8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

1/33

RECOMMENDED PRACTICE

DET NORSKE VERITAS

DNV-RP-C201

BUCKLING STRENGTH OF

PLATED STRUCTURES

OCTOBER 2010

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

2/33

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

Det Norske Veritas

Any comments may be sent by e-mail to rules@dnv.com

For subscription orders or information about subscription terms, please use distribution@dnv.com

Computer Typesetting (Adobe Frame Maker) by Det Norske Veritas

If any person suffers loss or damage which is proved to have been caused by any negligent act or omission of Det Norske Veritas, then Det Norske Veritas shall pay compensation to suchperson for his proved direct loss or damage. However, the compensation shall not exceed an amount equal to ten times the fee charged for the service in question, provided that the maximumcompensation shall never exceed USD 2 million.In this provision "Det Norske Veritas" shall mean the Foundation Det Norske Veritas as well as all its subsidiaries, directors, officers, employees, agents and any other acting on behalf of DetNorske Veritas.

FOREWORD

DET NORSKE VERITAS (DNV) is an autonomous and independent foundation with the objectives of safeguarding life,property and the environment, at sea and onshore. DNV undertakes classification, certification, and other verification andconsultancy services relating to quality of ships, offshore units and installations, and onshore industries worldwide, and carriesout research in relation to these functions.

DNV service documents consist of amongst other the following types of documents:

Service Specifications. Procedual requirements.

Standards. Technical requirements.

Recommended Practices. Guidance.

The Standards and Recommended Practices are offered within the following areas:

A) Qualification, Quality and Safety Methodology

B) Materials Technology

C) Structures

D) Systems

E) Special Facilities

F) Pipelines and Risers

G) Asset Operation

H) Marine Operations

J) Cleaner Energy

O) Subsea Systems

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

3/33

DETNORSKE VERITAS

CHANGES

GeneralAs of October 2010 all DNV service documents are primarily published electronically.

In order to ensure a practical transition from the print scheme to the electronic scheme, all documents having incorporatedamendments and corrections more recent than the date of the latest printed issue, have been given the date October 2010.

An overview of DNV service documents, their update status and historical amendments and corrections may be foundthrough http://www.dnv.com/resources/rules_standards/.

Main changesSince the previous edition (October 2002), this document has been amended, most recently in October 2008. All changes havebeen incorporated and a new date (October 2010) has been given as explained under General.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

4/33

Recommended Practice DNVRP-C201, October 2010

Page4

Det Norske Veritas AS 1997

Data processed and typeset by Division Technology and Products, Det Norske Veritas AS

Printed in Norway by Det Norske Veritas AS

28/10/2010 12:09:00 - Rp-C201.doc

x.9x.x000 (to be filled in by DTP 274)

Para 13 will be inserted by DTP 274

CONTENTS

Part 1. Buckling Strength of Plated

Structures - Conventional BucklingCode......................................................6

1 Introduction.................................................................6 1.1 General..........................................................................61.2 Symbols ............................................................. ...........62 Safety format...............................................................73 General design considerations for flat plate

structures.....................................................................8 3.1 Introduction...................................................................83.2 Definitions ................................................................... .83.3 Failure modes................................................................83.4 Tolerance requirements.................................................83.5 Serviceability limit states..............................................83.6 Validity ........................................................................ .84 Analysis Strategies ....................................................114.1 General........................................................................114.2 Plated structure assumed to resist shear only..............114.3 Consideration of shear lag effects...............................114.4 Determination of buckling resistance based upon linear

elastic buckling stress .................................................115 Lateral loaded plates ................................................116 Buckling of unstiffened plates..................................126.1 General........................................................................126.2 Buckling of unstiffened plates under longitudinally

uniform compression ..................................................126.3 Buckling of unstiffened plates with transverse

compression................................................................126.4 Buckling of unstiffened plate with shear ....................136.5 Buckling of unstiffened biaxially loaded plates with

shear............................................................................13 6.6 Buckling of unstiffened plates with varying

longitudinal stress. Internal compression elements.....146.7 Buckling of outstand compression elements...............15

6.8 Buckling of unstiffened plates with varying transversestress..................... ...................................................... 15

6.9 Buckling of unstiffened plate with longitudianal andtransverse varying stress and with shear stress........... 15

7 Buckling of stiffened plates...................................... 177.1 General ............................................................... ........ 177.2 Forces in the idealised stiffened plate ........................ 177.3 Effective plate width .................................................. 187.4 Resistance of plate between stiffeners........................ 187.5 Characteristic buckling strength of stiffeners............. 197.6 Resistance of stiffened panels to shear stresses.......... 207.7 Interaction formulas for axial compression and lateral

pressure ............................................................... ....... 217.8 Check for shear force ................................................. 228 Buckling of girders ................................................... 23

8.1 General ............................................................... ........ 238.2 Girder forces............................................................... 238.3 Resistance parameters for girders............................... 248.4 Effective widths of girders ......................................... 248.5 Torsional buckling of girders ..................................... 259 Local buckling of stiffeners, girders and brackets 269.1 Local buckling of stiffeners and girders..................... 269.2 Buckling of brackets................................................... 2610 Commentary ............................................................. 27

Part 2. Buckling Strength of Plated

Structures - PULS Buckling Code...30

1 Introduction .............................................................. 301.1 General ............................................................... ........ 301.2 Purpose....................... ................................................ 311.3 Theoretical background.......................... .................... 311.4 Code principles........................................................... 311.5 Safety formats ............................................................ 311.6 PULS software features.............................................. 331.7 References.................................................................. 33

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

5/33

Recommended Practice DNVRP-C201, October 2010

Page 5

DETNORSKE VERITAS

Introduction

This document describes two different, but equally acceptable methods, for buckling and ultimate strength assessment of

plated structures.

The first method, as given in Part 1, is a conventional buckling code for stiffened and unstiffened panels of steel. It is anupdate and development of the stiffened flat plate part of previous DNV Classification Note No. 30.1 Buckling Strength

Analysis. Recommendations are given for plates, stiffeners and girders.

The second method, as given in Part 2, is a computerised semi-analytical model called PULS (Panel Ultimate Limit State). Itis based on a recognized non-linear plate theory, Rayleigh-Ritz discretizations of deflections and a numerical procedure forsolving the equilibrium equations. The method is essentially geometrically non-linear with stress control in critical positionsalong plate edges and plate stiffener junction lines for handling material plasticity. The procedure provides estimates of theultimate buckling capacity to be used in extreme load design (ULS philosophy). The buckling limit is also assessed as it maybe of interest in problems related to functional requirements, i.e. for load conditions and structural parts in which elastic

buckling and thereby large elastic displacements are not acceptable (SLS philosophy). The PULS code is supported by officialstand alone DNV Software programs. It is also implemented as a postprocessor in other DNV programs.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

6/33

Recommended Practice DNVRP-C201, October 2010

Page6

DETNORSKE VERITAS

Part 1.Buckling Strength of Plated Structures - Conventional Buckling Code

1 Introduction1.1 GeneralThis document gives design recommendations to flat steelplate structures intended for marine structures. The RP isintended to supplement the DNV Offshore standards DNV-OS-C101 and is intended to be used for design of structuresaccording to this standard.

