Common Structural Rules for
Bulk Carriers, January 2006

Background Document
CHAPTER 7 – DIRECT STRENGTH ANALYSIS

NOTE:- This TB is published to improve the transparency of CSRs and increase theunderstanding of CSRs in the industry.
- The content of the TB is not to be considered as requirements.
- This TB cannot be used to avoid any requirements in CSRs, and in cases
where this TB deviates from the Rules, the Rules have precedence.
- This TB provides the background for the first version (January 2006) of theCSRs, and is not subject to maintenance.

COMMON STRUCTURAL RULES FOR BULK CARRIERS − BACKGROUND DOCUMENT
PAGE 2 – CHAPTER 7: DIRECT STRENGTH ANALYSIS© IACS - the International Association of Classification Societies and theInternational Association of Classification Societies Limited
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COMMON STRUCTURAL RULES FOR BULK CARRIERSCOMMON STRUCTURAL RULES FOR BULK CARRIERS − BACKGROUND DOCUMENT
PAGE 3 – CHAPTER 7: DIRECT STRENGTH ANALYSISTABLE OF CONTENTS:
SECTION 1 – DIRECT STRENGTH ASSESSMENT OF PRIMARY SUPPORTING MEMBERS..... 5
1

GENERAL.................................................................................................................................... 5
1.1
1.2
1.3
1.4
1.5

Application.............................................................................................................................. 5
Computer program ................................................................................................................. 5
Submission of analysis report ................................................................................................. 5
Net scantling ........................................................................................................................... 5
Applied loads .......................................................................................................................... 6

SECTION 2 – GLOBAL STRENGTH FE ANALYSIS OF CARGO HOLD STRUCTURES................ 7
1

GENERAL.................................................................................................................................... 7
1.1

2

ANALYSIS MODEL .................................................................................................................... 7
2.1
2.2
2.3
2.4
2.5

3

Application.............................................................................................................................. 7
Extent of model ....................................................................................................................... 7
Finite element modeling.......................................................................................................... 8
Boundary conditions ............................................................................................................... 8
Loading conditions.................................................................................................................. 9
Consideration of hull girder loads .......................................................................................... 9

ANALYSIS CRITERIA .............................................................................................................. 11
3.1
3.2
3.3
3.4

General .................................................................................................................................. 11
Yield strength assessment ..................................................................................................... 11
Buckling and ultimate strength assessment .......................................................................... 12
Deflection of primary supporting members.......................................................................... 12

SECTION 3 – DETAILED STRESS ASSESSMENT.......................................................................... 141

GENERAL.................................................................................................................................. 14
1.1

2

ANALYSIS MODEL .................................................................................................................. 14
2.1
2.2
2.3
2.4
2.5

3

Application............................................................................................................................ 14
Areas to be refined ................................................................................................................ 14
Refining method.................................................................................................................... 14
Modeling ............................................................................................................................... 14
Loading conditions................................................................................................................ 14
Boundary conditions ............................................................................................................. 15

ANALYSIS CRITERIA .............................................................................................................. 15
3.1

Allowable stress .................................................................................................................... 15

SECTION 4 – HOT SPOT STRESS ANALYSIS FOR FATIGUE STRENGTH ASSESSMENT ....... 16
1

GENERAL.................................................................................................................................. 16
1.1

2

ANALYSIS MODEL .................................................................................................................. 16
2.1
2.2
2.3

3

Application............................................................................................................................ 16
Modeling ............................................................................................................................... 16
Loading conditions................................................................................................................ 16
Boundary conditions ............................................................................................................. 17

HOT SPOT STRESS .................................................................................................................. 17
3.1
3.2

Definition .............................................................................................................................. 17
Evaluation of hot spot stress ................................................................................................. 17

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PAGE 4 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

3.3

Simplified method for the bilge hopper knuckle part ........................................................... 17

APPENDIX 1 – LONGITUDINAL EXTENT OF THE FINITE ELEMENT MODELS....................... 19
1

LONGITUDINAL EXTENT....................................................................................................... 19
1.1 ...................................................................................................................................................... 19

