solid shell connection in finite element analysis

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Marine Structures 20 (2007) 143–163

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Study on shell–solid coupling FE analysis for fatigueassessment of ship structure

N. Osawa�, K. Hashimoto, J. Sawamura, T. Nakai, S. Suzuki

Department of Naval Architecture and Ocean Engineering, Osaka University,

2-1 Yamadaoka, Suita, Osaka 565-0871, Japan

Received 26 March 2007; received in revised form 30 March 2007; accepted 4 April 2007

Abstract

A simple, robust and high-precision method for shell–solid coupling has been demonstrated. The

coupling is achieved by a fictitious shell plane perpendicular to the original shell plane. The guidelines

for this coupling technique for ship structural analysis are established by examining the local stress of

the stool-like welded joint models. The surface stresses in the vicinity of the weld of a small corner

joint model and a large ship structure model calculated by the proposed technique are in good

agreement with those obtained from the multipoint constraint-based coupling models and the strain

measurements. This demonstrates the effectiveness of the proposed technique in the local approach

fatigue assessment of actual ship structures.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Fatigue; Local approach; Finite element method; Coupling analysis

1. Introduction

The approaches to the fatigue strength assessment have been further developed duringrecent years (e.g. [1]). In addition to the conventional nominal stress approach, localapproaches such as the (structural) hot-spot stress (HSS) approach and the notch stressapproach have reached the stage of practical application, and recommendationsconcerning stress determination for these approaches have been presented (e.g. [2,3]).

Because ships are plate structures, global shell FE models are usually employed forsimplicity and low cost. In the local approach, the modeling of weld is a problem in the

see front matter r 2007 Elsevier Ltd. All rights reserved.

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nding author. Tel.: +8166879 7576; fax: +81 66879 7594.

dress: [email protected] (N. Osawa).

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ARTICLE IN PRESSN. Osawa et al. / Marine Structures 20 (2007) 143–163144

shell FE analysis. For the HSS approach, Fricke [3] showed that the welds had to bemodeled for the cases where the results were affected by local bending, e.g. due to an offsetbetween plates. Regarding the modeling of weld in shell FE models, Niemi [2] presented aseries of means by which the weld detail could be modeled, but did not give a clearrecommendation.When a solid FE model is employed, modeling of welds is easily possible and the stress

field in the vicinity of the weld can be investigated with a high degree of precision. Thenotch stress needs to be evaluated by solid analysis for the case where the deformation nearthe critical point cannot be approximated by two-dimensional stress state. It is needed toextend the scope of application of solid FE analysis in order to make the local approachesmore advanced.Ship hulls are redundant structures, and various loads, such as external sea pressure,

inner cargo load, etc. act on them simultaneously with phase differences. In these cases, itis difficult to define the boundary conditions for the local FE model if the extent of themodel is limited in the vicinity of the welded joint. These boundary conditions can beevaluated by global (whole-ship or hold) shell FE models. As has been discussed, it isdesirable to use solid local FE models. It is needed to transfer the angular rotations or themoments of the global shell elements to the translational displacements or forces of thelocal solid elements.There are some studies on the solid-based local approach considering the ship’s global

load effect (e.g. [4]). The ‘‘submodeling’’ technique is generally employed in these studies.The conversions of the rotations/moments to the translational displacements/forces areperformed manually in this technique. The fatigue assessment of a ship structure requires ahuge number of load cases because the stress calculation is repeated for various headingangles, encountering wavelengths and loading conditions. The manual conversionsmentioned above increase the analysis time inadmissibly.The need for rotation/moment conversions is eliminated by using shell–solid coupling

FE models in the local analysis. The submodeling procedures themselves becomeunnecessary when the solid local model is embedded in the global shell model. It ispossible to reduce the analysis time for the solid-based local approach assessmentdrastically by employing the shell–solid coupling FE analysis.In this study, a simple, robust and high precision shell–solid coupling technique is

demonstrated. The coupling is achieved by a fictitious shell plane perpendicular to theoriginal shell plane. The guidelines for this coupling technique for ship structural analysisare established by examining the local stress of the stool-like welded joint models. Thesurface stresses in the vicinity of the weld of a small corner joint model and a large shipstructure model calculated by the proposed technique are compared with those obtainedfrom the conventional multipoint constraint based coupling models and the strainmeasurements, and the effectiveness of the proposed technique in the local approachfatigue assessment of actual ship structures is investigated.