1.2 SymbolsThe following symbols apply to this document:

A cross sectional areaAe effective area

Af cross sectional area of flangeAG cross-sectional area of girder

As cross sectional area of stiffenerAw cross sectional area of webC factorCx buckling factor for stresses in x-directionCxs effective width factor due to stresses in x-

direction

Cys effective width factor due to stresses in y-direction

C0 factorE Youngs modulus of elasticity, 2.1105MPaG shear modulusI moment of inertiaIp polar moment of inertiaIpo polar moment of inertia = dAr

2 where r is

measured from the connection between thestiffener and the plate

Is moment of inertia of stiffener with full platewidth

Iz moment of inertia of stiffener about z-axisL length, distanceLP length of panelLG length of girder

LGk buckling length of girderLGT distance between lateral support of girderLGT0 limiting distance between lateral support of

girderMp,Rd design bending moment resistance on plate

side

Mpl,Rd design plastic bending moment resistanceMRd design bending moment resistance

MSd design bending moment

Ms,Rd design bending moment resistance onstiffener side

Mst,Rd design bending moment resistance onstiffener side in tension

NE Euler buckling strengthNks,Rd design stiffener induced axial bucklingresistance

Nkp,Rd design plate induced axial buckling resistance

NSd design axial forcePSd design lateral forceQ FactorVRd design shear resistanceVSd design shear forceW elastic section modulus

WeG effective section modulus on girder flangeside

Wep

effective section modulus on plate sideWes effective section modulus on stiffener sideb width of flangebe effective widthc length of plate outstand, Factorci interaction factor

ef flange eccentricityfcr elastic plate buckling strength

fd design yield strengthfE Euler buckling strengthfEpx Euler buckling strength for plate due to

longitudinal stressesfEpy Euler buckling strength for plate due to

transverse stressesfEp Euler buckling shear strength for plate

fET torsional elastic buckling strength

fETG torsional elastic buckling strength for girdersfEy, fEz Euler buckling strength corresponding to the

member y and z axis respectivelyfk characteristic buckling strength

fr characteristic strengthfT characteristic torsional buckling strengthfTG characteristic torsional buckling strength for

girdersfy characteristic yield strengthh height

hw height of stiffener webhwG height of girder webi radius of gyrationie effective radius of gyrationk, kg buckling factorkc factorkp reduction factor for plate buckling due to

lateral pressurek buckling factor for unstiffened platesl length, element lengthle effective lengthC stiffener buckling lengthll length of longitudinal web stiffener

lt length of transverse web stiffenerlT distance between sideways support of

stiffenerl1 length to reference point

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

7/33

Recommended Practice DNVRP-C201, October 2010

Page 7

DETNORSKE VERITAS

pf lateral pressure giving yield in outer-fibre of acontinuous stiffener using elastic sectionmodules

pSd design hydrostatic pressure, design lateralpressure

p0 equivalent lateral pressureqSd design lateral line loadr radius, factors plate width, stiffener spacingse effective width of stiffened platet thicknesstb bracket thicknesstf flange thickness

tw web thicknesszp, zt distancez

* distance

Factorf partial factor for actions

M resulting material factor Factor reduced slenderness, column slenderness

parameter

e reduced equivalent slendernessG reduced slendernessp reduced plate slendernessT reduced torsional slendernessTG reduced torsional slenderness for girders reduced slenderness coefficient, geometric parameter

Poissons ratio

j,Sd design von Mises equivalent stressy1,Sd larger design stress in the transverse direction,with tensile stresses taken as negative

y2,Sd smaller design stress in the transversedirection, with tensile stresses taken asnegative

ceg, cel elastic buckling strengthcrg, crl critical shear stressRd design resistance shear stressSd design shear stress, x, y factors

2 Safety formatThis Recommended Practice is written in the load andresistance factor design format (LRFD format) to suit the

DNV Offshore Standard DNV-OS-C101. This standard make

use of material (resistance) and loadfactors as safety factors.

This Recommended Practice may be used in combination

with a working stress design format (WSD) by the followingmethod. For the formulas used in this standard use a material

factor M = 1.15 . The checks should be made using amodified allowable usage factor taken as UF1.15, where UFis the allowable usage factor according to the WSD standard.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

8/33

Recommended Practice DNVRP-C201, October 2010

Page8

DETNORSKE VERITAS

3 General design considerations for flatplate structures

3.1 IntroductionThe structural stability shall be checked for the structure as awhole and for each structural member.

Buckling strength analyses shall be based on thecharacteristic buckling strength for the most unfavourablebuckling mode.

The characteristic buckling strength shall be based on thelower 5thpercentile of test results. In lieu of more relevantinformation or more refined analysis, characteristic bucklingstrength may be obtained from this note.

3.2 DefinitionsNotation of plate elements are shown in Figure 3-1. The platepanel may be the web or the flange of a beam, or a part ofbox girders, bulkheads, pontoons, hull or integrated plateddecks.

Figure 3-1 Stiffened plate panel

3.3 Failure modesThis recommended practice addresses failure modes forunstiffened and stiffened plates, which are not covered by thecross sectional check of members. (See DNV-OS-C101

Sec.5 A 400.) Such failure modes are:

Yielding of plates in bending due to lateral load. Buckling of slender plates (high span to thickness

ratio) due to in-plane compressive stresses or shearstresses.

Guidance for determining resistance is given both forindividual plates (unstiffend plates), stiffened plates and forgirders supporting stiffended plate panels. For stiffenedpanels the recommendations cover panel buckling, stiffenerbuckling as well as local buckling of stiffener and girderflanges, webs and brackets. See Table 3-1.

3.4

Tolerance requirementsThe recommendations are applicable for structures builtaccording to DNV-OS-C401 Fabrication and Testing ofOffshore Structures or normal ship classification standards.See also Commentary Chapter 10.

3.5 Serviceability limit statesCheck of serviceability limit states for slender plates relatedto out of plane deflection may normally be omitted if the

smallest span of the plate is less than 120times the platethickness. See also Commentary to 6 in Chapter 10.

3.6 ValidityThis Recommended Practice is best suited to rectangularplates and stiffened panels with stiffener length being largerthan the stiffener spacing ( l> s ). It may also be used forgirders being orthogonal to the stiffeners and with the girder

having significant larger cross-sectional dimensions than thestiffeners.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

9/33

Recommended Practice DNVRP-C201, October 2010

Page 9

DETNORSKE VERITAS

Table 3-1 Reference table for buckling checks of plates

Description Load Sketch Clause

reference

Limiting value

Unstiffenedplate

Longitudinalcompression

x,Sdx,Sd

- t -

l

s

6.2 s < lBuckling check notnecessary if

42t

s

Unstiffenedplate

Transversecompression

y,Sd

- t - s

l

y,Sd

6.3 s < l

Buckling check notnecessary if

5.4t

s

Unstiffenedplate

Shear stressSd

s - t -

l

6.4 s < l

Buckling check notnecessary if

70t

s

Unstiffenedplate

Linear varyinglongitudinalcompression

- t -

x,Sdl

s

x,Sd

x,Sd x,Sd

6.6 s < l

Buckling check not

necessary if

42t

s

Unstiffenedplate

Linear varyingtransversecompression

- t -

y,Sd

l

sl1

6.8 s < l

Buckling check notnecessary if

5.4t

s

y235/f= = 1.0 for fy= 235 MPa = 0.814 for fy= 355 MPa

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

10/33

Recommended Practice DNVRP-C201, October 2010

Page10

DETNORSKE VERITAS

Description Load Sketch Clause

reference

Limiting value

Unstiffenedplate

Combinedlongitudinalandtransversecompressionand shear

-t- s

Sd

y,Sd

x,Sd

6.5 s < l

Buckling check not

necessary if5.4

t

s

Unstiffenedplate

Uniformlateral loadand in-planenormal and

shearstresses

x,Sd PSdSd

y,Sd

s - t -

l

5and

6.5

s < l

Buckling check not

necessary if

5.4t

s

Longitudinalstiffened platepanel

Longitudinalandtransversecompressioncombinedwith shear

and lateralload

GL

Sd

y,Sd

x,Sd SdP

- t -

l

5 and 7

Girdersupportingstiffened

panel

Longitudinalandtransverse

compressioncombined

with shearand lateralload

L

Sd

y,Sd

x,Sd SdP

l

- t -

l

G

5 and 8

Stiffeners togirder webs

Longitudinalandtransversecompression

combinedwith shearand lateral

load

s

lt

ll

s

9.1

Brackets 9.2

y235/f= = 1.0 for fy= 235 MPa = 0.814 for fy= 355 MPa

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

11/33

Recommended Practice DNVRP-C201, October 2010

Page 11

DETNORSKE VERITAS

4 Analysis Strategies4.1 GeneralThe design check of plated structures are normally madewith linear elastic finite element analyses for determinationof load effects. Flat plate structures will redistributecompressive stresses to the edges as the load approaches the

resistance of the plate and the plate will cease to behavelinearly. Linear finite element analyses will generally beadequate as long as the resistance is checked for theresultants from the integrated stresses in the analyses.