2

TYPICAL MESH ........................................................................................................................ 19
2.1 ...................................................................................................................................................... 19

APPENDIX 2 – DISPLACEMENT BASED BUCKLING ASSESSMENT IN FINITE ELEMENTANALYSIS......................................................................................................................................... 20
1

INTRODUCTION ..................................................................................................................... 20
1.1 ...................................................................................................................................................... 20

2

DISPLACEMENT METHOD .................................................................................................... 20
2.1
2.2

General .................................................................................................................................. 20
Calculation of buckling stresses and edge stress ratios ......................................................... 21

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PAGE 5 – CHAPTER 7: DIRECT STRENGTH ANALYSISSECTION 1 – DIRECT STRENGTH ASSESSMENT OF
PRIMARY SUPPORTING MEMBERS
1

GENERAL

1.1

Application

1.1.1
1.1.1.a

Chapter 7 describes the calculation methods for direct strength assessment of bulk
carriers. The three types of analysis required by CSR and given below are
mentioned in the rules, but only the description common to the three types ofanalysis in Section 1 is given here.

1.1.2
1.1.2.a

1.2

The analysis required in Sec.2 will give the scantling of primary supporting
members, the sec.3 is for the local reinforcement of stress concentration area and theanalysis of Sec. 4 is for the calculation of fatigue damage factors.

Computer program

1.2.1
1.2.1.a

1.3

A program with adequate reliability including not only a solver for FEA but also a
pre/post processor should be used because scantlings will be increased/decreased
according to the results of direct strength assessment. However, since rules do not
approve or specify any program in particular, a statement such as the above has
been made.

Submission of analysis report

1.3.1
1.3.1.a

1.4

Some of data can be submitted as electric style such as application data for used Prepost processor or CAE application.

Net scantling

1.4.1
1.4.1.a

The corrosion amount given in Chapter 3 is the maximum expected value by which
the thickness is considered to decrease during the design life of a ship. Deducting
the required wastage amount from all the members of the hold model used wasconsidered excessive; it was considered more appropriate to deduct the average
wastage amount. The average wastage amount was set as half the required amount,and it was decided as the deduction amount for the model.
However, 100% corrosion in each panel is fairly appropriate when determining thecritical buckling stress, and therefore, the corrosion deduction is taken as 100% inthis case.

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PAGE 6 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

1.5

Applied loads

1.5.1

Design loads

1.5.1.aApplicable loads will be calculated according to the formulae in Ch. 4. Actual
combination of design load and loading condition are described in Appendixes of
Ch. 4.

1.5.2

Structural weight

1.5.2.a

The hull structure inertia force by dynamic acceleration is not considered FEA.

1.5.3

Loading conditions

1.5.3.aActual loading cases to be applied are described in Appendixes of Ch. 4.

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PAGE 7 – CHAPTER 7: DIRECT STRENGTH ANALYSISSECTION 2 – GLOBAL STRENGTH FE ANALYSIS OF CARGOHOLD STRUCTURES
1

GENERAL

1.1

Application

1.1.1
1.1.1.aCoarse mesh analysis is performed with the aim of deciding the scantlings (plating
thickness) of primary parts of primary structural members. Studies on the bearing
strength to withstand fatigue collapse and details of partial reinforcements will be
carried out in the detailed stress assessment of Section 3 or the hot-spot stressanalysis for fatigue strength assessment of Section 4.

1.1.2
1.1.2.a

The requirements show that the CSR set the three criteria, yield, buckling anddeformation. Finally, designer should clearly show that the scantling of the ship areclear all criteria.