2. Shell–solid coupling techniques

2.1. Coupling via multipoint constraint (MPC) equations

The coupling of shell and solid elements is usually achieved via MPC equations (e.g. [5]).This provides a way of modeling transitions with little or no stress perturbation in the

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vicinity of the dimensional interfaces. This approach can be applied for the geometricallynon-linear analysis [6] and the structures with anisotropic, elastic–plastic and compositematerials [7–9]. However, the evaluation of the coupling equation is so troublesome that itis difficult to practice the precise MPC coupling in the practical design.

In the practical analysis, shell–solid coupling by rigid link, such as RBE2 function ofMSC. Nastran [10] is frequently employed as an alternative to the precise MPC coupling.The ease of the rigid-link definition makes it suitable for the practical analysis. However,superfluous constraint in the direction of the plate thickness applied by the rigid linksometimes causes a large stress perturbation near the interface. Adding to this, because ofthe interaction of in-plane displacements caused by the shell rotations, the perturbationoften sharply increases at the junctions where two interfaces with different orientationsmeet. There is considerable uncertainty as to the accuracy of local stress evaluated by rigid-link technique.

2.2. Coupling by fictitious perpendicular shell planes

As an alternative to the MPC coupling or rigid link, sometimes solid elements arecoupled with shell elements by a fictitious shell plane perpendicular to the original shellplane as shown in Fig. 1. Hereafter, this technique is called ‘perpendicular shell couplingmethod (PSCM)’. The advantages in PSCM comparing with the MPC coupling and rigidlink is as follows:

(a)
The shell–solid coupling is achieved with incredible ease even for complex-shaped solidparts. It is usable by anyone with a minimal skill in shell and solid FE modeling.
(b)
The superfluous constraint in the direction of the thickness can be relieved by the in-plane deformation of the fictitious shell, resulting in less stress perturbation near theinterface in comparison to that for the rigid-link case.
(c)
The interaction of in-plane displacements at the junctions of interfaces can be absorbedby the in-plane and out-of-plane deformations of the fictitious shell, resulting in lessperturbation at the junctions of the interfaces.
The problem in PSCM is how to decide the stiffness of the fictitious shell. In this study, itis assumed that the elastic properties, Young’s modulus E and Poison’s ratio n, of thefictitious shell are equal to those of the original shell plate. Hereafter, let tS denote the

Fig. 1. The concepts of perpendicular shell coupling method (PSCM).


thickness of the fictitious shell. The stiffness of the fictitious shell can be controlled bychanging tS.The inhibitory action against the stress perturbation near the interfaces and the

junctions is reduced when thick tS is employed, while the transfer of the angular rotationsor moments becomes insufficient and unreal concentrated deformation arises on theinterface section of the solid part when tS is excessively thin. The value of tS which givesreasonable results depends on the model and the boundary conditions. Such optimumvalue of tS has to be established empirically, but there is no report on this problem for shipstructures so far as the authors know.

3. Optimization of the fictitious shell thickness for ship structural analysis

To optimize the fictitious shell thickness tS, simple welded joint models subjected todifferent loading conditions are examined. The models are shaped like the base joints ofthe stools in bulk carriers, and the extent of them are chosen so that the entire model canbe modeled with solid elements. The analyses of the shell–solid coupling FE models createdby PSCM with various tS produce results that serve comparisons with those obtained bythe whole-solid models.In this paper, all FE analyses are based on the infinitesimal deformation formulation

and the theory of isotropic linear elasticity.

3.1. FE models of the stool joints

The perpendicular and inclined welded joints of lower stools in bulk carriers are modeledas shown in Fig. 2. The thickness of the tank top plate is 10mm (hereafter, referred to as t).The models are subjected to three types of load and boundary conditions illustrated inFig. 3. The all displacement components are constrained on the back end of the model, andthe ‘tensile’ (Fig. 3a), ‘vertical’ (Fig. 3b) and ‘pressure’ (Fig. 3c) loads are applied on thefront end or the upper face of the tank top plate.Figs. 4 and 5 show the whole-solid and shell–solid coupling FE models of the

perpendicular and inclined joints. For the coupling models, the three-dimensional solidpart extends to a minimum of ten times of plate thickness from the hot spot (theintersection of tank top, solid floor, girder and stool). The weld beads are not modeled inthese models. The shell–solid couplings are achieved by PSCM. The thickness of thefictitious shell is chosen so that tS/t ¼ 1.0, 0.1 and 0.01. The analysis of the coupling modelcreated by rigid-link technique is also performed for comparison. The number of man-hours needed to set up the shell–solid coupling is less than 0.25 (15min) for PSCM while itis more than 10 for rigid link.The analyses are performed by MSC.Marc 2005r2. The solid parts comprise of three-