As slender plates under compressive loading will tend toredistribute stresses to the edges, an analysis where the partof the structure subject to buckling is given reduced stiffnessmay lead to more efficient structures. The adjoining structureneed to be checked on the basis of the same model.

4.2 Plated structure assumed to resist shearonly

The following design philosophy may be used for platepanels which main function is to carry in-plane shear loads.These plated structures may be analysed and checked byconsidering the plates as pure shear panels. Such panels maybe decks or walls in topside modules. Then all axialmembrane stresses need to be carried by the adjoiningframing only which should be analysed and checkedaccordingly. The analysis may be carried out with the platepanels modelled with elements that are only given shear

stiffness.

4.3 Consideration of shear lag effectsIf the stresses are determined from beam theory, the effect ofshear deformations of wide flanges need to be considered.See also Commentary to 7 in Chapter 10.

4.4 Determination of buckling resistance basedupon linear elastic buckling stress

The buckling resistance may be based on linear elasticbuckling stress provided the following effects are accounted

for:

Material non-linearities Imperfections Residual stresses Possible interaction between local and global buckling

modes

See also Commentary Chapter 10.

5 Lateral loaded platesFor plates subjected to lateral pressure, either alone or in

combination with in-plane stresses, the stresses may bechecked by the following formula:

+

x

2

y

2

M

y

Sd s

s

t

f4.0p

l

(5.1)

where

pSd= design lateral pressure

2

y

Sd

2

y

Sdx,

2

y

Sdj,

y

f

3

f

4

31

f

1

=

(5.2)

2

y

Sd

2

y

Sdy,

2

y

Sdj,

x

f

3

f

4

31

f

1

=

(5.3)

2

SdSdy,Sdx,

2

Sdy,

2

Sdx,Sdj, 3 + (5.4)

This formula for the design of a plate subjected to lateralpressure is based on yield-line theory, and accounts for thereduction of the moment resistance along the yield-line dueto applied in-plane stresses. The reduced resistance iscalculated based on von Mises equivalent stress. It is

emphasised that the formulation is based on a yield patternassuming yield lines along all four edges, and will giveuncertain results for cases where yield-lines can not bedeveloped along all edges. Furthermore, since the formuladoes not take account of second-order effects, platessubjected to compressive stresses shall also fulfil the

requirements of Chapter 6 and 7 whichever is relevant.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

12/33

Recommended Practice DNVRP-C201, October 2010

Page12

DETNORSKE VERITAS

6 Buckling of unstiffened plates6.1 GeneralThis section presents recommendations for calculating thebuckling resistance of unstiffened plates.

For plates that are part of a stiffened panel, the plate arechecked as part of the buckling checks according to Chapter

7. Then additional check of the plate according to this sectionis not required.

Buckling checks of unstiffened plates in compression shallbe made according to the effective width method. Thereduction in plate resistance for in-plane compressive forcesis expressed by a reduced (effective) width of the plate whichis multiplied by the design yield strength to obtain the design

resistance, see Figure 6-1 .

See also Commentary Chapter 10.

S

Se

S

.

Real stress distribution

Effectivestress distribution

Figure 6-1 Effective width concept

6.2 Buckling of unstiffened plates underlongitudinally uniform compression

The design buckling resistance of an unstiffened plate underlongitudinal compression force may be calculated as:

M

y

xRdx,

fC =

(6.1)

where

1Cx = when 673.0p (6.2)

( )2

p

p

x

22.0C

= when 673.0p>

where pis the plate slenderness given by:

E

f

t

s525.0

f

f

y

cr

y

p = (6.3)

in which

s = plate width

t = plate thickness

fcr = critical plate buckling strength

The resistance of the plate is satisfactory when:

Rdx,Sdx, (6.4)

x,Sdx,Sd

- t -

l

s

Figure 6-2 Plate with longitudinal compression

6.3 Buckling of unstiffened plates withtransverse compression

The design buckling resistance of a plate under transverse

compression force may be found from:

M

Ry,Rdy,

= (6.5)

pyyy

Ry, kff

Et3.11

f

Et3.1

=

ll

(6.6)

where:

0.1= for c0.2

= 2c22c2cc

2 411

2

1

for 0.2

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

13/33

Recommended Practice DNVRP-C201, October 2010

Page 13

DETNORSKE VERITAS

t = plate thickness

l = plate length

s = plate width

The reduction factor due to lateral load kp may, in lieu of

more accurate results, be calculated as:

0kbut,s

t2

f

ph0.1k

otherwise

fs

t2pfor1.0k

p

2

y

Sdp

y

2

Sdp

(6.10)

where

50.7t

s0.05h but h0 (6.11)

The resistance of the plate is satisfactory when:

Rdy,Sdy, (6.12)

y,Sd

- t - s

l

y,Sd

Figure 6-3 Plate with transverse compression

6.4 Buckling of unstiffened plate with shearShear buckling of a plate can be checked by

RdSd (6.13)

3

f

C y

M

Rd =

(6.14)

where

0.8for1.0C w

( ) 2.18.0for,8.0625.00.1C = w

w

(6.15)

lkE

f

t

s795.0

y

w =

(6.16)

sfor,4s

34.5

sfor,s434.5k

2

2

If either of x,Sd and y,Sd or both is in tension (negative), thenci= 1.0.

x,Rd is given by eq. (6.1) and y,Rd is given by eq. (6.5). In

case of tension, apply fy/M.

Rdis given by eq. (6.19) in cases where y,Sd is positive(compression) and by eq. (6.14) in cases where y,Sd is zero

or negative (in tension).

3

f

C y

M

eRd

(6.19)

8.0for0.1Ce w

25.18.0for,8.08.00.1Ce ww 25.1for,

0.1C

2e >w

w

(6.20)

-t- s

Sd

y,Sd

x,Sd

Figure 6-4 Biaxially loaded plate with shear

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

14/33

Recommended Practice DNVRP-C201, October 2010

Page14

DETNORSKE VERITAS

6.6 Buckling of unstiffened plates with varyinglongitudinal stress. Internal compression

elements

The buckling resistance of an unstiffened plate with varyinglongitudinal stress may be found from:

M

y

xRdx,

fC=

(6.21)

where

1Cx= when 673.0p (6.22)

2p

px

3055.0C

+= when 673.0p> (6.23)

where pis the plate slenderness given by:

k4.28

1

t

s

f

f

cr

y

p (6.24)

in which

s = plate width

= 2/ 1 Stress ratio. 1 is largest stress with

compressive stress taken as positive.

t = plate thickness

fcr = critical plate buckling strength

=yf

235

k =

05.1

2.8

+

when 10

= 7.81-6.29+9.782 when 01 0

e1effe2

effe1

eff

bbb

b-5

2b

bCb

=x

b

b be1 e2

12

b bc t

< 0

effe2

effe1

ceff

b0.6b

b0.4b

1

bCbCb

==

= xx

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

15/33

Recommended Practice DNVRP-C201, October 2010

Page 15

DETNORSKE VERITAS

6.7 Buckling of outstand compression elementsThe buckling resistance of an outstand compression elementwith varying or constant longitudinal stress may be found

from:

