD format) to suitthe DNV Offshore Standard DNV-OS-C101. This standard makes use of material- (resistance) and loadfactorsas safety factors. DNV-RP-C202 may be used in combination with working stress design format (WSD) by the followingmethod: For the formulas used in DNV-RP-C202, including eq. 3.1.3, use a material factor γM=1.15. The utilisationchecks should be made using a modified permissible usage factor ηp=1.15βη0, see DNV-OS-C201 Sec.2 TableE1 for η0 and Sec. 5 Table C1 for β.

Guidance note:

β given in DNV-OS-C201 Sec.5 C corresponds to 1.15/γM, γM as given in eq. 3.1.3.

---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---

r

sRING FRAME

LONGITUDINALSTIFFENER

L

l

2

1

X

l

l

N

T

Q1

θ M1

M2

Q2

σx

σh

τ

P



1.3 Design applying numerical methodsIn cases where recommendations for buckling checks are not explicit given herein, the analysis may be carriedout by numerical analysis (analyses by finite element method, FEM) provided that the following effects areaccounted for:

— imperfections— material non-linearities— residual stresses— possible interaction between local and global buckling.

Mesh sensitivity analyses shall be carried out. The analysis methodology including calibration factors shall bechecked against known design cases according to this RP.

1.4 Symbols and Definitions

1.4.1 Symbols

The following symbols are used and may not have a specific definition in the text where they appear:

A cross-sectional area of a longitudinal stiffener (exclusive of shell flange)Ac cross sectional area of complete cylinder section; including longitudinal stiffeners/internal bulkheads if anyAf cross sectional area of flange (=btf)AR cross-sectional area of a ring frame (exclusive of shell flange)AReq required cross sectional area (exclusive of effective plate flange) of ring frame to avoid panel ring bucklingAw cross sectional area of web (=htw)C reduced buckling coefficientC1 coefficientC2 coefficientE Young's modulus = 2.1 ⋅ 105 N/mm2

G shear modulus,

I moment of inertia of a longitudinal stiffener (exclusive of shell flange)Ic moment of inertia of the complete cylinder section (about weakest axis), including longitudinal stiffeners/internal

bulkheads if anyIpo polar moment of inertiaIR effective moment of inertia of a ring frameIsef moment of inertia of longitudinal stiffener including effective shell width seIt stiffener torsional moment of inertia (St. Venant torsion).Iz moment of inertia of a stiffeners neutral axis normal to the plane of the plateIh minimum required moment of inertia of ringframes inclusive effective shell flange in a cylindrical shell subjected

to external lateral or hydrostatic pressureIx minimum required moment of inertia of ringframes inclusive effective shell flange in a cylindrical shell subjected

to axial and/or bendingIxh minimum required moment of inertia of ringframes inclusive effective shell flange in a cylindrical shell subjected

to torsion and/or shearL distance between effective supports of the ring stiffened cylinderLc total cylinder lengthLH equivalent cylinder length for heavy ring frameMSd design bending momentM1, Sd design bending moment about principal axis 1M2, Sd design bending moment about principal axis 2NSd design axial forceQSd design shear forceQ1,Sd design shear force in direction of principal axis 1Q2,Sd design shear force in direction of principal axis 2TSd design torsional moment

, curvature parameter


( )ν+=

12

EG

22

L ν1rt

LZ −=

22

ν-1rt

=Zl

l




a Factorb flange width, factorbf flange outstandc Factore distance from shell to centroid of ring frame exclusive of any shell flangeef flange eccentricityfak reduced characteristic buckling strengthfakd design local buckling strengthfE elastic buckling strengthfEa elastic buckling strength for axial force.fEh elastic buckling strength for hydrostatic pressure, lateral pressure and circumferential compression.fEm elastic buckling strength for bending moment.fET elastic buckling strength for torsion.fEτ elastic buckling strength for shear force.fk characteristic buckling strengthfkc characteristic column buckling strengthfkcd design column buckling strengthfks characteristic buckling strength of a shellfksd design buckling strength of a shellfr characteristic material strengthfT torsional buckling strengthfy yield strength of the materialh web heighths distance from stiffener toe (connection between stiffener and plate) to the shear centre of the stiffener.i radius of gyrationic radius of gyration of cylinder sectionih effective radius of gyration of ring frame inclusive affective shell flangek effective length factor, column bucklingl distance between ring framesle equivalent length of conical shelllef effective width of shell platingleo equivalent lengthlT torsional buckling lengthpSd design lateral pressurer shell radius (midtplane)re equivalent radius of conical shellrf radius of the shell measured to the ring flangerr radius (variable)r0 radius of the shell measured to the neutral axis of ring frame with effective shell flange, leos distance between longitudinal stiffenersse effective shell widtht shell thicknesstb thickness of bulkheadte equivalent thicknesstf thickness of flangetw thickness of webw initial out-of-roundnesszt distance from outer edge of ring flange to centroid of stiffener inclusive effective shell platingα angleα, αA coefficientsαB, αC coefficientsβ coefficientδ0 initial out-of-roundness parameter

22

s ν1rt

sZ −=



1.4.2 Definitions

A general ring frame cross section is shown Figure 1-2.