2

ANALYSIS MODEL

2.1

Extent of model

2.1.1
2.1.1.aSince the past, conditions of symmetry at both ends of the model were introduced in
the 1/2+1/2 or 1/2+1+1/2 hold length model and boundary conditions were usedso as to support bulkheads in the direct calculation of bulk carriers. Analysis using
such modeling extent and boundary conditions has the following issues:(1) Bulkhead stress condition must be analyzed correctly since bulkhead parts arealso subject to strength assessment in the common rules. However, large support
reactions are generated in bulkheads with the boundary conditions used forbulkhead support.
(2) Since bulkheads are supported, stress components due to hull girder loads thatwere being ignored until now can no longer be ignored because stresses that
include hull girder loads are now being calculated and assessed. Hull girder loads
due to design loads applied in the boundary conditions until now cannot becalculated easily.
(3) In the past, the 1/2 hold length part was also assessed by introducing conditionsof symmetry. However, the centers of the hold (in the longitudinal direction) andthe double bottom may be offset because of the effect of the stool (often extends
over 1 transverse spacing) installed at the lower part of the bulkhead, and the stresscondition of the double bottom, which is important, may not be analyzed correctly.
Considering these issues, the modeling extent in CSR has been taken as 3 holds. The
central hold and the bulkheads forward and aft of it are the extent of assessment,
and the supports at the bulkheads to be assessed are eliminated.

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PAGE 8 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

2.1.2
2.1.2.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

2.1.3
2.1.3.a

2.2

Generally, all structure members that will be assessed by the requirements of CSR
should be modeled. Stiffeners on outer and inner shell or watertight wall should bemodeled since these elements will transfer the pressure to supporting structure.

Finite element modeling

2.2.1
2.2.1.a

The types of finite elements used in the model are described here. The analysis ofholds of bulk carriers performed conventionally is clarified.
Regulation has been established for using two-dimensional anisotropic elements.
The method for performing analysis of rigidity of longitudinals including plating
has been approved.

2.2.2
2.2.2.aSince stiffeners will be boundary condition of buckling panel in buckling strength
assessment, mesh boundaries are to simulate the stiffening system. If meshboundaries do not coincide to actual stiffening systems, stresses can be modified
according to Appendix 2 of this Chapter.

2.2.3
2.2.3.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

2.2.4
2.2.4.a

The criteria value of yield strength is decided considering the mesh size which
described in the requirement.
Primary supporting members in double bottom or double hull system are divided
into three elements heights since stress due to bending of the double hull/bottom
structure will be clearly shown in the model.
Considering the accuracy of the analysis, aspect ration of each shell elements shouldbe less than 4 as far as practicable. Aspect ratio more than 4 can be acceptable for theelements not to be stress evaluated.

2.2.5
2.2.5.a

2.3

The requirements describe for the orthotropic elements.

Boundary conditions

2.3.1
2.3.1.aThe support conditions were set for both ends of the model as shown in the Fig.1.
The model will be simply supported at both end and hull girder stress will be
modified superimposing to element stress or applying hull girder force at both ends.
COMMON STRUCTURAL RULES FOR BULK CARRIERSCOMMON STRUCTURAL RULES FOR BULK CARRIERS − BACKGROUND DOCUMENT
PAGE 9 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

Fig. 7.1 Support condition for hold model

2.4

Loading conditions

2.4.1

General2.4.1.a

The loading conditions to be applied for global analysis are described in Chapter 4
Appendix 2.

2.5

Consideration of hull girder loads

2.5.1

General2.5.1.a

Hull girder loads are considered for each loading condition, but two kinds ofanalysis are required by this regulation for combining bending and shear loads:
bending moment analysis where bending moment is the primary component andshear analysis where shear load is the main component.
The hull girder loads are required to be adjusted during bending moment analysis
such that the bending moments occurring in the model at the center of the target
hold become target values, and during shear analysis such that the shear loads ofthe model at the position of either the forward or aft bulkhead become target values.
Hull girder loads occurring in the model are calculated assuming that the model is abeam simply supported at both ends. Refer to the requirement of 2.5.4 of the Rules.

2.5.2

Vertical bending moment analysis

2.5.2.a

The analysis considers only still water and wave bending moments. No shearing
force except shearing force due to local loads that are applied to the extent of themodel will be considered.
The combination of MS and MW for each load case is shown in the Appendix 2 ofChapter 4.

2.5.3

Vertical shear force analysis

2.5.3.aThe shearing force analysis is required only for the case that the sharing force will
be dominant, such as heavy ballast or alternate loading. Bending moment should beconsidered in spite of no shearing force considering in bending moment analysis,
but not full.