M

y

xRdx,

fC=

(6.26)

where

1Cx= when 749.0p (6.27)

2p

px

188.0C

= when 749.0p> (6.28)

where pis the plate slenderness given by:

k4.28

1

t

s

f

f

cr

y

p (6.29)

in which

s = plate width

t = plate thickness

fcr = critical plate buckling strength

=yf

235

For outstand with largest compression stress at free edge:

k = 0.57 - 0.21 + 0.07 2 when 13

For outstand with largest compression stress at supportededge:

k =34.0

578.0

+ when 10

k = 1.7 - 5 + 17.1 2 when 01

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

16/33

Recommended Practice DNVRP-C201, October 2010

Page16

DETNORSKE VERITAS

Table 6-2 Effective width for outstand compression plate elements with largest stress at free edge

Stress distribution (compression positive) Effective width beff

c

21

b eff

0< 1 cCbeff x

b2

1

bt

eff

bc

< 0

-1

cCbCb ceff

= xx

Table 6-3 Effective width for outstand compression plate elements with largest stress at supported edge

Stress distribution (compression positive) Effective width beff

c

12

beff

0< 1 cCbeff x

b

1

2

beff

bc t

< 0

-1

cCbCb ceff

= xx

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

17/33

Recommended Practice DNVRP-C201, October 2010

Page 17

DETNORSKE VERITAS

7 Buckling of stiffened plates7.1 GeneralThis chapter deals with stiffened plate panels subjected toaxial stress in two directions, shear stress and lateral load.

There are different formulas for stiffeners being continuous(or connected to frames with their full moment resistance)

and simple supported (sniped) stiffeners.

An example of a stiffened plate panel is shown in Figure 3-1.

The stiffener cross section needs to fulfil requirements toavoid local buckling given in Chapter 9.

For shear lag effects see Commentary Chapter 10.

The plate between stiffeners will normally be checkedimplicitly by the stiffener check since plate buckling is

accounted for by the effective width method. However, in

cases where y,Sdstress is the dominant stress it is necessaryto check the plate resistance according to eq. (7.19).

For slender stiffened plates the load carrying resistance in the

direction transverse to the stiffener may be neglected. Then

y,Sdstresses may be assumed to be carried solely by thegirder. In such cases the effective girder flange may bedetermined by disregarding the stiffeners, and the stiffener

with plate may be checked by neglecting y,Sdstresses(method 2 in sec. 8.4). See also Commentary to 8 in Chapter

10.

7.2 Forces in the idealised stiffened plateStiffened plates subjected to combined forces, see Figure 7-1should be designed to resist an equivalent axial forceaccording to eq. (7.1) and an equivalent lateral loadaccording to eq. (7.8).

The equivalent axial force should be taken as:

ststAN tfsSdx,Sd + (7.1)

where

As = cross sectional area of stiffener

s = distance between stiffeners

t = plate thickness

x,Sd = axial stress in plate and stiffener withcompressive stresses as positive

crgSdtf forM

crlSd

>

and tension field action is allowed

(7.2)

0 tf= otherwise (7.3)

Assumption of tension field action implies that no (ornegligible) resistance of the plate against transversecompression stresses (y )can be assumed. See alsoCommentary Chapter 10.

crg= critical shear stress for the plate with the stiffenersremoved, according to eq. (7.4).

crl= critical shear stress for the plate panel between twostiffeners, according to eq. (7.6).

2

gcrg

tE0.904k

l

(7.4)

where:

G

2

G

G

2

G

g

Lfor,4L

34.5

Lfor,L

434.5k

>

=

+

ll

ll

(7.5)

LG= Girder length see Figure 3-1

2

crs

tE0.904k

= ll

(7.6)

where:

sfor,4s

34.5

sfor,s

434.5k

2

2

-1.5 (7.9)

0p0= if -1.5 (7.10)

p0= 0 in case y,Sdis in tension along the whole length of thepanel.

stEk

mfWC

2c

cyes

0 = (7.11)

Sdy1,

Sdy2,

=

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

18/33

Recommended Practice DNVRP-C201, October 2010

Page18

DETNORSKE VERITAS

y1,Sd = larger design stress in the transverse direction,with tensile stresses taken as negative

y2,Sd = smaller design stress in the transverse direction,with tensile stresses taken as negative

Wes = section modulus for stiffener with effective plateat flange tip

mc = 13.3 for continuous stiffeners or,

= 8.9 for simple supported stiffeners (sniped stiffeners)

+st

I10.9112k

3

sc

(7.12)

Is= moment of inertia of stiffener with full plate width

STIFFENED PLATE BEAM COLUMN

y1,SdNSd

N ( x,Sd, Sd)

x,Sd

Sd,

Sdp

y2, Sd

Sd

N

(p p )Sd

q =q

Sd =

o

Figure 7-1 Strut model

7.3 Effective plate widthThe effective plate width for a continuous stiffener subjectedto longitudinal and transverse stress and shear is calculatedas:

ysxse CCs

s = (7.13)

The reduction factor due to stresses in the longitudinaldirection, Cxs, is

673.0if1.0,

673.0if,

0.22C

p

p2

p

p

xs

=

>

=

(7.14)

where

E

f

t

s525.0

y

p = (7.15)

and the reduction factor for compression stresses in thetransverse direction, Cys, is found from:

+

Ry,yxs

Sdy,Sdx,i

2

Ry,

Sdy,ys

fC

c

1C

(7.16)

where

120t

sfor0c

120t

sfor

t120

s1c

i

i

>

y,R is calculated according to eq. (6.6).

In case of linear varying stress, y,Sdmay be determined asdescribed in sec. 6.8

The reduction factor for tension stresses in the transverse

direction, Cys, is calculated as:

0.1Cbut,f

f

34

2

1C ys

y

Sdy,

2

y

Sdy,

ys

+

=

(7.17)

Tensile stresses are defined as negative.

The effective width for varying stiffener spacing see Figure

7-2.

Figure 7-2 Effective widths for varying stiffener spacing

7.4 Resistance of plate between stiffenersThe plate between stiffeners shall be checked for:

M

yRdSd

3

f

=

(7.18)

Rdy,spSdy, k (7.19)

where:

2

y

Sdsp

f

30.1k

(7.20)

and y,Rdis determined from eq. (6.5).

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

19/33

Recommended Practice DNVRP-C201, October 2010

Page 19

DETNORSKE VERITAS

When this check and stiffener check according to sec. 7.7 iscarried out it is not necessary to check the plate betweenstiffeners according to Chapter 6.

See also Commentary Chapter 10.

7.5 Characteristic buckling strength ofstiffeners

7.5.1 GeneralThe characteristic buckling strength for stiffeners may befound from:

1f

f

r

k = when 2.0 (7.21)

( )2

2222

r

k

2

411

f

f

++++=

when 2.0>

(7.22)

where

E

r

f

f=

(7.23)

2

k

e2

E

iEf

=

l

(7.24)

for check at plate side

( )0.2i

z08.034.0

e

p

+=

(7.25)

for check at stiffener side

( )0.2i

z08.034.0

e

t

+=

(7.26)

where:

fr = fy for check at plate side

fr = fy for check at stiffener side ifT0.6

fr = fT for check at stiffener side ifT> 0.6,

fTmay be calculated according to sec. 7.5.2

T see eq. (7.30)

lk see eq. (7.74)

e

e

eA

Ii = , effective radius of gyration

Ie effective moment of inertia

Aeeffective area

zp, zt is defined in Figure 7-3

A

B

tw

c

e

t

zp

z t

f

b

C

h

wt

tf

c

tw

b

wt

b

f

t

.

c

A = centroid of stiffener with effective plate flange.