Figure 1-2Cross sectional parameters for a ring frame

1.5 Buckling modesThe buckling modes for stiffened cylindrical shells are categorised as follows:

a) Shell buckling: Buckling of shell plating between rings/ longitudinal stiffeners.

b) Panel stiffener buckling: Buckling of shell plating including longitudinal stiffeners. Rings are nodal lines.

γM material factorη coefficient⎯λ reduced column slenderness⎯λs reduced shell slenderness⎯λT reduced torsional slendernessμ Coefficientθ circumferential co-ordinate measured from axis 1ρ Coefficientν Poisson's ratio = 0.3σa,Sd design membrane stress in the longitudinal direction due to uniform axial forceσh,Sd design membrane stress in the circumferential directionσhR,Sd design membrane stress in a ring frameσhm,Sd design circumferential bending stress in a shell at a bulkhead or a ringframeσj,Sd design equivalent von Mises’ stressσm,Sd design membrane stress in the longitudinal direction due to global bendingσx,Sd design membrane stress in the longitudinal directionσxm,Sd design longitudinal bending stress in a shell at a bulkhead or a ringframeτSd design shear stress tangential to the shell surface (in sections x = constant and θ = constant)τT,Sd design shear stress tangential to the shell surface due to torsional momentτQ,Sd design shear stress tangential to the shell surface due to overall shear forcesξ coefficientψ coefficientζ coefficient

A Centroid of ring frame with effective shell flange, leoB Centroid of ring frame exclusive any shell flangeC Centroid of free flange

A

B

twe

tz t

f

e

bCf

h

teo

r

r

r f

0

bf

l



c) Panel ring buckling: Buckling of shell plating including rings. Longitudinal stiffeners act as nodal lines.d) General buckling: Buckling of shell plating including longitudinal stiffeners and rings.e) Column buckling: Buckling of the cylinder as a column.

For long cylindrical shells it is possible that interaction between local buckling and overall column bucklingmay occur because second order effects of axial compression alter the stress distribution calculated fromlinear theory. It is then necessary to take this effect into account in the column buckling analysis. This isdone by basing the column buckling on a reduced yield strength, fkc, as given for the relevant type ofstructure.

f) Local buckling of longitudinal stiffeners and rings. Section 3.10

The buckling modes and their relevance for the different cylinder geometries are illustrated in Table 1-1.

Table 1-1 Buckling modes for different types of cylindersBuckling mode Type of structure geometry

Ring stiffened(unstiffened circular)

Longitudinal stiffened Orthogonally stiffened

a) Shell buckling

Section 3.4 Section 3.3 Section 3.3

b) Panel stiffener buckling

Section 3.6 Section 3.7c) Panel ring buckling

Section 3.5 Section 3.7

d) General buckling

Section 3.7e) Column buckling

Section 3.8 Section 3.8 Section 3.8


Recommended Practice DNV-RP-C202, January 2013Sec.2 Stresses in Closed Cylinders – Page 10

2 Stresses in Closed Cylinders

2.1 General

The stress resultants governing the stresses in a cylindrical shell is normally defined by the followingquantities:

Any of the above quantities may be a function of the axial co-ordinate x. In addition pSd may be a function ofthe circumferential co-ordinate θ, measured from axis 1. pSd is always to be taken as the difference betweeninternal and external pressures, i.e. pSd is taken positive outwards.

Actual combinations of the above actions are to be considered in the buckling strength assessments.

2.2 Stresses

2.2.1 General

The membrane stresses at an arbitrary point of the shell plating, due to any or all of the above five actions, arecompletely defined by the following three stress components:

2.2.2 Longitudinal membrane stress

If the simple beam theory is applicable, the design longitudinal membrane stress may be taken as:

where σa,Sd is due to uniform axial force and σm,Sd is due to bending.

For a cylindrical shell without longitudinal stiffeners:

For a cylindrical shell with longitudinal stiffeners it is usually permissible to replace the shell thickness by theequivalent thickness for calculation of longitudinal membrane stress only:

2.2.3 Shear stresses

If simple beam theory is applicable, the membrane shear stress may be taken as:

where τT,Sd is due to the torsional moment and τQ,Sd is due to the overall shear forces.

where the signs of the torsional moment and the shear forces must be reflected. Circumferential andlongitudinal stiffeners are normally not considered to affect τSd.

NSd = Design axial forceMSd = Design bending momentsTSd = Design torsional momentQSd = Design shear forcepSd = Design lateral pressure

σx,Sd = design membrane stress in the longitudinal direction (tension is positive)σh,Sd = design membrane stress in the circumferential direction (tension is positive)τSd = design shear stress tangential to the shell surface (in sections x = constant and θ = constant)

(2.2.1)

(2.2.2)

(2.2.3)

(2.2.4)

(2.2.5)

(2.2.6)

(2.2.7)

Sdm,Sda,Sdx, σσσ +=

trπ2

Nσ Sd

Sda, =

cosθtπr

Msinθ

tπr

Mσ

2Sd2,

2Sd1,

Sdm, −=

s

Att e +=

SdQ,τSdT,τSdτ +=

trπ2

Tτ

2Sd

SdT, =

cosθtrπSd2,Q

sinθtrπ

Sd1,Q

SdQ,τ +−=



2.2.4 Circumferential membrane stress

For an unstiffened cylinder the circumferential membrane stress may be taken as:

provided pSd is constant (gas pressure) or a sine or cosine function of θ (liquid pressure).