dimensional arbitrarily distorted brick elements (Element 7 [11]). The number of divisionin the direction of the thickness is 8, and the element size in the vicinity of the joint is t/8� t/8� t/8. The shell parts are comprised of 4-node or 3-node bilinear thick-shellelements (Element 75 or 138 [11]). The number of the integration point in the direction ofthe thickness is 9 (default). The material properties of steel (E ¼ 200GPa, n ¼ 0.3) aregiven to the models. We consider the coordinates shown in Fig. 2, x in the ship’slongitudinal direction (directed from the stool to the inside of the hold), y in the transversedirection and z in the vertical direction. The origin is at the hot spot.

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Fig. 2. The models of the stool joints.

N. Osawa et al. / Marine Structures 20 (2007) 143–163 147

3.2. Analysis results

3.2.1. Perpendicular joint (tensile and vertical load cases)

A comparison of the out-of-plane (z-direction) displacements of the tank topplate computed by the whole-solid model and the PSCM-based shell–solid couplingFE models with tS/t ¼ 0.01 and tS/t ¼ 1.0 for the perpendicular joint model is plottedin Fig. 6. Fig. 6(a) shows the displacements on the longitudinal (x-direction) center-line, and Fig. 6(b) those on the transverse shell–solid interface and the transverse freeedge. The deformations calculated from the coupling models are in good agreementwith the results from the whole-solid model without distinction of the thickness of thefictitious shell.

Because there are no significant differences between the principal stress and the stresscomponent in the direction perpendicular to the weld line (x-direction), x-components ofthe surface and mid-plane stresses are evaluated. The origin of the stress evaluation path is

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Fig. 3. The load and boundary conditions applied to the stool joint models.

N. Osawa et al. / Marine Structures 20 (2007) 143–163148

at the notch (the intersection of the plate surfaces of the stool and the tank top) on thelongitudinal centerline for the surface stress, and the intersection of the mid-planes ofplates for the mid-plane stress. The evaluation path lies along the x-direction. The surfaceand mid-plane stresses of the solid part are evaluated by the linear interpolation orextrapolotion of the element stresses.A comparison of the x-component of the surface stresses, sSX, on the tank top plate

computed by the whole-solid model and the PSCM-based shell–solid coupling FE modelswith tS/t ¼ 0.01, 0.1 and 1.0 are plotted in Fig. 7. The results from the rigid-link-basedcoupling model are also plotted in this figure. The transverse axis in this figure is thedistance of the read-out-points (ROPs) from the notch, d. The shell–solid interface lies atd ¼ 95mm. The surface stresses calculated from the coupling models are in goodagreement with the results from the whole-solid model regardless of tS excluding thevicinity of the interface (60mmpdp100mm). The stress of the elements right on the

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Fig. 4. FE models of the perpendicular joint.


interface (d ¼ 90mm) is larger than the reference value (the result from the whole-solidmodel) for the rigid-link model while it is smaller than the reference for the PSCM-basedmodels. Stress perturbation appears in the vicinity of the interface. The error (differencebetween the stress from coupling models and that from the whole-solid model) of thesurface stress of the PSCM-based models is much larger than that from the rigid-linkmodel when tS/tp0.1. However, this error becomes negligible and smaller than that of therigid-link model when tS/t ¼ 1.0.

A comparison of the x-component of the mid-plane stresses (stress on the mid-plane ofthe plate), smX, of the tank top plate computed by the whole-solid model and the PSCM-based coupling FE models with tS/t ¼ 0.01, 0.1 and 1.0 are plotted in Fig. 8. The transverseaxis in this figure is the distance of the ROPs from the intersection of the mid-planes, x.The shell–solid interface lies at x ¼ 100mm. The mid-plane stresses calculated from thecoupling models agree well with the results from the whole-solid model apart from thevicinity of the interface. Strong stress fluctuation appears in the vicinity of the interface,and it becomes severe as tS/t decreases. The concentrated nodal force on the mid-planetransferred from the shell part and the finite bending rigidity of the fictitious shell causethis stress fluctuation. The stress fluctuation that occurs on the mid-plane decays rapidlywith the distance in the direction of the thickness, resulting in less stress perturbation onthe surface. The error of the stress is exceeding acceptable level when tS/tp0.1 while itbecomes negligible excluding the element right on the interface (x ¼ 95mm) whentS/t ¼ 1.0.