B = centroid of stiffener exclusive of any plate flange.

C = centroid of flange.

a

w wh

whwh

ef

Figure 7-3 Cross-sectional parameters for stiffeners andgirders

7.5.2 Torsional buckling of stiffenersThe torsional buckling strength may be calculated as:

0.1f

f

y

T = when 6.0T (7.27)

2T

2T

22T

2T

y

T

2

411

f

f

=

when 6.0T >

(7.28)

where

( )6.035.0 T= (7.29)

ET

y

Tf

f =

(7.30)

Generally fETmay be calculated as:

2

Tpo

z

2

s2

po

tET

I

IEh

I

GIf

l+=

(7.31)

For L- and T-stiffeners fET may be calculated as:

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

20/33

Recommended Practice DNVRP-C201, October 2010

Page20

DETNORSKE VERITAS

2Tf

W

z22

w

W

fW

f

2

W

fW

ET

A3

A

EI

h

tG

A3A

At

tA

f

l

++

+

+=

(7.32)

W

f

f2

f

2

fz

A

A1

AebA

12

1I

++=

(7.33)

For flatbar stiffeners fET may be calculated as:

2

w

w

2

T

wET

h

tG

h2f

+=

l

(7.34)

where

= 1.0,or may for stocky plates alternatively be calculated as

per eq. (7.35) for s l

Af = cross sectional area of flange

AW= cross sectional area of web

G = shear modulus

Ipo = polar moment of inertia= dAr2

where r is measured from the connection between thestiffener and the plate

It = stiffener torsional moment of inertia (St. Venanttorsion)

Iz = moment of inertia of the stiffeners neutral axis normalto the plane of the plate

b = flange width

ef = flange eccentricity, see Figure 7-3

hw = web height

hs = distance from stiffener toe (connection between

stiffener and plate) to the shear centre of the stiffener

lT = distance between sideways supports of stiffener,distance between tripping brackets (torsional bucklinglength).

t = plate thickness

tf = thickness of flange

tW = thickness of web

where

0.2C

0.23C

++= (7.35)

( )1t

t

s

hC

3

w

w

=

(7.36)

where:

0.1f

ep

Sdj, = (7.37)

2

SdSdy,Sdx,

2

Sdy,

2

Sdx,Sdj, 3 ++= (7.38)

4

e

y

ep

1

ff

+=

(7.39)

c

1c

Ep

Sd

c

Epy

Sdy,

c

Epx

Sdx,

Sdj,

y2e

f

f

f

f

+

+

=

(7.40)

where

l

s2c =

(7.41)

2

Epxs

t3.62Ef

= (7.42)

2

Epy s

t0.9Ef

= (7.43)

2

Eps

t5.0Ef

= (7.44)

x,Sdand y,Sdshould be set to zero if in tension

7.6 Resistance of stiffened panels to shearstresses

The resistance towards shear stressesRdis found as theminimum of Rdy, Rdland Rdsaccording to the following:

M

y

Rdy3

f

= (7.45)

M

cr

Rd

l

l= (7.46)

M

scr

Rds

= (7.47)

where crl is obtained from eq. (7.6) and crs is obtainedfrom:

4 3

sp2crsII

ts

E36 =

l (7.48)

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

21/33

Recommended Practice DNVRP-C201, October 2010

Page 21

DETNORSKE VERITAS

with:

10.9

stI

3

p

= (7.49)

and Is= moment of inertia of stiffener with full plate width.

7.7 Interaction formulas for axial compressionand lateral pressure

7.7.1 Continuous stiffenersFor continuous stiffeners the following four interactionequations need to be fulfilled in case of:

Lateral pressure on plate side:

1u

N

N1M

zNM

N

N

E

Sd

Rds1,

*SdSd1,

Rdks,

Sd

+ (7.50)

1u

N

N1M

zNM

N

N2

N

N

E

SdRdp,

*SdSd1,

Rd

Sd

Rdkp,

Sd

+ (7.51)

1u

N

N1M

zNM

N

N2

N

N

E

SdRdst,

*SdSd2,

Rd

Sd

Rdks,

Sd

+ (7.52)

1u

N

N1M

zNM

N

N

E

SdRdp,

*SdSd2,

Rdkp,

Sd

+

(7.53)

Lateral pressure on stiffener side:

1u

N

N1M

zNM

N

N2

N

N

E

SdRdst,

*SdSd1,

Rd

Sd

Rdks,

Sd

+ (7.54)

1u

N

N1M

zNM

N

N

E

SdRdp,

*SdSd1,

Rdkp,

Sd

+

(7.55)

1u

N

N1M

zNM

N

N

E

SdRds2,

*SdSd2,

Rdks,

Sd

+ (7.56)

1u

N

N1M

zNM

N

N2

N

N

E

SdRdp,

*SdSd2,

Rd

Sd

Rdkp,

Sd

+ (7.57)

where

2

Rd

Sd

u

= (7.58)

When tension field action is assumed according to eq. (7.2)then u = 0.

For resistance parameters see sec. 7.7.3 for stiffener and sec.8.3 for girders.

M1,Sd =12

q2

Sdl for continuous stiffeners with equal spans

and equal lateral pressure in all spans= absolute value of the actual largest supportmoment for continuous stiffeners with unequal spans

and/or unequal lateral pressure in adjacent spans

M2,Sd =24

q2

Sdl for continuous stiffeners with equal spans

and equal lateral pressure in all spans= absolute value of the actual largest field momentfor continuous stiffeners with unequal spans and/or

unequal lateral pressure in adjacent spans

qsd is given in eq. (7.8)

l = span length

z*is the distance from the neutral axis of the effective sectionto the working point of the axial force. z*may be varied inorder to optimise the resistance. z*should then be selected sothe maximum utilisation found from the equations (7.50) to(7.53) or (7.54) to (7.57) is at its minimum, see also

Commentary Chapter 10. The value of z*is taken positivetowards the plate. The simplification z*= 0 is always

allowed.

7.7.2 Simple supported stiffener (sniped stiffener)Simple supported stiffener (sniped stiffener):

Lateral pressure on plate side:

1u

N

N1M

zN8

q

N

N2

N

N

E

SdRdst,

*Sd

2Sd

Rd

Sd

Rdks,

Sd

+

l

(7.59)

1u

N

N1M

zN8

q

N

N

E

SdRdp,

*Sd

2Sd

Rdkp,

Sd

+

l

(7.60)

Lateral pressure on stiffener side:

if*

Sd

2Sd zN8

q l then:

1u

N

N1M

zN8

q

N

N

E

SdRds2,

*Sd

2Sd

Rdks,

Sd

+

+

l

(7.61)

1u

N

N1M

zN8

q

N

N2

N

N

E

SdRdp,

*Sd

2Sd

Rd

Sd

Rdkp,

Sd +

+

l

(7.62)

if *Sd

2Sd zN8

q 0.5 VRdthen the stiffener section modulus andeffective area need to be reduced to account for theinteraction of the shear with the moment and axial force in

the stiffener.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

23/33

Recommended Practice DNVRP-C201, October 2010

Page 23

DETNORSKE VERITAS

8 Buckling of girders8.1 GeneralThe check for girders is similar to the check for stiffeners ofstiffened plates in equations (7.50) to (7.57) or (7.59) to

(7.64) for continuous or sniped girders, respectively. Forcesshall be calculated according to sec. 8.2 and cross section

properties according to 8.4. Girder resistance should befound from sec. 8.3. Torsional buckling of girders may beassessed according to sec. 8.5.

In the equations (7.50) to (7.57) or (7.59) to (7.62) u = 0 forgirders.