For a ringstiffened cylinder (without longitudinal stiffeners) the circumferential membrane stress midwaybetween two ring frames may be taken as:

where

ζ and λeo may also be obtained from Figure 2-1.For simplification of the analysis the following approximation may be made:

or whichever is the smaller.

For the particular case when pSd is constant and σx,Sd is due to the end pressure alone, the above formula maybe written as:

Figure 2-1The parameters leo and ζ

2.2.5 Circumferential stress in a ring frame

For ring stiffened shells the circumferential stress in a ring frame at the distance rr (rr is variable, rr = rf at ringflange position and rr = r at shell) from the cylinder axis may be taken as:

(2.2.8)

(2.2.9)

(2.2.10)

(2.2.11)

(2.2.12)

(2.2.13)

(2.2.14)

(2.2.15)

t

rSdpSdh,σ =

−

+−= Sdx,σν

t

rSdp

1α

ζα

t

rSdpSdh,σ

0ζbut,2βsin2βSinh

sinβCoshβcosβSinhβ2ζ ≥

++

=

tr1.56β

l=

t

Aα

eo

R

l=

+−=

2βsin2βSinh

2βcos2βCosh

βeo

ll

ll =eo tr56.1eo =l

+

−

−=1α

ζ2

ν1α

1t

rpσ Sd

Sdh,

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

β

rt56.1e0l

ζ

+

−=

rSdx,

SdSdhR, r

r

α1

1νσ

t

rpσ



For the particular case when pSd is constant and σx,Sd is due to the end pressure alone, the above formula canbe written as:

For longitudinally stiffened shells α should be replaced by in eq. (2.2.15) and (2.2.16).

2.2.6 Stresses in shells at bulkheads and ring stiffeners

2.2.6.1 GeneralThe below stresses may be applied in a check for local yielding in the material based on a von Mises’ equivalentstress criterion. The bending stresses should also be accounted for in the fatigue check, but may be neglectedin the evaluation of buckling stability.

2.2.6.2 Circumferential membrane stressThe circumferential membrane stress at a ring frame for a ring stiffened cylinder (without longitudinalstiffeners) may be taken as:

In the case of a bulkhead instead of a ring, AR is taken as , where tb is the thickness of the bulkhead. Forthe particular case when pSd is constant and σx,Sd is due to the end pressure alone, the above formula can bewritten as:

2.2.6.3 Bending stressBending stresses and associated shear stresses will occur in the vicinity of “discontinuities” such as bulkheadsand frames. The longitudinal bending stress in the shell at a bulkhead or a ring frame may be taken as:

where σh,Sd is given in (2.2.17) or (2.2.18).The circumferential bending stress in the shell at a bulkhead or a ring frame is:

(2.2.16)

(2.2.17)

(2.2.18)

(2.2.19)

(2.2.20)

r

SdSdhR, r

r

α12

ν1

t

rpσ

+

−=

t

AR

l

Sdx,Sdx,Sd

Sdh, νσα1

1νσ

t

rpσ +

+

−=

( )ν-1

tr b

++

−=

2

ν

α12

ν1

t

rpσ Sd

Sdh,

2Sdh,Sd

Sdxm,ν1

3σ

t

rpσ

−

−=

Sdxm,Sdm,h νσσ =


Recommended Practice DNV-RP-C202, January 2013Sec.3 Buckling Resistance of Cylindrical Shells – Page 13

3 Buckling Resistance of Cylindrical Shells

3.1 Stability requirement

The stability requirement for shells subjected to one or more of the following components:

— axial compression or tension— bending— circumferential compression or tension— torsion— shear

is given by:

σj,Sd is defined in Section 3.2, and the design shell buckling strength is defined as:

The characteristic buckling strength, fks, is calculated in accordance with 3.2.

The material factor, γM, is given as:

Shell structures may be subjected to global column buckling. Evaluation of global column buckling is found in3.8.

3.2 Characteristic buckling strength of shells

The characteristic buckling strength of shells is defined as:

where

(3.1.1)

(3.1.2)

(3.1.3)

(3.2.1)

(3.2.2)

(3.2.3)

(3.2.4)

(3.2.5)

(3.2.6)

σa,Sd = design axial stress in the shell due to axial forces (tension positive), see eq. (2.2.2)σm,Sd = design bending stress in the shell due to global bending moment (tension positive), see eq. (2.2.3).σh,Sd = design circumferential stress in the shell due to external pressure (tension positive), see eq (2.2.8), (2.2.9), or