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Fig. 5. FE models of the inclined joint.


3.2.2. Inclined joint (tensile and vertical load cases)

In the same manner, as the perpendicular joint model, the surface stresses cal-culated from the PSCM-based coupling models agree well with the results from thewhole-solid model regardless of tS excluding the vicinity of the interface, and thecalculation error in the stress from the coupling model becomes negligible whentS/t ¼ 1.0. A comparison of the surface stress, sSX, on the tank top plate com-puted by the whole-solid model and the PSCM-based shell–solid coupling FE modelwith tS/t ¼ 1.0 is plotted in Fig. 9. The transverse axis is the distance of ROPs from thenotch, d.

3.2.3. Perpendicular joint model subjected to the pressure load

A comparison of the surface stress, sSX, on the tank top plate of the perpendicular jointmodel computed by the whole-solid, PSCM-based (tS/t ¼ 1.0) coupling, and rigid-link-based coupling FE models for the pressure load case is plotted in Fig. 10. The transverseaxis is the distance of ROPs from the notch, d.In the same manner, as the tensile and vertical load cases, the surface stress of the PSCM

model agrees well with that of the whole-solid model and the stress fluctuation is negligiblewhen tS/t ¼ 1.0. On the other hand, a considerable stress perturbation occurs for the rigid-link-based coupling model. It is considered that this stress perturbation is caused by thesuperfluous constraint in the direction of the thickness and the interaction of in-planedisplacements at the junctions of interfaces. The stress perturbation does not exert an effect

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Dis

pla

cem

ent (m

m)

-200 -100 0 100 200

Longitudinal distance from the hot spot, x (mm)

Whole-solid

Coupling (PSCM , ts/t = 1.0)

Coupling (PSCM, ts/t = 0.01)

-8

-6

-4

-2

0

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ent (m

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Transverse free edge (x=250mm)

Transverse shell-solid interface (x = 100mm)

0.1

0.0

-0.1

-0.2

-0.3

-0.4

-0.5

Transversal distance from the hot spot, y (mm)

Fig. 6. Displacement of the perpendicular joint model subjected to the vertical load.


on the accuracy of the surface stress within the region of do3t in this case, but theperturbation can become so large that it becomes difficult to maintain the accuracy of thelocal stress in some loading conditions.

The calculation results obtained in this section can be summarized as follows:

(a)
PSCM technique can save enormous amounts of time for the shell–solid coupling incomparison with the rigid-link coupling.

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Coupling (Rigid-link)

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Surf

ace s

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sx (

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Fig. 7. Surface stress for the perpendicular joint model.


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σ mx (

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Distance from the intersection of mid-planes, x (mm)

Fig. 8. Mid-plane stress for the perpendicular joint model.


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Vertical Load Case

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Fig. 9. Surface stress for the inclined joint model.


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σ sx (

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100806040200

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Pressure Load Case

Whole-solid




Fig. 10. Surface stress for the perpendicular joint model subjected to the pressure load.


(b)
The accuracy of the local surface stress calculated by the PSCM-based coupling modelsis equal to or better than that of the rigid-link-based models when tS/t ¼ 1.0.
(c)
PSCM technique can inhibit the sharp stress perturbation caused by the superfluousconstraint and the interaction of in-plane displacements of the rigid-link-based models.
(d)
For the PSCM-based coupling models with tS/t ¼ 1.0, the influence of the mid-planestress fluctuation near the shell–solid interface on the solid local stress can be neglectedat a distance of two times of plate thickness or more from the interface.
3.3. Guidelines for PSCM-based shell– solid coupling FE modeling

In local approaches, the stress field within the region which extends to a minimum oftwo or three times the plate thickness from the hot spot, has to be evaluated with a highdegree of precision. Local approaches other than HSS, e.g., notch stress approach, or ananalysis of fatigue flaw growth based on fracture mechanics may be employed in theassessment. A high degree of precision of not only surface stress but also inner stress isrequired in these cases. Based on the results described above, it can be said that theserequirements can be satisfied by using PSCM-based shell–solid coupling FE modelscreated in accordance with the following guidelines for the welded joints in stool- orhopper-like structures:

(a)
The thickness of the fictitious shell is comparable to the plate thickness. (b) The three-dimensional solid part extends to a minimum of five times of plate thickness
from the hot spot.


