1 2 modelling of propulsion shaft 3 line and basic

22359(2008)3, 223-227

MODELLING OF PROPULSION SHAFT LINE AND SHAFTING ALIGNMENT... N. VULIĆ, A. ŠESTAN, V. CVITANIĆUDC 629.5.026

Nenad VULIĆ1

Ante ŠESTAN2

Vedrana CVITANIĆ3

Modelling of Propulsion Shaft Line and Basic Procedure of Shafting Alignment Calculation

Original scientifi c paper

The main propulsion shafting is exposed to various operating conditions throughout the entire lifetime of a modern ship. The necessary condition for the shafting to withstand and survive all possible situations is its proper dimensioning and manufacture, as well as its assembly and testing onboard. Its alignment is of utmost importance during the assembly process itself.

The aim of this paper is to present the shafting alignment calculation procedure in order to help the designer to understand the whole alignment process. Calculation presumptions, modelling of shafting parts, material properties and loading are given in detail. The advantages of the transfer matrix methods over the fi nite element methods in this particular case have been described. The important part is to establish the designed shafting elastic line onboard the ship, during the outfi tting in the shipyard. It is proposed in the conclusion that the presented matter be included into a future edition of the CRS Technical Rules.

Keywords: propulsion system, propulsion shafting, shaft alignment, transfer matrix method

Modeliranje i osnove proračuna centracije brodskog porivnog vratilnog voda

Izvorni znanstveni rad

Tijekom životnog vijeka suvremenog broda, porivni vratilni vod izložen je vrlo promjenjivim radnim stanjima. Osnovni su uvjeti da vratilni vod ispuni svoju funkciju u svim mogućim radnim uvjetima pravilno dimenzioniranje i izrada, kao i montaža i ispitivanje na brodu. U provođenju montaže posebnu važnost ima postupak centracije.

Cilj je ovoga rada prikazati metodologiju proračuna centracije vratilnog voda, sa svrhom da se projektantima omogući lakše razumijevanje cjelovitoga postupka centracije. Potanko su prikazane proračunske pretpostavke, modeliranje dijelova vratila, kao i značajke materijala i opterećenja. Opisana je prednost primjene metode početnih parametara u matričnom prikazu (tzv. metode pri-jenosnih matrica) u odnosu na metodu konačnih elemenata u ovom specifi čnom slučaju. Naglašena je važnost postizanja projektne elastične linije vratilnog voda, tijekom opremanja broda. U zaključku se predlaže da se prikazani pristup uključi u buduća izdanja Tehničkih pravila HRB-a.

Ključne riječi: porivni sustav, porivni vratilni vod, centracija vratilnog voda, metoda prijenosnih matrica

Authors' addresses:1 Croatian Register of Shipping,

Marasovićeva 67, HR-21000 Split,

Croatia,

E-mail: [email protected] Ante Šestan, Faculty of Mechanical

Engineering and Naval Architecture,

University of Zagreb, I. Lučića 5,

Zagreb, Croatia

E-mail: [email protected] 3 Vedrana Cvitanić, FESB, University

of Split, Ruđera Boškovića bb, HR-

21000 Split, Croatia

E-mail: [email protected]

Received (Primljeno): 2007-12-17Accepted (Prihvaćeno): 2008-02-11Open for discussion (Otvoreno za raspravu): 2009-30-09

1 Introduction

The purpose of the propulsion machinery (main engine, gearbox, propulsion shaft line, propeller and pertinent auxiliary systems) is to propel the ship and to control manoeuvring, thus enabling the navigator to be in control of the ship’s speed and course. The main propulsion shaft line is the essential part of a modern ship propulsion system, exposed to various conditions throughout the ship’s lifetime. It has to function properly under all possible operating conditions. Consequently, the shaft line preliminary and fi nal design, its static and dynamic behaviour shall be carefully considered by the designer and by the clas-sifi cation society.

Shafting alignment procedure considers static and pseudo-static loading of the shafting in order to determine its static response. This procedure consists of three phases: calculation,

assembly and validation of the assembled shaft line onboard the ship. The main goal of this procedure is to determine and ensure onboard achievement of the bearings designed positions in athwart direction in order to comply with the loading criteria for propulsion system and shafting parts. For this purpose the shaft line is usually modelled as a continuous multi-span beam on several supports. They may be modelled as absolutely stiff or linearly elastic (in the case of static and pseudo-static response), or even as real radial journal bearings (in the case of dynamic response).