(2.2.14). For ring stiffened cylinders shall only stresses midway between rings be usedτSd = design shear stress in the shell due to torsional moments and shear force, see eq. (2.2.5)

ksdSdj, fσ ≤

M

ksksd γ

ff =

1.0λfor1.45γ

1.0λ0.5forλ0.600.85γ

0.5λfor 1.15γ

sM

ssM

sM

>=

≤≤+=

<=

4s

yks

+1

ff

λ=

+++=

Eτ

Sd

Eh

Sdh0,

Em

Sdm0,

Ea

Sda0,

Sdj,

y2s

f

τ

f

σ

f

σ

f

σ

σ

fλ

( ) ( ) 2Sd

2Sdh,Sdh,Sdm,Sda,

2Sdm,Sda,

Sdj,

3τσσσσσσ

σ

+++−+

=

<≥

−=

0σ if

0σ if

σ

0σ

Sda,

Sda,

Sda,Sda0,

<≥

−=

0σ if

0σ if

σ

0σ

Sdm,

Sdm,

Sdm,Sdm0,

<−≥

=pressurenet ext. 0,σ if σ

pressurenet internal 0,σ if 0σ

Sdh,Sdh,

Sdh,

Sdh0,



fEa, fEm, fEh and fEτ are the elastic buckling strengths of curved panels or circular cylindrical shells subjectedto axial compression forces, global bending moments, lateral pressure, and torsional moments and/or shearforces respectively, where:

These may be calculated in accordance with 3.3 to 3.7 taking the appropriate buckling coefficients into account.

3.3 Elastic buckling strength of unstiffened curved panels

3.3.1 General

This section deals with buckling of shell plate between stiffeners.

The buckling mode to be checked is:

a) Shell buckling, see 3.3.2.

3.3.2 Shell buckling

The characteristic buckling strength is calculated from 3.2.

The elastic buckling strength of curved panels with aspect ratio l/s > 1 is given by:

A curved panel with aspect ratio l/s < 1 may be considered as an unstiffened circular cylindrical shell withlength equal to l, see 3.4.2.

The reduced buckling coefficient may be calculated as:

The values for ψ, ξ and r are given in Table 3-1 for the most important load cases.

The curvature parameter Zs is defined as:

3.4 Elastic buckling strength of unstiffened circular cylinders

3.4.1 General

The buckling modes to be checked are:


b) Column buckling, see 3.8.

fEa = elastic buckling strength for axial forcefEm = elastic buckling strength for bending momentfEh = elastic buckling strength for hydrostatic pressure, lateral pressure and circumferential compressionfEτ = elastic buckling strength for torsion and shear force

(3.3.1)

(3.3.2)

Table 3-1 Buckling coefficient for unstiffened curved panels, mode a) Shell buckling

ψ ξ ρ

Axial stress 4

Shear stress 0.6

Circumferential compression 0.6

(3.3.3)

2

2

2

E s

t

)-12(1

ECf

=

νπ

2

ψ

ρξ+1ψ=C

0 702. Zs 0 5 10 5

..

+

−r

150t

2s

434.5

+

l3/4sZ

s856.0

l22

s1

+

l sZs

04.1l

22

s ν-1rt

s=Z



3.4.2 Shell buckling

The characteristic buckling strength of unstiffened circular cylinders is calculated from 3.2. The elasticbuckling strength of an unstiffened circular cylindrical shell is given by:


The values for ψ, ξ and r are given in Table 3-2 for the most important load cases.

The curvature parameter Z is defined as:

For long cylinders the solutions in Table 3-2 will be pessimistic. Alternative solutions are:

— Torsion and shear force

If then the elastic buckling strength may be calculated as:

— Lateral/hydrostatic pressure

If then the elastic buckling strength may be calculated as:

3.5 Ring stiffened shells

3.5.1 General

The buckling modes to be checked are:


b) Panel ring buckling, see 3.5.2.

c) Column buckling, see 3.8.

(3.4.1)

(3.4.2)

(3.4.3)

(3.4.4)

(3.4.5)

Table 3-2 Buckling coefficients for unstiffened cylindrical shells, mode a) Shell buckling

ψ ξ ρ

Axial stress 1

Bending 1

Torsion and shear force 5.34 0.6

Lateral pressure1)4 0.6

Hydrostatic pressure2)2 0.6

NOTE 1: Lateral pressure coefficient ψ = 4, accounts for lateral pressure on the cylinder shell only. NOTE 2: Hydrostatic pressure, ψ = 2, accounts for the effect of the lateral pressure on the cylinder shell and the end cap (i.e. axial stresses due to pressure on the end cap shall not be included in the calculation of axial stress, σa).

2t

)2ν-12(1

E2πCEf

=

l

2

ψ

ρξ+1ψ=C

22

ν-1rt

=Zl

l

t

r3.85

r>l

23

Eτ r

tE25.0f

=

t

r2.25

r>l

2

Eh r

tE25.0f

=

l Z702.0 0 5 10 5

..

+

−r

150t

l Z702.0 0 5 10 5

..

+

−r

300t

4/3 Z856.0 l

lZ04.1

lZ04.1



3.5.2 Panel ring bucklingThe rings will normally be proportioned to avoid the panel ring buckling mode. This is ensured if the followingrequirements are satisfied.