(a) is the condition to inhibit the surface stress perturbation near the shell–solidinterfaces, (b) eliminates the effect of the mid-plane stress fluctuation near the interface on
the inner stress in the vicinity of the hot spot; the extent of the solid part is decided as thesum of the extents of the local stress evaluation region (three times of plate thickness) andthe buffer zone which absorbs the mid-plane stress fluctuation (two times of platethickness).
4. Local stress analysis of welded joint models

To confirm the effectiveness and reliability of the proposed approach for shell–solidcoupling, welded joint models are examined. The models are selected so that the strainmeasurements and three-dimensional FE analysis results are available. The surface stressesin the vicinity of the weld are calculated using PSCM-based shell–solid coupling FEmodels following the guidelines shown in the previous section, and they are compared withthe measured stresses and the results of the three-dimensional FE analyses reported in theliteratures.The shell–solid coupling FE analyses in this section are performed by MSC.Marc

2005r2. The same elements as those used in Section 3.1 (solid: Element 7, shell: Element 75or 138) are employed. The material properties of E ¼ 206,000MPa, n ¼ 0.3 are given tothe models.In the analyses, the surface stress component in the direction perpendicular to the weld

line (x-direction) sSX is evaluated. The origin of the stress evaluation path is at the weldtoe. The evaluation path lies in the direction perpendicular to the weld line. The surfacestress of the solid part is evaluated by the linear extrapolation of the solid element stresses.

4.1. Perpendicular corner joint model

Sugimura et al. [12] examined the local stress in the vicinity of the weld of theperpendicular corner joint model shown in Fig. 11. The hot spot is the intersection of theupper horizontal flange and the inner vertical flange. The surface stresses of the upperhorizontal flange were measured by strain gages, and they were also calculated by MPC-based shell–solid coupling FE analysis. The solid part of Sugimura’s model extended toabout 20 times of plate thickness in flange and five in web. The plate thickness of flangeand web plates, t, is 20mm. The number of division in the direction of the thickness is 8,and the minimum solid element size is t/9� t/9� t/8.The PSCM-based coupling model employed is shown in Fig. 12. The solid part extends

to a minimum of seven times of plate thickness from the hot spot. The number of divisionin the direction of the thickness is 10, and the minimum solid element size is t/10� t/10�t/10. Weld beads are modeled in the solid part. At the corner of the weld bead, the beadand base plate are bonded together using MSC.Marc’s ‘glued contact’ function [13]. Thefictitious shell thickness is chosen so that tS/t is 1.0.A comparison of the surface stress, sSX, on the centerline of the upper flange computed

by the PSCM-based coupling FE model with that from Sugimura’s MPC-based couplingmodel and the strain measurement is plotted in Fig. 13. The transverse axis in this figure isthe distance of ROPs from the weld toe, d. In this figure, it is shown that the surfacestresses calculated from PSCM-based coupling model are in good agreement with thoseobtained from the MPC-based model and the strain measurement. This good correlation

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Fig. 11. Perpendicular corner joint [12].


of the results demonstrates the effectiveness of the proposed PSCM technique in the localstress evaluation of small joint models.

4.2. VLCC bilge knuckle model

Ship Research Panel 245 examined the local stress in the vicinity of the welded joint of aVLCC bilge knuckle model shown in Fig. 14 [14,15]. The model was a bilge knuckle sectionfor a double-hull VLCC in approximately 1/3 scale, and it was about 6m in length, 5m inwidth, 3.6m in height and 20 tons in steel weight. To make stress distributions of the modelsimilar to those of the actual ship, a three-floor space in the longitudinal direction wasmodeled. The model was fixed to a rigid wall at the double-hull side, with the ship’s bottombeing upside and the inner bottom being downside. The load was applied by threesyncronized hydraulic jacks on the centerline of the double bottom as shown in Fig. 14.The thickness of the inner bottom plate, t, was 10mm. The model was built from mildsteel, and the welding was performed in accordance with NK rules. The hot spot is theintersection of the inner bottom plate, the inclined inner hull plate, the floor and the sidegirder. The flank angle of the fillet weld bead at the hot spot was about 451. Because theplates intersect at an angle of 451, the joint can be modeled as shown in Fig. 15.

The surface stress of the inner bottom plate was measured by strain gages, and they werealso calculated by MPC-based shell–solid coupling FE analysis. The solid part of SR245’smodel extended to about eight times of plate thickness from the hot spot in the innerbottom plate. Adaptive P-method and a shape function suitable for the treatment of stresssingularity were applied to the solid analysis. The solid surface stress was evaluated bycalculating the strain components at the corner points of solid elements.