The goal of this paper is to provide designers with the basic information how to model real shafting systems in order to perform shaft alignment calculations. The paper aims to present the conventional shafting alignment calculation procedure and its presumptions. Modelling of shafting parts, material proper-ties and loading is given in detail, in order to help the designer

224 59(2008)3, 223-227

N. VULIĆ, A. ŠESTAN, V. CVITANIĆ MODELLING OF PROPULSION SHAFT LINE AND SHAFTING ALIGNMENT...

understand the whole calculation process. The advantage of the transfer matrix method over the fi nite element method in this particular case is briefl y described. The important part is to establish the designed shafting elastic line onboard the ship, during the outfi tting phase in the shipyard. The results of a real life calculation example are presented in the end.

2 Shafting alignment calculation procedure

The shafting alignment calculation comprises evaluation of the shafting elastic line and the reaction forces of supports for the pre-determined offsets of supports. In case of propulsion systems with gearboxes (mainly small, medium and high-speed four-stroke diesel engines) the scope of the analysis is restricted to the shaft line from the propeller to the output shaft of the gearbox, together with its bearings and the bull gear. The remaining shaft line parts (clutches, input shaft, as well as the engine itself) need not be taken into account. A typical shaft line layout of this kind is schematically shown in Figure 1.

In the case of directly coupled engines (mainly large slow-speed two-stroke diesel engines) the shafting alignment analysis takes into account the model and the static behaviour of the engine crankshaft. The complete crankshaft need not be modelled in detail, as almost every slow-speed diesel engine manufacturer provides drawings describing this model as a girder system avail-able to the shaft line designers.

Figure 1 Schematic of a typical marine shaft line including a gearbox [1]

Slika 1 Shematski prikaz tipičnog brodskog vratilnog voda s reduktorom [1]

2.1 Input data and modelling of the system

The data describing dimensions, material and loading of the shafts, together with the data describing the bearings concept (slide or roller), bearing clearances and lubrication means are to be available for shafting alignment calculations. This real system is modelled as a statically indeterminate system of variable sec-tion beams with multiple supports. The shaft line elements are modelled by means of circular section model elements, and the shaft line bearings are modelled by means of absolutely stiff or linearly elastic supports. In general, the cross-section varies from one beam to another.

In general, model elements are of conical shape. A special case of conical element is the element of cylindrical shape, as a cone with equal diameters on both ends.

Elements are made of homogenous material, of specifi c density ρ, submerged (completely, partially, or not at all) into sea-water of specifi c density ρ

w. The shaft material elastic prop-

erties are described by means of Young modulus of elasticity E and shear modulus G.

As the calculation presumes the ship afl oat, after assembling all the parts of shaft line, loading of elements consists of: • self-weight of the element; • buoyancy in sea water (for submerged elements);

• external concentrated force F in the centre of the cross section of the left element end, [N];

• external concentrated moment T in the centre of the cross section of the left element end, [Nm];

• external uniformly distributed load q along the element (ow-ing to other possible forces, additional to the shaft self weight and buoyancy), [N/m].A general element model, together with the support at its right

end is shown in Figure 2.

Figure 2 General model of shaft line element [3]Slika 2 Općeniti model elementa vratilnog voda [3]

All the calculations are to be performed for the vertical plane, where the infl uence of self-weight and buoyancy shall be taken into consideration within the loading of the model. In the case of propulsion systems with gearboxes, where gearing forces in horizontal direction have a signifi cant infl uence, the separate calculations for the horizontal plane are also needed.

2.2 Calculation presumptions

The calculations are based upon the real element dimensions, and the following presumptions: • Propeller is completely or semi-submerged into water;• Volumetric forces (self-weights and buoyancy) are uniformly

distributed along each element; • All the bearings may be modelled by means of absolutely

rigid or linearly elastic supports; • The infl uence of shear forces and deformations is to be taken

into account; • The axial position of each reaction force is on the half way

of the bearing length. If necessary, the inclination of shafting with respect to the

ship waterline (horizontal plane) may be taken into account by calculating of components (for concentrated forces) and correc-tion of gravity constant (for volumetric forces).