3.5.2.1 Cross sectional area.The cross sectional area of a ring frame (exclusive of effective shell plate flange) should not be less than AReq,which is defined by:

3.5.2.2 Moment of inertiaThe effective moment of inertia of a ring frame (inclusive effective shell plate flange) should not be less thanIR, which is defined by:

Ix, Ixh and Ih are defined in eq.(3.5.5), (3.5.7) and (3.5.8), (see also 3.5.2.7), the effective width of the shell plateflange is defined in 3.5.2.3.

3.5.2.3 Effective widthThe effective width of the shell plating to be included in the actual moment of inertia of a ring frame shall betaken as the smaller of:

and

3.5.2.4 Calculation of Ix

The moment of inertia of ring frames inclusive effective width of shell plate in a cylindrical shell subjected toaxial compression and/or bending should not be less than Ix, which is defined by:

where

A = cross sectional area of a longitudinal stiffener.

3.5.2.5 Calculation of Ixh

The moment of inertia of ring frames inclusive effective width of shell plate in a cylindrical shell subjected totorsion and/or shear should not be less than Ixh, which is defined by:

(3.5.1)

(3.5.2)

(3.5.3)

(3.5.4)

(3.5.5)

(3.5.6)

(3.5.7)

t06.0Z

2A

2Req ll

+≥

hxhxR IIII ++=

r

t121

rt1.56ef

+=l

ll =ef

( )lE 500

40rAα1tSdx,σ

xI+

=

ts

AαA =

ltLrL

r

Ε

τI 0

5/10

5/8Sd

xh

=



3.5.2.6 General calculation of Ih for external pressure

The moment of inertia of ring frames inclusive effective width of shell plate in a cylindrical shell subjected toexternal lateral pressure should not be less than Ih, which is conservatively defined by:

The ringframe deforms into an oval which corresponds to the buckling mode of long cylinders or heavy ringframes.The characteristic material resistance, fr, shall be taken as:

— For fabricated ring frames:fr= fT

— For cold-formed ring frames:fr= 0.9fT

The torsional buckling strength, fT, may be taken equal to the yield strength, fy, if the following requirementsare satisfied:

— Flat bar ring frames:

— Flanged ring frames (ef = 0, for ef ≠ 0 see 3.10):

Otherwise fT may be obtained from 3.9. zt is defined in Figure 1-2. For σhR,Sd see 2.2.5 and for pSd see 2.1.The assumed mode of deformation of the ring frame corresponds to ovalization, and the initial out-of-roundness is defined by:

Alternatively the capacity of the ring frame may be assessed from 3.5.2.7.

3.5.2.7 Special calculation of Ih for external pressure

If a ring stiffened cylinder of short or moderate length, is effectively supported at the ends, the followingprocedure may be used to calculate required moment of inertia Ih. For design it might be recommended to startwith equation (3.5.8) to arrive at an initial geometry. (The reason is that Ih is implicit in the present procedurein equations (3.5.23) and (3.5.27)). The required Ih shall be used in the check of moment of inertia in 3.5.2.2in cases where the shell is also exposed to axial loads and/or shear stresses.When a ring stiffened cylinder is subjected to external pressure the ring stiffeners should satisfy:

and

(3.5.8)

(3.5.9)

(3.5.10)

(3.5.11)

(3.5.12)

(3.5.13)

(3.5.14)

−+=

SdR,hr2

0

0t20Sd

h

2

fr

zE35.1

E3

rrpI

σ

δl

hR,Sd

r

2

f σ>

yW f

Et0.4h ≤

yW f

Et1.35h ≤

r

h

f

E10

h7b

y

+≥

θδ 2cosw 0=

r005.00 =δ

−

+

≤

21r

t

A1rt

f75.0p

2

eo

Rf

M

kSd νγ

l



where

where

The values for the parameters fr, fE and m may be taken as:

The characteristic material strength, fr, may be taken equal to the yield strength, fy, if the followingrequirements are satisfied:


— Flanged ring frames (ef = 0, for ef ≠ 0 see section 3.10):

Otherwise fr should be set to fT. fT may be obtained from 3.9.

where

pSd = design external pressuret = shell thicknessrf = radius of the shell measured to the ring flange, see Figure 1-2.r = shell radiusleo = smaller of and lAR = cross sectional area of ring stiffener (exclusive shell flange)fk = is the characteristic buckling strength found from:

(3.5.15)

(3.5.16)

(3.5.17)

(3.5.18)

(3.5.19)

(3.5.20)

(3.5.21)

(3.5.22)

(3.5.23)

(3.5.24)

(3.5.25)

(3.5.26)

(3.5.27)

rt56.1

2λ2

2λ422λμ12λμ1

rfkf −

++−++

=

E

r

f

fλ =

yW f

Et0.4h ≤

yW f

Et1.35h ≤

r

h

f

E10

h7b

y

+≥

2

2

2

1E L

t

)-12(1

ECf

=

νπ

( )

+−+

+

+=

Bα1Bα

Bα+1L Z0.27

1α1Bα12

1C

22

L 1tr

LZ ν−=

( )3

h2

Bt

I112α

l

ν−=

teo

RAα

l=

−

−=

21

1

C

C1

r

r

i

z

1

2

eo

f2h

0t

νδμ

l

l

0.005r0δ =

tA

Ii

eoR

h2h l+

=



L = distance between effective supports of the ring stiffened cylinder. Effective supports may be:

— End closures, see Figure 3-1a.— Bulkheads, see Figure 3-1b.— Heavy ring frames, see Figure 3-1c.