Girders may be checked for shear forces similar to stiffenerssee sec. 7.8.

8.2 Girder forcesThe axial force should be taken as:

( )GSdy,Sdy, AtN += l (8.1)

The lateral line load should be taken as:

( )l0SdSd ppq += (8.2)

where

pSd= design lateral pressure

p0 = equivalent lateral pressure

AG= cross sectional area of girder

The calculation of the additional equivalent lateral pressure

due to longitudinal compression stresses and shear shall becalculated as follows:

For compression in the x-direction:

( )SdSdx,

2

Gy

G

wG

s

0 C

L

E

f

L

s1h

s

At0.4

p +

+

= l

But not less than ( )SdSdx,

s

Cs

At

0.02 ++

l

(8.3)

where

2

crl

crgSd2

s57QC

=l

forSd> crg(8.4)

0C= for Sdcrg (8.5)

Q = 2.0G , but not less than 0 and not greater than 1.0

G =EG

y

f

f fEGis given in eq. (8.11)

crg= critical shear stress of panel with girders removed,calculated from eq.(8.6)with

calculated using

ce = ceg. If the stiffener is not continuousthrough the girder crg = 0.

crl= critical shear stress of panel between girders calculatedfrom eq. (8.6)with

calculated using ce = cel

1for,f

0.6

1for,0.6f

y2

ycr

>=

= (8.6)

=ce

y

0.6f

with

ceg= 2P

2cel

L

l

cel=75.0

s

2 s

tI

t

E18

l

LP = length of panel

hwG = web height of girder

As = cross sectional area of stiffener

LG= girder span

s = stiffener spacing

Is = moment of inertia of stiffener with full plate width

For linear variation of x,Sd, the maximum value within0.25LGto each side of the midpoint of the span may be used.

Sdshould correspond to the average shear flow over the

panel.

L

L

s

l

Stiffener

Girder

P

G

Figure 8-1 Panel geometry definitions

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

24/33

Recommended Practice DNVRP-C201, October 2010

Page24

DETNORSKE VERITAS

For tension in the x-direction:

Sd

2

Gy

G

wG

s

0 CL

E

f

Ls1h

s

At0.4

p

+

=l

(8.7)

8.3 Resistance parameters for girdersThe resistance of girders may be determined by theinteraction formulas in sec. 7.7 using the followingresistance

( )M

y

eGRd

ftAN l+=

(8.8)

( )M

k

eGRdks,

ftAN l+=

(8.9)

where

fkis calculated from sec. 7.5 using according to eq. (7.26).

( )M

keGRdkp,

ftAN l+=

(8.10)

where:

fk is calculated from sec. 7.5 using according to eq.(7.25) using:

fr = fy for check at plate side

fr = fTG for check at girder flange side

2

Gk

Ge2

EGL

iEf

=

(8.11)

LGk= buckling length of girder equal LGunless further

evaluations are made

fTG may be obtained from eq. (8.27)

AG= cross sectional area of girder

le = effective width of girder plate, see sec. 8.4.

M

reGRds1,

fWM = (8.12)

with frcalculated using lt = 0.4 LGor distance between lateralsupport if this is less.

M

reGRds2,

fWM = (8.13)

with frcalculated using lt = 0.8 LGor distance between lateralsupport if this is less.

M

y

eGRdst,

fWM = (8.14)

M

y

epRdp,

fWM =

(8.15)

Wep =

p

e

z

I , effective elastic section modulus on plate

side, see Figure 7-3

WeG =

t

e

z

I , effective elastic section modulus on girder

flange side, see Figure 7-3

2

Ge

Gk

Ge2

E

i

L

EAN

=

(8.16)

where

Ge

GeGe

A

Ii = (8.17)

8.4 Effective widths of girders8.4.1 GeneralThe effective width for the plate of the girder is taken equal

to:

GyGxG

e CCC =l

l

(8.18)

For the determination of the effective width the designer isgiven two options denoted method 1 and method 2. Thesemethods are described in sec. 8.4.2 and 8.4.3 respectively:

8.4.2 Method 1Calculation of the girder by assuming that the stiffened plate

is effective against transverse compression (y) stresses. Seealso Commentary Chapter 10 and sec. 7.1.

In this method the effective width may be calculated as:

2

kx

Sdx,

xGf

1C

=

(8.19)

where:

yxskx fCf = (8.20)

Cxsis found from eq. (7.14).

If the ystress in the girder is in tension due to the combinedgirder axial force and bending moment over the total span of

the girder CyGmay be taken as:

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

25/33

Recommended Practice DNVRP-C201, October 2010

Page 25

DETNORSKE VERITAS

CyG= 1.0(8.21)

If the ystress in the plate is partly or complete incompression CyGmay be found from eq. (7.16).

2

y

SdG

f

31C

=

(8.22)

le should not be taken larger than 0.3 LGfor continuousgirders and 0.4 LGfor simple supported girders whencalculating section modules Wepand WeG.

8.4.3 Method 2Calculation of the girder by assuming that the stiffened plate

is not effective against transverse compression stresses (y).See also Commentary Chapter 10 and Sec. 7.1.

In this case the plate and stiffener can be checked with ystresses equal to zero.

In method 2 the effective width for the girder should becalculated as if the stiffener was removed.

then:

2

y

Sdx,

xGf

1C

=

(8.23)

where

x,Sdis based on total plate and stiffener area in x-direction.

673.0if1.0,

673.0if

0.22C

G

G2G

GyG

>= (8.24)

where

E

f

t525.0

y

G

l=

(8.25)

2

y

SdG

f

31C

= (8.26)

8.5 Torsional buckling of girdersThe torsional buckling strength of girders may be determinedas:

yTG ff = if 6.0TG

2

TG

2TG

22TG

2TG

yTG

2

411ff

if 6.0TG>

(8.27)

ETG

yTG

f

f =

(8.28)

( )6.035.0 TG= (8.29)

where

2GT

wf

z2

ETG

L3

AA

EIf

+= (8.30)

LGT = distance between lateral supports

Af, Aw= cross sectional area of flange and web of girder

Iz = moment of inertia of girder (exclusive of plate flange)about the neutral axis perpendicular to the plate

Torsional buckling need not to be considered if trippingbrackets are provided so that the laterally unsupported lengthLGT, does not exceed the value LGT0defined by:

+

=

3

AAf

EAC

b

L

wfy

fGT0 (8.31)

where

b = flange width

C = 0.55 for symmetric flanges

1.10 for one sided flanges

Tripping brackets are to be designed for a lateral force PSd,which may be taken equal to (see Figure 8-2 ):

+=3

AA0.02P wfSdy,Sd

(8.32)

y,Sd = compressive stress in the free flange

PSd

Af

Tripping bracket1/3 Aw

Figure 8-2 Definitions for tripping brackets

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

26/33

Recommended Practice DNVRP-C201, October 2010

Page26

DETNORSKE VERITAS

9 Local buckling of stiffeners, girders andbrackets

9.1 Local buckling of stiffeners and girders9.1.1 GeneralThe methodology given in Chapter 7 and Chapter 8 is onlyvalid for webs and flanges that satisfy the the following

requirements or fulfils requirements to cross section type IIIdefined in Appendix A of DNV-OS-C101.

Flange outstand for T or L stiffeners or girders should

satisfy:

c 14 tf for welded sections

c 15 tf for rolled sections

(9.1)

For definition of c see Figure 7-3 .

Web of stiffeners and girders should satisfy:

hw42 tw (9.2)

=yf

235

In lieu of more refined analysis such as in Chapter 7, web

stiffeners should satisfy the requirements given in sec. 9.1.2and sec. 9.1.3.