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PAGE 10 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

2.5.4

Influence of local loads

2.5.4.a

The formulae of bending moment and shearing force by local load are calculatedusing a simple beam theory. Sign conventions coincide to Ch4. Sec.3 [1.1.1].

2.5.5

Methods to account for hull girder loads

2.5.5.a

CSR specifies either of two methods to perform analysis - the "superimposition
method" that calculates stresses for assessment by superimposing the results ofanalysis by local loads only with the hull girder stresses, and the "direct method" inwhich adjusting loads (mainly bending moment) are applied to both ends of themodel before the analysis so that the required bending moment is obtained at theconsidered position.
Discussions were held in IACS on the adoption of both methods to CSR. It was
concluded that the actual results until now should be considered. Trial calculation
results of both methods have shown no difference in the required plating thickness.
Therefore, both superimposition method and indirect method were approved for
adoption in the CSR for bending moment analysis.

1.6
1.2
Usage Factor

Direct method
Direct method
Superimposition method
Superimposition method

Results ofyielding check

1.4
1.0
0.8
0.6
0.4
0.2
00
1.6

6

8

9

10

11

12

13

11

12

13

N o .5 L o a d e d h o ld

Results ofbuckling check

1.4
1.2
Usage Factor

7

1.0
0.8
0.6
0.4
0.2
00
mm
24

6

7

8

9

10

N o .5 L o a d e d h o ld

Required
thicknes

22

Original
Direct method
Superimposition method

20
18
16
14
12
10
6 Center⇔Side
7
8

9Center⇔side
10
11

Bhd

Mid-holdCenter
side
12
13
Bhd

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PAGE 11 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

Fig. 7.2 Results of comparison of required plating thickness by direct method and by
superimposition method
Fig. 7.2 shows the comparison of results calculated by the direct method and thesuperimposition method for ballast holds in existing double side skin Capesize BC.
The assessment factor (usage factor: designed value/permissible value) varies
slightly in the two methods; however, when the required plating thickness wasdetermined, it was seen that this difference was within 0.5 mm.
It is required to perform shear analysis only by the direct method.

2.5.6

Direct method

2.5.6.a

The bending moments which will apply to the both ends of the model will leadbased on the simple beam theory.

2.5.7

Superimposition method

2.5.7.a

Hull girder stress will be superimposed to the assessed stress at stress evaluation.
The stress is calculated based on the beam theory and considering only hull girder
bending moment. The moment is considered specified target moment and moment
due to local loads applied to the model.

3

ANALYSIS CRITERIA

3.1

General3.1.1

Assessment holds

3.1.1.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

3.1.2
3.1.2.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

3.2

Yield strength assessment

3.2.1

Reference stresses

3.2.1.a

3.2.2

Equivalent stress3.2.2.a3.2.3

Allowable stress3.2.3.a

The permissible value was taken as the yield strength of the material because thedesign loads were considered for the severest conditions of operation in a 25-year
period in the North Atlantic route. In case of high tensile steel, it was decided tocorrect the permissible value using the material constant so that the permissible
value was not the yield strength of material but a value smaller than that because

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PAGE 12 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

the fatigue strength does not change for mild steel and because stress may increase
with the increase in deformation.
The permissible value when anisotropic elements were used was set referring to thevalues of criteria of classification societies with experience of using the sameelements.

3.3

Buckling and ultimate strength assessment

3.3.1

General3.3.1.a

3.3.2

Stress of panel

3.3.2.a

The regulation for buckling and ultimate strength in Section 3 Chapter 6 applicableto the hold FEA results have been developed in the first place, as a regulation
applicable to results of analysis by frame element model. In frame analysis, even if
members corresponding to flanges, such as in the double bottom are continuous, theincrease in stress due to the Poisson effect occurring from flange continuity is notincluded because the floor or girder has been modeled as a single frame.
On the other hand, in the CSR analysis method, stress increase due to the Poisson
effect in FEA of plating elements is included. For this reason, the FEA method
becomes more severe if the analysis results according to this chapter are applied asis to buckling and ultimate strength of Section 3, Chapter 6 if the same safety factor
is used. It was therefore decided to use the criteria in Section 3, Chapter 6 after
performing the correction of excluding the Poisson effect from the FEA results. The
correction is to be limited only to cases where stress increases due to the Poisson
effect. Therefore, it is applicable only when the two axial stresses in a panel are bothcompressive stresses.