The moment of inertia of a heavy ring frame has to comply with the requirement given in 3.5.2.2 with Ix, Ixhand Ih defined in eq. (3.5.5), (3.5.7) and (3.5.8) and with l substituted by LH, which is defined in Figure 3-1d.

Figure 3-1Definition of parameters L and LH

3.6 Longitudinally stiffened shells

3.6.1 GeneralStiffened shells where may be designed according to the requirements given below or as anequivalent flat plate taking into account the design transverse stress, normally equal to pSd r/t. The bucklingmodes to be checked are:

a) Shell buckling, see 3.6.2.b) Panel stiffener buckling, see 3.6.3.c) Column buckling, see 3.8.

Lightly stiffened shells where will behave basically as an unstiffened shell and can be calculatedas an unstiffened shell. The stiffeners may be neglected both in the stress and capacity calculations. Thebuckling modes to be checked are:

a) Shell buckling, see 3.2 for characteristic buckling strength and 3.4.2 for elastic buckling strength

zt = distance from outer edge of ring flange to centroid of stiffener inclusive effective shell plating, see Figure 1-2.

(3.5.28)

a.

b.

c

d.

L2 Z27.012C +=

r/t3s/t ≤

r/t3s/t >



b) Column buckling, see 3.8.

3.6.2 Shell bucklingThe characteristic buckling strength is found from 3.2 and the elastic buckling strengths are given in 3.3.2.

3.6.3 Panel stiffener buckling

3.6.3.1 GeneralThe characteristic buckling strength is found from 3.2. It is necessary to base the strength assessment oneffective shell area. The axial stress σa,Sd and bending stress σm,Sd are per effective shell width, se is calculatedfrom 3.6.3.3.Torsional buckling of longitudinal stiffeners may be excluded as a possible failure mode if the followingrequirements are fulfilled:

— Flat bar longitudinal stiffeners:

— Flanged longitudinal stiffeners:

If the above requirements are not fulfilled for the longitudinal stiffeners, an alternative design procedure is toreplace the yield strength, fy, with the torsional buckling strength, fT, in all equations.

and fT may be found in 3.9.

3.6.3.2 Elastic buckling strengthThe elastic buckling strength of longitudinally stiffened cylindrical shells is given by:


The values for ψ, ξ and ρ are given in Table 3-3 for the most important load cases.

where

(3.6.1)

(3.6.2)

(3.6.3)

(3.6.4)

Table 3-3 Table 3.6-1Buckling coefficients for stiffened cylindrical shells, mode b) Panel stiffener buckling

ψ ξ ρ

Axial stress 0.5

Torsion and shear stress 0.6

Lateral Pressure 0.6

(3.6.5)

(3.6.6)

A = area of one stiffener, exclusive shell plateIsef = moment of inertia of longitudinal stiffener including effective shell width se, see eq. (3.6.7)

yW f

Et0.4h ≤

6.0T ≤λ

Tλ

2

2

2

Et

)-12(1

ECf

=

lνπ

2

ψ

ρξ1ψC

+=

ts

A1

α1

e

C

+

+

l Z702.0

1/3C

3/4

αs

82.134.5

+l 3/4 Z856.0 l

( )Cα112 ++ lZ04.1

22

1tr

Z ν−= ll

( )3

sef2

Cts

I112α

ν−=



3.6.3.3 Effective shell widthThe effective shell width, se, may be calculated from:

where:

3.7 Orthogonally stiffened shells

3.7.1 GeneralThe buckling modes to be checked are:

a) Shell buckling (unstiffened curved panels), see 3.7.2.b) Panel stiffener buckling, see 3.6.c) Panel ring buckling, see 3.7.3.d) General buckling, see 3.7.4.e) Column buckling, see 3.8.

3.7.2 Shell bucklingThe characteristic buckling strength is found from 3.2 and the elastic buckling strengths are given in 3.3.2.

3.7.3 Panel ring bucklingConservative strength assessment following 3.5.2.

3.7.4 General bucklingThe rings will normally be proportioned to avoid the general buckling mode. Applicable criteria are given in3.5.

3.8 Column buckling

3.8.1 Stability requirementThe column buckling strength should be assessed if

where

The stability requirement for a shell-column subjected to axial compression, bending and circumferentialcompression is given by:

(3.6.7)

fks = characteristic buckling strength from 3.2 (based on fE from 3.3.2 / 3.4.2)σj,sd = design equivalent von Mises stress, see eq. (3.2.3), using total area of shellσx,Sd = design membrane stress from axial force and bending moment, see eq. (2.2.1)fy = yield strength

(3.8.1)

k = effective length factorLC = total cylinder lengthiC = = radius of gyration of cylinder section

IC = moment of inertia of the complete cylinder section (about weakest axis), including longitudinal stiffeners/internal bulkheads if any

AC = cross sectional area of complete cylinder section; including longitudinal stiffeners/internal bulkheads if any

(3.8.2)

yf

Sdx,

Sdj,

ksf

ses σ

σ=

yf

E2.5

2

ci

ckL

≥

CC/AI

01.