9.1.2 Transverse web stiffeners:

E

fs2

s2.5ts0.3I

y

t

tW

2

ts

>

l

ll

(9.3)

Is = moment of inertia of web stiffener with full web plateflange s

lt = length of transverse web stiffener

s = distance between transverse web stiffeners

s

lt

Figure 9-1 Definitions for transverse web stiffeners

9.1.3 Longitudinal web stiffener:( )

E

fstA0.25I

y

Ws

2

s +> ll (9.4)

Is = moment of inertia of web stiffener with full web plateflange s.

As = cross sectional area of web stiffener exclusive web

plating.

ll = length of longitudinal web stiffener

s = distance between longitudinal web stiffeners

ll

s

Figure 9-2 Definitions for longitudinal web stiffeners

9.2 Buckling of bracketsBrackets should be stiffened in such a way that:

y

b0f

E0.7td

(9.5)

y

b1f

E1.65td

(9.6)

y

b2f

E1.35td

(9.7)

tb = plate thickness of bracket.

Stiffeners as required in eq. (9.6)or eq. (9.7) may bedesigned in accordance with Chapter 7. See Figure 9-3.

Figure 9-3 Definitions for brackets

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

27/33

Recommended Practice DNVRP-C201, October 2010

Page 27

DETNORSKE VERITAS

10 CommentaryCommentary to 3.4 Tolerance requirements

An important factor for the buckling strength is theimperfections that are permitted. As a basis the formulas aredeveloped on the basis that the imperfections are similar to

what is allowed in the DNV-OS-C401 Fabrication andTesting of Offshore Structures. There are differences in thisstandard and what is allowed in DNV Classification Rulesfor Ships and IACS Shipbuilding and Repair QualityStandard-Part A. However, the formulas is seen as beingrelevant for both typical ship with normal good practice and

offshore structures even if an nonlinear FEM analysis of thepanel including the worst combination of allowableimperfections may yield less resistance than obtain from theRP formulas. The reason why this is seen as acceptable is the

following:

The resistance of stiffened plate structures is dependenton imperfections in several elements. Both theimperfection size and pattern for both the plate andstiffener are important and it is less probable that theyhave their maximum at the same time.

The resistance is dependent on more than one element. Itis less probable that all elements have their mostdetrimental imperfection pattern and size at the sametime.

The importance of the imperfection is largest for smallslenderness plate and stiffeners while the likelihood ofdeviations are largest for large slenderness plates.

There are several supportive effects in a real stiffenedplate structure that are disregarded in the resistanceformulations that will in many cases mean a capacity

reserve that is larger than the effect from imperfections.

For structures where these elements are less valid it may benecessary to evaluate the effect of imperfections separately.An example may be a short stocky sniped stiffenerconstructed according to ship rules fabrication tolerances andwhere redistribution of stresses are not possible. Ship rulestolerances are given with a tolerances that are independent of

the member length. This will imply that the tolerances arelarger than the basis for this Recommended Practise and thecapacity of short members may be over-predicted.

Commentary to 4.4 Determination of buckling

resistance based upon linear elastic buckling stress

Linear elastic buckling stress found from literature or byFEM eigenvalue analyses may be used as basis fordetermination of buckling resistance. In order to account formaterial non-linearity, residual stresses and imperfection asuitable buckling curve may be used by calculating the

reduced slenderness parameter defined as:

cr

y

pf

f =

where fcr is linearised buckling stress.

The linearised buckling stress should be carefully selected tobe maximum compressive stress in the analysis. From the

reduced slenderness a buckling resistance may be determined

by using an appropriate buckling curve. Normally a columnbuckling curve as defined in eq. (7.21) and eq. (7.22) can beused unless it is evident that a plate buckling curve asdefined in eq. (6.2) and eq. (6.6) or a shear buckling curve asin eq. (6.17) can be used.

In case of interaction effects e.g. between local and global

buckling the interaction effects can be conservativelyaccounted for by calculated a combined linearised buckling

stress according to the following formula:

crlocalcrglobalcrcomb f

1

f

1

f

1 +

Commentary to 6 Buckling of unstiffened plates

Slender plates designed according to the effective widthformula utilise the plates in the post critical range. Thismeans that higher plate stresses than the buckling stressaccording to linear theory or the so-called critical bucklingstress are allowed. Very slender plates, i.e. span to thickness

ratio greater than 120, may need to be checked forserviceability limit states or fatigue limit states. Examples of

failure modes in the serviceability limit states are reduced

aesthetic appearance due to out of plane distortions or snapthrough if the plate is suddenly changing its out of planedeformation pattern. As the main source for the distortionswill be due to welding during fabrication, the most effectiveway to prevent these phenomena is to limit the slenderness ofthe plate. The likelihood of fatigue cracking at the weld

along the edges of the plate may increase for very slenderplates if the in plane loading is dynamic. This stems frombending stresses in the plate created by out of planedeflection in a deflected plate with in plane loading. Forplates with slenderness less than 120, ordinary fatigue checkswhere out of plane deflections of plate are disregarded will

be sufficient.

Commentary to 7 Buckling of stiffened plates

For wide flanges the stresses in the longitudinal directionwill vary due to shear deformations, (shear lag). For bucklingcheck of flanges with longitudinal stiffeners shear lag effectsmay be neglected as long as the flange width is less than 0.2

L to each side of the web (bulkhead). L being length betweenpoints of counterflexure.

Commentary to 7.2 Forces in the idealised stiffened

plate

With tension field action is understood the load carryingaction in slender webs beyond the elastic buckling load.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

28/33

Recommended Practice DNVRP-C201, October 2010

Page28

DETNORSKE VERITAS

Commentary to 7.4 Resistance of plate between

stiffeners

If secondary stiffeners are used to stabilise the plate fieldbetween ordinary stiffeners the secondary stiffeners need to

be checked as a plate stiffener and the ordinary stiffeners asgirders according to sec. 7.5 and Chapter 8, respectively.

Commentary to 7.7 Interaction equations for axial

compression and lateral pressure

The equations (7.50) and (7.51) may be seen as interactionformulas for the stiffener and plate side respectively for asection at the support. Equations (7.52) and (7.53) arelikewise interaction checks at the mid-span of the stiffener.See also Figure 10-1.

l/2 l/2

Figure 10-1 Check points for interaction equations

With the lateral load on the stiffener side, the stresses changesign and the equations (7.54) to (7.57) shall be used. Thesections to be checked remain the same.

The eccentricity z* is introduced in the equations to find themaximum resistance of the stiffened panel. In the ultimate

limit state a continuos stiffened panel will carry the load inthe axis giving the maximum load. For calculation of theforces and moments in the total structure, of which thestiffened panel is a part, the working point for the stiffenedpanel should correspond to the assumed value of z*. In most

cases the influence of variations in z* on global forces andmoments will be negligible. See also Figure 10-2.

Figure 10-2 Definition of z*. Positive value shown

The maximum capacity will be found for the value of z*when the largest utilisation ratio found for the four equationsis at its minimum. See Figure 10-3.

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-80 -60 -40 -20 0 20 40 60 80

z* (mm)

Utilisationratio

Check point 1

Check point 2

Check point 3

Check point 4

Figure 10-3 Utilisation ratios for the four interactionequations with varying z*

Commentary to 8 Buckling of girders

When a stiffened panel supported by girders is subjected tolateral loads the moments from this load should be includedin the check of the girder. If the girder is checked according

to method 1, the stiffener and plate should also be checked

for the ystresses imposed by the bending of the girder. In

method 2, the ystresses imposed by the bending of thegirder can be neglected when checking plate and stiffener.