3.3.3

Boundary conditions

3.3.3.a

3.4

Deflection of primary supporting members

3.4.1
3.4.1.aAccording to this regulation, adequate rigidity can be given to the double bottom
structure by restricting the depth in 6.1.3, Section 6, Chapter 3. However, it was
decided to control the relative deflection of the double bottom according to the FEA
results with the aim of confirming adequate rigidity from the results of holdanalysis. Fig. 7.2 shows the relative deflection (δmax/li) of the double bottom
structure obtained from analysis results based on CSR for conventional ships.

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PAGE 13 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

H eavy C argo H old

δ/li*1000

Light C argo H old

7.000
6.000
5.000
4.000
3.000
2.000
1.000
0.000
C apesize

H andyH andyD1

D2

D3

二重船側B CC apesize C apesize P anam ax
D4

S1

S2

H andyC apesize

S3

S4

単船側B C

Fig. 7.2 Relative deflection of double bottom of existing ships according to common
rules

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PAGE 14 – CHAPTER 7: DIRECT STRENGTH ANALYSISSECTION 3 – DETAILED STRESS ASSESSMENT
1

GENERAL

1.1

Application

1.1.1
1.1.1.aThe detailed analysis described in this section is implemented mainly to study theneed for partial reinforcements.

2

ANALYSIS MODEL

2.1

Areas to be refined

2.1.1
2.1.1.a

The purpose of detailed analysis is to confirm the strength of cross section ofprimary support members. When the global hold analysis is carried out using
isotropic element with longitudinally frame space size mesh, detailed FEA will be
required for the higher stressed case only due to the mesh size relatively small.

2.1.2
2.1.2.a

2.2

In case orthotropic elements used, detailed analysis according to the section is
mandatory since mesh size is larger than isotropic elements.

Refining method

2.2.1
2.2.1.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

2.3

Modeling

2.3.1

Element type2.3.1.aOrthotropic elements cannot be used due to detailed analysis.

2.3.2

Mesh

2.3.2.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

2.4

Loading conditions

2.4.1
2.4.1.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

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PAGE 15 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

2.5

Boundary conditions

2.5.1
2.5.1.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

2.5.2
2.5.2.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

3

ANALYSIS CRITERIA

3.1

Allowable stress

3.1.1
3.1.1.a

Detailed stress assessment requires the use of meshes having sizes of about onefourth the longitudinal space in the overall assessment analysis of cargo holds. The
results of actual detailed assessment of trial ships showed that the stress in the finemesh parts was about 1.2 times the stress in coarse mesh analysis. Thus, thepermissible stress for mild steel was specified as 1.2 times 235 N/mm2, that is, 280
N/mm2.

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PAGE 16 – CHAPTER 7: DIRECT STRENGTH ANALYSISSECTION 4 – HOT SPOT STRESS ANALYSIS FOR FATIGUESTRENGTH ASSESSMENT
1

GENERAL

1.1

Application

1.1.1
1.1.1.aVery fine mesh analysis is performed to evaluate directly the hot spot stress forfatigue strength assessment.

1.1.2
1.1.2.aStandard loading conditions to be considered in fatigue strength assessment isreferred to Appendix 3, Chapter 4 of the Rules.

2

ANALYSIS MODEL

2.1

Modeling

2.1.1
2.1.1.aSince it is necessary to assess hot spot stress which represents the stressconcentration due to structural discontinuity for the fatigue assessment, it ispreferable to evaluate hot spot stress by the model where a very fine mesh ismounted in the global cargo hold model.

2.1.2
2.1.2.aSince the stress concentration due to structural discontinuity depends on thesurrounding structural arrangements, area where a very fine mesh is mounted is to
be extended so that the effect of surrounding structural arrangements on stressconcentration can be represented.