0.52

E2fSda0,σ

1

Sdm2,σ

2

E1fSda0,σ

1

Sdm1,σ

akdf

1

kcdfSda0,σ

≤

−

+

−

+



where

3.8.2 Column buckling strengthThe characteristic buckling strength, fkc, for column buckling may be defined as:

where

In the general case eq. (3.1.1) shall be satisfied. Hence fak may be determined (by iteration of equations (3.1.1)to (3.2.6)) as maximum allowable σa0,Sd (σa,Sd) where the actual design values for σm,Sd, σh,Sd and τSd havebeen applied. For the special case when the shell is an unstiffened shell the following method may be used to calculate fak.

σa0,Sd = design axial compression stress, see eq. (3.2.4)σm,Sd = maximum design bending stress about given axis, see eq. (2.2.3)fakd = design local buckling strength, see 3.8.2fkcd = design column buckling strength, see eq. (3.8.4)fE1,fE2 = Euler buckling strength found from eq. (3.8.3):

(3.8.3)

(3.8.4)

γM = material factor, see eq. (3.1.3)fkc = characteristic column buckling strength, see eq. (3.8.5) or (3.8.6)

(3.8.5)

(3.8.6)

(3.8.7)

(3.8.8)

(3.8.9)

(3.8.10)

(3.8.11)

(3.8.12)

σh,Sd = design circumferential membrane stress, see eq. (2.2.8) or (2.2.9), tension positivefy = yield strengthγM = material factor, see eq. (3.1.3)fEa, fEh = elastic buckling strengths, see 3.4

2,1i,

cA2

ic,Lik

ic,EI2π

Eif =

=

M

kckcd γ

ff =

1.34for f]28.00.1[f ak2

kc ≤−= λλ

1.34for f9.0

f ak2kc >= λλ

Eakf

ciπckL

Efakf

λ ==

2a

4acbbf

2

ak

−+=

2Ea

2y

f

f1a +=

Sdh,EhEa

2y

σ1ff

2fb

−=

2y2

Eh

2sdh,

2y2

Sdh, ff

σfσc −+=

M

akakd γ

ff =



3.9 Torsional bucklingThe torsional buckling strength may be found from:

— if :

— if :

where:

Generally fET may be found from:

For L and T stiffeners fET may, when eqs. (3.10.4) and (3.10.5) are satisfied, be found from:

For flat bar ring stiffeners fET may be found from:

For flat bar longitudinal stiffeners fET may be found from:

(3.9.1)

(3.9.2)

(3.9.3)

(3.9.4)

(3.9.5)

(3.9.6)

(3.9.7)

(3.9.8)

(3.9.9)

β = 1.0, or may alternatively be calculated as per eq. (3.9.10)Af = cross sectional area of flangeAW = cross sectional area of webG = shear modulusIpo = polar moment of inertia = where r is measured from the connection between the stiffener and the plateIt = stiffener torsional moment of inertia (St. Venant torsion)Iz = moment of inertia about centroid axis of stiffener normal to the plane of the platelT = for ring stiffeners:

distance (arc length) between tripping brackets. lT need not be taken greater than for the analysis;for longitudinal stiffeners: distance between ring frames

b = flange widthef = flange eccentricity, see Figure 1-2h = web heighths = distance from stiffener toe (connection between stiffener and plate) to the shear centre of the stiffener

6.0λT ≤

0.1f

f

y

T =

6.0λT >

( )2T

2T

22T

2T

y

T

λ2

λ4λμ1λμ1

f

f −++−++=

( )6.0λ35.0μ T −=

ET

yT f

fλ =

2Tpo

z2s2

po

tET

I

IEhπ

I

GIβf

l+=

2Tf

W

z22

W

fW

f

2

W

fW

ET

A3

AEIπ

h

tG

A3A

At

tA

βfl

+

+

+

+

=

W

f

f2f

2fz

A

A1

AebA

12

1I

++=

2

wET h

tG

r

h2.0βf

+=

2w

2

TET h

tG

h2βf

+=

l

dAr2

rhπ



and

σj,Sd may be found from eq. (3.2.3) and fks may be calculated from eq. (3.2.1) using the elastic bucklingstrengths from 3.3.2 or 3.4.2.Ring frames in a cylindrical shell which is not designed for external lateral pressure shall be so proportionedthat the reduced slenderness with respect to torsional buckling, , is not greater than 0.6.