2.3 Selection of calculation method (FEM vs. transfer-matrix method)

The most appropriate modelling and calculation procedures in this case are the method of initial parameters in its matrix form (the so called: transfer-matrix method) and fi nite element method (FEM). Practically equivalent results may be obtained by means of either of these two methods, except in the case of trapezoidal loading along the element itself.

However, the transfer matrix method is chosen and preferred, as it requires linear systems of signifi cantly smaller ranges to be solved. Particularly, FEM requires solving of 2m equations (where m is the number of shaft line elements). On the other hand, the

22559(2008)3, 223-227

MODELLING OF PROPULSION SHAFT LINE AND SHAFTING ALIGNMENT... N. VULIĆ, A. ŠESTAN, V. CVITANIĆ

transfer-matrix method requires solving only of z+2 equations (where z is the total number of stiff supports between the system ends). In addition to this, the transfer matrix method is purely analytical, implementing the solutions to differential equations for beams in bending and shear.

There is an additional advantage of the transfer-matrix method over FEM in this case. FEM calculation results (solutions) are valid in the nodes only, whereas the transfer-matrix method al-lows the user to obtain defl ections, slopes, bending moments and shear forces along the element itself (i.e. between the nodes) on the basis of calculated results in the nodes.

Consequently, the calculation model based on transfer ma-trices in a single (e.g. vertical) plane is chosen and described further on.

2.4 Element transfer matrices and selection of initial parameters

For calculation purposes the whole shaft line is modelled as a system of multi-span beams, supported in rigid (absolute stiff) or linearly elastic supports. Each beam has a uniform circular cross-section (solid or hollow). Conical shafting elements are modelled as cylindrical with mid-section diameters, for the evaluation of stiffness and loading by volumetric forces.

The basic goal of the transfer matrix method is to determine the state vectors v

i in each section of the whole system. It is

necessary to determine these vectors at each end section of each element:

(1)

Considering the system element (i), the state vector (vi+1

) at the right section of the element right end is related with the state vector (v

i) at the right section of the element left end as

follows:

(2)

In the equation (2) Li = L

i,support ·L

i,elem denotes the total transfer

matrix of the element i (including the support at its right end). It may also be written in the expanded form (3):

(3)

The quantities in the equations (1) to (3) have the following meaning:li– element length, [mm]

EIi – element bending stiffness, [Nm2]

GAi – element shear stiffness, [N]

κi = f

(d

u/d

v) – shear form factor for the circular (solid or hollow)

section, κi =1,11 … 1,45

qi – uniform distributed external loading along the element (see

Figure 2), [N/m]

Fi – concentrated force at the element left end (see Figure 2),

[N]T

i – concentrated moment at the element left end (see Figure

2), [Nm]R

i+1 –reaction of the support (if any), at the element right end, positive downwards, [N].

w, β – displacement components (defl ection, [m] and slope, [m/m]),

M, Q – internal forces (bending moment, [Nm] and shear force, [N]).

Note: In case there is no support at the element right end, the transfer matrix L

i,elem = L

i for the sole element is obtained from

(3), taking Ri+1

=0.

The initial parameters to be selected are the two additional unknowns at the whole system left end. They are fi nally deter-mined from the two known parameters at the system right end, together with the reactions in all rigid supports. The system ends may either be free, propped, or fi xed. Any case may be chosen, however, the most common situation is that both of the ends are free. In the case of free left end of the system the unknown initial parameters are:

w1 – defl ection of the system left section;

β1 – slope of the system left section.

These parameters, together with all the reaction forces in rigid supports (R

1, R

2, … , R

z) are determined from the known boundary

conditions at the right end of the system, i.e.

Mm+1

=0; Qm+1

=0 – in case of free right end;w

m+1=0; M

m+1=0 – in case of propped right end;

wm+1

=0; βm+1

=0 – in case of fi xed right end.

The total number of equations to be solved is thus z+2 only.

2.5 Calculation of infl uence coeffi cients, initial reac-tions of supports and system response

The whole elastic system is described by means of the system matrix A, and the system vector b. Both of them are assembled on the basis of the boundary conditions at each fi xed support and at the rightmost end of the system, by means of span transfer matrices. For each span these span transfer matrices are simply matrix products of transfer matrices that relate the state vector in the section of one stiff support (or system leftmost end) to the next one:

(4)

where:

k – number of elements in the present span.