2.1.3
2.1.3.a

2.2

Since the stress concentration due to structural discontinuity is remarkable in thelocal area, the size of very fine mesh is to be determined so that the behavior ofstress concentration can be represented. The standard size of vary fine mesh isreferred to the Guidance of IIW (Fatigue Design of Welded Joints and Components).

Loading conditions

2.2.1
2.2.1.a

Referred to Appendix 3, Chapter 4 of the Rules.

2.2.2
2.2.2.a

Referred probability level of the assessed stress response is determined so that therealistic stress can be assessed.

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PAGE 17 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

2.3

Boundary conditions

2.3.1
2.3.1.a

Refer to 2.3 of Section 2, Chapter 7.

3

HOT SPOT STRESS

3.1

Definition

3.1.1
3.1.1.aSince most of the fatigue life is occupied by the crack initiation life, surface stress at
a hot spot is to be assessed.

3.1.2
3.1.2.a

3.2

Hot spot stress used in the fatigue assessment should include all of the load
components. The hull girder stress is to be superimposed when the hot spot stressis assessed according to the superimposition method.

Evaluation of hot spot stress

3.2.1
3.2.1.aHot spot stress is obtained using a linear extrapolation based on the stress located at
0.5 times and 1.5 times the plate thickness so that the effect of abrupt stress increase
near the intersection of shell elements can be excluded.

3.2.2
3.2.2.a

The hot spot stress is obtained linearly extrapolating the surface stress to theposition of node of shell elements. In this case, especially in the case of right angle,
evaluated stress tends to be overestimated. Degree of overestimation is mainly
depending on the difference between the actual hot spot location and assumed
location and the difference of surface stress gradient depending on the angle
between two plates. Especially in the case of right angle, degree of overestimation isremarkable since a steep stress gradient is observed. Therefore, the correction factor
for the hot spot stress evaluated by the shell elements FE analysis was examinedbased on the results of solid elements EF analysis.

3.2.3
3.2.3.a

3.3

Stress evaluated by the shell element is generally represented by the stress at centerof the element. When assessing the stress along free edge of non-welded area, thesestresses evaluated by the shell element FEA are to be extrapolated to the evaluation
point.

Simplified method for the bilge hopper knuckle part

3.3.1
3.3.1.a

To assess hot spot stress, evaluation by multiplying the nominal stress with thestress concentration factor is an alternative method other than the evaluationaccording to the very fine FEA.
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PAGE 18 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

3.3.1.b

To assess hot spot stress, evaluation by multiplying the nominal stress with thestress concentration factor is an alternative method other than the evaluationaccording to the very fine FEA.

3.3.2
3.3.2.aIn order to apply stress concentration factor, the nominal stress at hot spot locationis necessary to define. The definition of nominal stress at hot spot location is definedaccording to the results of global hold FEA. Refer to the Section 2, Chapter 7.

3.3.3
3.3.3.aStress concentration factors applicable to the hot spot at an intersection of plates
such as the bilge knuckle structure, the lower stool structure etc., which are thetypical structural details to be fatigue assessed, were examined based on thetheoretical solution and the results of FEA.

3.3.3.b

The geometrical stress concentration depends on the differences of structural details.
The effects of some typical cases of structural detail on the degree of stressconcentration were evaluated according to the results of FEA to introduce thecorrection factors depending on the difference of structural details.

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PAGE 19 – CHAPTER 7: DIRECT STRENGTH ANALYSISAPPENDIX 1 – LONGITUDINAL EXTENT OF THE FINITEELEMENT MODELS
1LONGITUDINAL EXTENT

1.1
1.1.1
1.1.1.a2

Refer to Chapter 7 Sec.2 2.1.

TYPICAL MESH

2.1
2.1.1
2.1.1.a

It is considered that for this topic, no information in addition to that shown in theRules is necessary to explain the background.

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PAGE 20 – CHAPTER 7: DIRECT STRENGTH ANALYSISAPPENDIX 2 – DISPLACEMENT BASED BUCKLINGASSESSMENT IN FINITE ELEMENT ANALYSIS
1

INTRODUCTION

1.1
1.1.1
Information obtained from a finite element calculation cannot always be useddirectly for buckling analyses, as the subdivision of the finite element does notnecessarily account for the buckling demands. Appropriate stresses and edge stressratios must be evaluated for the buckling panel.