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Fig. 12. PSCM-based shell–solid coupling FE model of the perpendicular corner joint.


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Fig. 14. Bilge knuckle model (Panel SR245 [15]).

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Surf

ace s

tress,

σ sx (

MP

a)

35302520151050

measured


Coupling (MPC, Sugimura etal. [12])

Distance from the notch, d (mm)

Fig. 13. Surface stress of the upper flange of the perpendicular corner joint.

Fig. 15. Simplified model of the weld joint in the VLCC bilge knuckle model.


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Fig. 16. Global shell FE mesh of the VLCC bilge knuckle model.


The surface stress of the inner bottom plate is calculated by submodeling techniqueusing a PSCM-based shell–solid coupling local model. The global shell FE model is shownin Fig. 16. Considering the symmetry, one half-symmetric model is employed. The globalshell FE analysis is performed by MSC.Nastran 2005. The global model is modeled by4-node shell elements (QUAD4 [10]), and the element size in the vicinity of the hot spotis t� t.The PSCM-based shell–solid coupling local FE model is shown in Fig. 17. The solid part

extends to a minimum of eight times of plate thickness from the hot spot. The number ofdivision in the direction of the thickness is 8, and the minimum solid element size is t/8� t/8� t/8. The nodal displacement and angular rotation calculated by the global shell modelis transferred to this local model. The fictitious shell thickness is chosen so that tS/t is 1.0.A comparison of the surface stresses, sSX, on the inner bottom plate computed by the

PSCM-based coupling FE model with those from SR245’s MPC-based coupling modeland the strain measurement are shown in Fig. 18. The transverse axis in this figure is thedistance of ROPs from the weld toe, d. In this figure, it is shown that the surface stressescalculated from PSCM-based coupling model are again in good agreement with thoseobtained from the MPC-based model and the strain measurement. The hot spot of thismodel is in a complex multiaxial stress state. Though the guidelines in Section 3.3 arederived based on the simple load cases, the local stress of this model is evaluated with veryhigh accuracy. Therefore, it is considered that the proposed guidelines are applicable tocomplex loading modes. This means that the local approach fatigue assessment of actualship structures can be performed by following the proposed guidelines.

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Fig. 17. PSCM-based local shell–solid coupled FE model for the welded joint in the VLCC bilge knuckle model.


5. Conclusions

A simple, robust and high precision method for shell–solid coupling has beendemonstrated. The coupling is achieved by a fictitious shell plane perpendicular to theoriginal shell plane. The guidelines for this coupling technique for ship structural analysisare established by examining the local stresses of the stool-like welded joint models. Thesurface stresses in the vicinity of the weld of a small corner joint model and a large shipstructure model are calculated by the proposed technique and compared with thoseobtained from the MPC-based coupling models and the strain measurements. As results,the followings are found:

(1)
In the proposed coupling approach, the accuracy of the calculated local stress is assuredwhen the thickness of the fictitious shell is comparable to the plate thickness and thesolid part extends to a minimum of five times of plate thickness from the hot spot.
(2)
For both a small corner joint model and a large ship structure model, the surfacestresses calculated by the proposed coupling technique are in good agreement withthose obtained from the MPC-based models and the strain measurements. Thisdemonstrates the effectiveness of the proposed technique in the fatigue assessment ofactual ship structures by the local approaches.

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800

600

400

200

0

Surf

ace s

tress,

σ Sx [M

Pa]

50403020100


Measured

Coupling (P-method, MPC, SR245[15])

Distance from the notch, d [mm]

Fig. 18. Surface stress of the inner bottom plate of the VLCC bilge knuckle model.


Acknowledgments

The authors gratefully acknowledge Mr. Tadashi Sugimura (Mitsubishi HeavyIndustries Co., Ltd.) and Mr. Yoshiteru Tanaka (National Maritime Research Institute,Japan) for providing experimental and calculation results on the models analyzed in thispaper. The authors would like to express to Mr. Kenji Nakata (Mitsui Engineering &Shipbuilding Co., Ltd.) our deepest gratitude for his cooperation in the FE modeling andanalyses. Also, the authors thank Mr. Tetsuji Fukuoka (Mitsui Engineering &Shipbuilding Co., Ltd.), Mr. Masao Morikawa (Universal Shipbuilding Co., Ltd.) andMr. Tetsuya Nakamura (Universal Shipbuilding Co., Ltd.) for their guidance orcomments.

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