In case of both the left and right ends free, the vector of un-knowns k consists of the two initial parameters (w

0 and β

0) and

of the reaction forces in stiff supports (Rj), as follows:

(5)

In this particular case, the boundary conditions used to as-semble matrix A and vector b are: ■ The zero displacements of the fi xed supports (forming the

fi rst z equations). This condition may also be expressed by

v i i i i i

Tw M Q= −[ ]β 1

Li

ii

i

i

i

i i

i

i

i

i i

EI EI GA EI

T F

=

− − ⋅ +12 6 2

2 3 4κ ii i i i i

ii

i iq

GAF

q

6 24 2

0 1

2

+⎛⎝⎜

⎞⎠⎟

+ ⋅ +⎛⎝⎜

⎞⎠⎟

κ

ii

i

i

i

i

ii

i i i i

EI EI EIT

F q2 2

2 2 6

0 0 1

− ⋅ + +⎛⎝⎜

⎞⎠⎟

i i i ii i

i i i i

T Fq

F q R

− + ⋅ +⎛⎝⎜

⎞⎠⎟

− + +( +

2

1

2

0 0 0 1 ))

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥0 0 0 0 1

v L v L L L vjR

span j jL

k k jL

+ −= ⋅ = ⋅ ⋅ ⋅ ⋅1 1 1( )

,( ) ( )…

k = −⎡⎣ ⎤⎦w R R Rz

T

0 0 1 2β …

v L v Liright

i port ileft

i por+ += ⋅ =1 1( )

,( )

,sup sup tt i elem ileft

i ileft⋅ ⋅ = ⋅L v L v,

( ) ( )

226 59(2008)3, 223-227

N. VULIĆ, A. ŠESTAN, V. CVITANIĆ MODELLING OF PROPULSION SHAFT LINE AND SHAFTING ALIGNMENT...

the null-vector p0 of initial offsets of supports (p

0=0).

■ Mm+1

=0 and Qm+1

=0, used to form the remaining 2 equations (i.e. the 2 rows of A and the 2 components of b).

The best practice is to calculate the infl uence coeffi cients prior to the calculation of components of k. That is the essential part of the complete analysis. The infl uence coeffi cient h

ij quantitatively

expresses the change of reaction force (in N) in the movement direction of the support i, when the support j moves for 1 mm in that direction.

The matrix of infl uence coeffi cients H, which is independent of the actual support offsets, is determined as:

(6)

The vector of unknowns k0, containing the initial reactions

in the stiff supports (i.e. those for zero support offsets) then becomes:

(7)

Once the components of the vector k0 are known, the state

vectors in each section of the system may be easily found by a simple matrix multiplication, beginning from the known state vec-tor at the leftmost end of the system. This is the system bending and shear response in terms of defl ection, slope, bending moment and shear force at both ends of each element.

2.6 Calculation of bearing reactions and the system response for designed support offsets

Designed support offsets are to be determined in advance so that the system response satisfi es certain criteria for the fi nal calculated case. This fi nal case may be the static response of the assembled shaft line during the ship outfi tting, or even the pseudo-static response of the shaft line in operation. If the external forces have not changed meanwhile, and transferring from the system with zero support offset to the present one, the vector of unknowns k will be:

(8)

Bending and shear response of the present system with the designed support offsets is determined according to the same procedure described in 2.4 for the system with zero support offsets.

2.7 Design acceptance criteria and their verifi cation

Detailed description of the design acceptance criteria would be beyond the scope of this paper, so they will be just briefl y outlined here. These criteria are to be met for the pseudo-static response of the shaft line in operation, both for cold and hot working conditions, as follows [3]: ■ The stresses in shafts are to be below the prescribed permis-

sible limits. This criterion may be applied either to the normal stresses or the equivalent stresses.

■ Loading of the bearings is to be within prescribed limits. In case of vertical plane calculations, bearing reactions are to be directed upwards (to avoid overloading of the neighbouring bearings) with the rule of thumb criterion for the minimal

reaction value as 20% of the left and right total load of the span. Maximal reaction values shall not exceed the ones al-lowable by the specifi c pressure in the bearings, dependent upon the bearing material in question.