3.10 Local buckling of longitudinal stiffeners and ring stiffeners

3.10.1 Ring stiffenersThe geometric proportions of ring stiffeners should comply with the requirements given below (see Figure 1-2for definitions):


— Flanged ring frames:

If the requirements in eqs. (3.10.1) and (3.10.2) are not satisfied, the characteristic material resistance fr shallbe taken as fT (where fT is calculated in accordance with 3.9).

where:bf = flange outstand

t = shell thicknesstf = thickness of flangetW = thickness of web

(3.9.10)

where:

— for longitudinal stiffeners

— for ring frames

(3.9.11)

(3.10.1)

(3.10.2)

(3.10.3)

(3.10.4)

(3.10.5)

0.2C

0.23Cβ

++=

( )η1t

t

s

hC

3

w

−

=

( )η1t

thC

3

w0

−

=

el

ks

Sdj,

f

ση =

Tλ

yw f

Et4.0h ≤

yW f

Et1.35h ≤

yff f

E0.4tb ≤

yf

wf

w fAh

EAr

3

2

t

h ≤

f

wf

w

f

A

A

h

r

3

1

t

e≤



3.10.2 Longitudinal stiffenersThe geometric proportions of longitudinal stiffeners should comply with the requirements given below (seeFigure 1-2 for definitions):

— Flat bar longitudinal stiffeners:

— Flanged longitudinal stiffeners:

If the requirements in eqs. (3.10.6) and (3.10.7) are not satisfied, the characteristic material resistance fr shallbe taken as fT (where fT is calculated in accordance with 3.9).

(3.10.6)

(3.10.7)

(3.10.8)

yw f

Et4.0h ≤

yW f

Et1.35h ≤

yff f

E0.4tb ≤


Recommended Practice DNV-RP-C202, January 2013Sec.4 Unstiffened Conical Shells – Page 26

4 Unstiffened Conical Shells

4.1 IntroductionThis section treats the buckling of unstiffened conical shells, see Figure 4-1.

Buckling of conical shells is treated like buckling of an equivalent circular cylindrical shell.

Figure 4-1Conical shell (force and pressure shown is negative)

4.2 Stresses in conical shells

4.2.1 General

The loading condition governing the stresses in a truncated conical shell, Figure 4-1, is normally defined by thefollowing quantities:

Any of the above quantities may be a function of the co-ordinate x along the shell generator. In addition pSdmay be a function of the circumferential co-ordinate θ, measured from axis 1. pSd is always to be taken as thedifference between internal and external pressures, i.e. pSd is taken positive outwards.

The membrane stresses at an arbitrary point of the shell plating, due to any or all of the above seven actions,are completely defined by the following three stress components:

The loading condition and axes are similar as defined for cylindrical shells in Figure 1-1.

4.2.2 Longitudinal membrane stress

If simple beam theory is applicable, the longitudinal membrane stress may be taken as:

where σa,Sd is due to uniform axial compression and σm,Sd is due to bending.

For a conical shell without stiffeners along the generator:

NSd = design overall axial force exclusive of end pressureM1,Sd = design overall bending moment acting about principal axis 1M2,Sd = design overall bending moment acting about principal axis 2TSd = design overall torsional momentQ1,Sd = design overall shear force acting parallel to principal axis 1Q2,Sd = design overall shear force acting parallel to principal axis 2pSd = design lateral pressure

σx,Sd = design membrane stress in the longitudinal directionσh,Sd = design membrane stress in the circumferential directionτSd = design shear stress tangential to the shell surface (in sections x = constant and θ = constant)

(4.2.1)

(4.2.2)

NSd

pSd

r1

r2

l α

Sdm,Sda,Sdx, σσσ +=

e

Sd

e

SdSda, tr2

N

t2

rpσ

π+=


Recommended Practice DNV-RP-C202, January 2013Sec.4 Unstiffened Conical Shells – Page 27

wherete= t cos α

4.2.3 Circumferential membrane stressThe circumferential membrane stress may be taken as:

wherete= t cos α

4.2.4 Shear stressIf simple beam theory is applicable, the membrane shear stress may be taken as:

where τT,Sd is due to the torsional moment and τQ,Sd is due to the overall shear forces.

where the signs of the torsional moment and the shear forces must be reflected.

4.3 Shell buckling

4.3.1 Buckling strengthThe characteristic buckling strength of a conical shell may be determined according to the procedure given forunstiffened cylindrical shells, 3.4.The elastic buckling strength of a conical shell may be taken equal to the elastic buckling resistance of anequivalent unstiffened cylindrical shell defined by the nominal thickness and:

The buckling strength of conical shells has to comply with the requirements given in 3.4 for cylindrical shells.In lieu of more accurate analyses, the requirements are to be satisfied at any point of the conical shell, basedon a membrane stress distribution according to 4.2.

(4.2.3)

(4.2.4)

(4.2.5)

(4.2.6)

(4.2.7)

(4.3.1)

(4.3.2)

θπ

θπ

costr

Msin

tr

Mσ

e2

Sd2,

e2

Sd1,Sdm, −=

e

SdSdh, t

rpσ =

Q,SdT,SdSd τττ +=

trπ2

Tτ

2Sd

SdT, =

sinθtrπ

Qcosθ

trπ

Qτ Sd2,Sd1,

SdQ, +−=

αcos2

rrr 21

e

+=

αcose

ll =


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DNV-RP-C202: Buckling Strength of Shells · PDF fileDNV-RP-C202 Buckling Strength of Shells ... This RP treats the buckling stability of shell structures based on the load and resistance