Maximumcapacity

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

29/33

Recommended Practice DNVRP-C201, October 2010

Page 29

DETNORSKE VERITAS

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

30/33

Recommended Practice DNVRP-C201, October 2010

Page30

DETNORSKE VERITAS

Part 2.Buckling Strength of Plated Structures - PULS Buckling Code

1 Introduction1.1 General1.1.1 This part describes an accepted computerised semi-analytical model for ultimate and buckling strength

assessment of thin-walled unstiffened or stiffened flat plates.The code has the name PULS as a shortcut for PanelUltimate Limit State.

1.1.2 The PULS code assess the buckling strength ofdifferent types of elements classified according to their

structural action and position in large plated constructions,e.g. in ship hulls, offshore platforms etc., Figure 1.

1.1.3 The code can be used for both steel and aluminiummaterial. Special criteria are introduced for aluminium alloyswith respect to Heat affected zone effects (HAZ).

1.1.4 Application to other metallic materials than steel andaluminium is possible. Special care is needed with respect towelding effects, heat affected zone effects etc.

1.1.5 The PULS code is programmed in a Visual basic (VB)environment. Two separate user interfaces are availableusing the same input/output file format, see Sec. 1.6.

1.1.6 The PULS VB program will be revised and updatedwith respect to new element types, improved solutions, new

features etc. whenever appropriate. The latest officialprogram version with corresponding documentation is

available by contacting the authorised unit within DNV.

1.1.7 The PULS code is implemented into other userinterface applications and as postprocessors in different DNV

software.

Figure 1 Ship hulls with stiffened panels as main building block

Global hull sections

Stiffened anels

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

31/33

Recommended Practice DNVRP-C201, October 2010

Page 31

DETNORSKE VERITAS

1.2 Purpose1.2.1 The present computerised buckling code is introducedas an alternative to the more standard buckling code formatgiven in the first part of the present document. The purpose

is to assess the ultimate and buckling strength limits withgreater consistency than available with more empirical curvefitting approaches.

1.2.2 The code also facilitate reduced orthotropic stiffnessparameters of elastically deformed/buckled plates. Theseproperties are meant for application on large plated structuresanalysed using linear finite elements programs and coarsemeshing, typically FE plate and shell models with oneelement between stiffeners. The goal with such applicationsis to improve the assessment of the nominal stress flow in

ship hulls etc. accepting and accounting for local elastic platebuckling between stiffeners. The option of

anisotropic/orthotropic material models in standard FEprograms can be used. Details of such applications are notdescribed here.

1.3 Theoretical background1.3.1 The PULS models are based on a recognized non-linear thin-walled plate theory according to Marguerre and

von Karman see e.g. Ref. [1], [2]. A harmonic Rayleigh-Ritzdiscretization of deflections is used together with energyprinciples for establishing the non-linear elastic equilibriumequations. The equilibrium equations are solved usingincremental numerical procedures.

1.3.2 For stiffened panels the models are based on anorthotropic version of Marguerres plate theory in which thestiffeners are smeared out over the plate surface. The elasticlocal buckling, postbuckling and imperfection effects of each

component plate in the cross-section are lumped into a set ofreduced orthotropic stiffness coefficients. These reducedorthotropic coefficients are used for assessing the upperbound global elastic buckling limit.

1.3.3 In non-linear elastic buckling theory the internal stressdistribution is split in two categories i.e. the direct externalapplied stresses and a second order stress field due to the

combined effect of buckling and geometrical imperfections.The latter stress field is due to the non-linear geometrical

effect. These stress categories add together forming aredistributed stress field across the panel.

1.3.4 In the Puls code the redistributed stress field is used

for identifying the critical positions (hard corners) wherematerial yielding starts. The values of the external loads, at

which the redistributed membrane stresses reaches the yieldcondition, are used as indicators for the ULS strength (limitstate formulations). For stiffened panels the largestredistributed stresses will typically be along supported edgesor along plate-stiffener junction lines.

1.4 Code principles1.4.1 The ultimate load bearing capacity of plates willdepend on whether the considered plate and/or stiffenedpanel constitute an integrated part of a bottom, deck, ship

side or bulkhead construction or whether they are a part of a

girder web with free membrane boundary conditions.Integrated thin plates in a ship deck, bottom or ship sides etc.can carry loads far beyond the ideal elastic buckling load(over critical strength), while plates with free membranestress support has limited overcritical strength.

1.4.2 For integrated elements and extreme load design (ULSdesign), elastic buckling is accepted, i.e. large elasticdisplacements are accepted as long as the consequences are

controlled and accounted for.

1.4.3 Ideal elastic buckling stresses (eigenvalues) areindependent of the in-plane (membrane stiffness) support

from neighbouring elements. They are useful as referencevalues and can be used as upper limits in case of functionalconsiderations i.e. for load conditions and structural parts inwhich elastic buckling and thereby large elasticdisplacements are not acceptable (SLS philosophy). Idealelastic buckling stresses is also acceptable as the upper limitsfor web girder design, stringer decks etc.

1.4.4 The yield stress to be used in a code strengthprediction is to be the characteristic value as specified for

each material type in the rules.

1.4.5 The PULS ULS capacity assessment is based on theredistributed stress distribution and local material yield

criterion in highly stressed positions (hard corners) alongplate edges and stiffener plate junction lines. This will limitextensive damages and permanent sets.

1.4.6 There is no coupling between strength assessments ofdifferent element types, i.e. the strength evaluation of thestiffened panel is self-contained with no need for bucklingcheck of the individual plate elements of which it iscomposed.

1.4.7 The PULS ultimate capacity values, using the defaultsettings for imperfections, are consistent with the IACSShipbuilding and Quality Repair Manual and DNV-OS-C401

fabrication standard.

1.4.8 Required safety margin against ULS element failuredepends on type of construction, global redundancy,

probability of loads and consequence of failure. Requiredsafety margins are given in respective Ship rules andOffshore standards.

1.5 Safety formats1.5.1 The PULS code calculates the usage factor as ameasure of the available safety margin. The usage factorrepresents the ratio between the applied combined loads and

the corresponding ultimate strength values (ULS). It is aparameter used in DNV Ship Rule contexts.

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

32/33

Recommended Practice DNVRP-C201, October 2010

Page32

DETNORSKE VERITAS

1.5.2 For combined loads the usage factor is defined as theratio between the radius vector to the applied load point inload space and the corresponding radius vector to the ULScollapse boundary, Figure 2.

The usage factor is defined as

u0 L/L=

where the radius vectors L0and Luin load space are defined

as

)(L2

0K

2

0i

2

20

2

100 +++++=

)(L2

Ku

2

iu

2

u2

2

u1u +++++=

K is the maximum number of simultaneously actingindependent in-plane load components.

1.5.3 For a single load cases the definition of usage factor1.5.2 becomes

iu0i / = i = axial load, transverse load etc.

1.5.4 The ULS acceptance criterion is

allow

8/14/2019 BUCKLING STRENGTH OF PLATED STRUCTURES

33/33

Recommended Practice DNVRP-C201, October 2010

Page 33

1.5.5 In the DNV Offshore Standards the LRFD format isused. This implies that the acceptance criterion is on the form

dd RS <

Sdis the load effect including relevant problem dependentload factors. Rdis the design resistance, which is related tothe characteristic resistance as

mkd /RR =

The factor mis the material parameter given in therespective offshore standards.

1.5.6 The LRFD offshore strength format in the PULSterminology is

allow< wherek

d

R

S

= andm

allow

1

=

The following definitions for Sdand Rkin case of combinedloads apply:

Load effect 0d LS (design load effectinclusive load factors)

Characteristic resistance uk LR (ultimate strengthexclusive safety factors)

The ratio kd R/S is the same as the usage factor . It givesa consistent measure of the safety margin. The material

factor mis the inverse of the acceptable usage factor allow.

1.5.7 The ratio ( allow/ ) can be used as a measure of thesafety margin relative to the required strength margin, i.e.

1allow