2

DISPLACEMENT METHOD

2.1

General2.1.1.aIn the FE-technique the stresses of a plane stress element is described by thedisplacements of the nodal points of that element. This leads to the idea to get thestresses of a buckling panel from the displacements of the corner nodes. The stresses
in the buckling panel then can be evaluated for any location within the panel
according to the shape function used. A suitable stress-displacement relationship isgiven in [PRZ] page 92.
With this stress information the buckling strength can be assessed as stated in
Section 3, 3. “Buckling Criteria of Elementary Plate Panels”.
Full numerical accuracy of displacements must be used, as the stresses result fromdifferences of nodal displacements.
To obtain background information about finite Element techniques, reference ismade to the following Literature:[PRZ] J. S. Przemieniecki, “Theory of Matrix Structural Analysis”, McGraw-Hill, Inc.
1968
[ZIE] O. C. Zienkiewicz, “The Finite Element Method”, 3rd Edition, McGraw-Hill
Book Company (UK) Ltd., 1977
[BAT] K. J. Bathe, “Finite Elemte Procedures ”, Prentice Hall, 1995

2.1.2

4-node and 8-node panels

2.1.2.aThe stresses obtained by the shape function of a 4-node plane stress element are
only a coarse representation of the real stress distribution in a plate panel. It isconsidered to be sufficient, if the aspect ratio of the plate panel is not too big (i.e.
less than 3) and the stress variation is moderate. In an 8-node panel the stressvariation in longitudinal direction can be calculated more reliable.

2.1.3

Calculation of nodal displacements

2.1.3.a

If a node of the buckling panel is located on the edge of a plane stress element, thenits location can be described by:rrrr = (1 − α )ri + αr j
COMMON STRUCTURAL RULES FOR BULK CARRIERS(a.1)

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PAGE 21 – CHAPTER 7: DIRECT STRENGTH ANALYSIS

and the corresponding displacement follows the same formula:rrr
d = (1 − α )d i + αd j

(a.2)

where

r
r r
r = node of the buckling panel, ri , r j = nodes of the plane stress element,
r
r r r
r r
d = displacement at r , d i , d j = displacements at ri , r j ,α = interpolation factor in the interval [0,1]2.1.3.b

If a node of the buckling panel is located inside of a plane stress element, then its
location can be described by:

r
r
r
r
r
r = (1 − β )[(1 − α )ri + αr j ] + β [(1 − α )rl + αrk ](b.1)

and the corresponding displacement follows the same formula:

r
r
r
r
r
d = (1 − β )[(1 − α )d i + αd j ] + β [(1 − α )rl + αrk ](b.2)

where the indices i, j, k, l refer to the F.E. nodes, in circular notation.

α , β = interpolation factors, both in the interval [0,1]Equations a.1 and b.1 are used to determine α , β , and then the displacement can bereceived using b.1 and b.2 respectively.

2.1.4

Transformation in local system

2.1.4.a

For the transformation of displacements from the global FE-system into the local
system of the buckling panel apply to standard FE-literature, e.g.[Zie] page 333 ff.

2.2

Calculation of buckling stresses and edge stress ratios

2.2.1
2.2.1.a

It is considered that for this topic, no information in addition to that shown in the
CSR for Bulk Carriers, is necessary to explain the background.

2.2.2

4-node buckling panel

2.2.2.aThe stress displacement matrix is obtained from the relation given in [PRZ] page 92,
evaluated at the corner points. Owing to the stress convention for buckling stresses,
the sign is changed, to receive positive compressive stresses.
The stress modification to contribute for the poisson effect is commented in 3.2.3.a

2.2.3

8-node buckling panel

2.2.3.aThe stress displacement matrix is obtained from the relation given in [PRZ] page 92,
evaluated at the corner points and the mid-points of the longer edges. Owing to thestress convention for buckling stresses, the sign is changed, to receive positive
compressive stresses. The stress modification to contribute for the poisson effect iscommented in 3.2.3.a
COMMON STRUCTURAL RULES FOR BULK CARRIERS