■ Shaft line slope in the bearings is to be within allowable limits dependent upon the bearing pre-selected clearances, to avoid metal contact between the bearing and the shaft at the bearing ends. Otherwise, slope boring of the bearings will be unavoid-able. The rule of thumb states that the slope may “spend” up to 50% of the bearing clearance. Some classifi cation societies, e.g. [4], prescribe that the relative slope between the propeller shaft and the aftermost sterntube bearing is, in general, not to exceed 0.3 mm/m in the static condition.

■ The shaft line shall not overload the gearbox itself, in case of propulsion systems with gearboxes. The gearbox manufactur-ers usually prescribe the maximal allowable load transmitted by the shaft line to the gearbox. In the absence of this data, the rule of thumb will be to limit the difference in reaction forces in the two bearings of the gearbox output shaft to maximum 20% of the weight of the bull gear.

■ The shaft line is not to overload the main propulsion engine crankshaft or thrust shaft, in case of propulsion systems with directly coupled main engines. As a rule, the engine manufacturers usually prescribe the maximal allowable load transmitted by the shaft line to the engine fl ange in terms of shear forces and bending moments allowable range. The stated design acceptance criteria shall be explicitly veri-

fi ed in the calculation phase, after all the results (system response values) have been obtained.

3 Calculation example

The presented calculation procedure has been implemented in the computer program MarShAl (Marine Shafting Alignment), coded in MS Excel/VBA, dedicated to the presented calcula-tions. For illustration a few characteristic diagrams obtained by this computer code, which have been implemented and verifi ed on an inland navigation ship, are presented hereafter (Figure 3). The propulsion system consists of a four-stroke engine (279 kW), connected to the shafting by a reduction gearbox.

4 Conclusion

Shaft line is to be properly aligned in order to ensure its reli-able functioning throughout the complete ship lifetime. Careful calculation, as well as setting up of its results onboard (for the ship afl oat), as well as their validation during the assembly and the testing phase is essential. This paper describes details of the calculation procedure, promoting the advantages of (somewhat forgotten) transfer matrix methods.

Details of the design acceptance criteria, a real model of radial journal bearings and the review of classifi cation society requirements are beyond the scope of this paper. However, from the authors’ long-term experience with this matter, it should be important to enhance the existing technical rules requirements of classifi cation societies for shafting alignment to cover also the case of small size ships. A proposal of this kind is expected to be the matter of further work.

It is to be pointed out once again that this paper presents only the basic information related to shaft alignment calculations, in

H A= −1

k H b0 0= ⋅

k H p p k= ⋅ −( ) +0 0

22759(2008)3, 223-227

MODELLING OF PROPULSION SHAFT LINE AND SHAFTING ALIGNMENT... N. VULIĆ, A. ŠESTAN, V. CVITANIĆ

order to help designers, shipbuilders, or even engineering students understand the essential calculation concepts. The authors’ experience shows that it is important to make such information, presented in simple terms, widely available to the public. Com-prehensive further information regarding the complex subject of shaft alignment calcula-tion, validation and criteria may be found elsewhere in literature, e.g. [4], [5], [6] and [7], together with the detailed information about specialised and extremely powerful software.

References

[1] KOZOUSEK, W.M., DAVIES, P. G.: “Analysis and Survey Procedures of Propulsion Systems: Shafting Align-ment”, LR Technical Association, Paper No. 5, London 2000.

[2] VULIĆ, N.: “Advanced Shafting Alignment: Behaviour of Shafting in Operation”, Brodogradnja 52(2004)3, p. 203-212.

[3] VULIĆ, N.: “Shaft Alignment Compu-ter Calculation (in Croatian)”, Bulletin Jugoregistar, No. 1, 1988, p. 1-13.

[4] …: “Rules and Regulations for the Classifi cation of Ships”, Part 5, Chap-ter 8 – Shaft Vibration and Alignment (edition July 2006), Lloyd’s Register, London, 2006.

[5] …: “Guidance Notes on Propulsion Shafting Alignment”, American Bu-reau of Shipping, Houston, 2004.

[6] …: “Guidelines on Shafting Align-ment”, Nippon Kaiji Kyokai ,Tokyo, 2006.

[7] BATRAK, Y.: “Intellectual Maritime Technologies - ShaftMaster Software”, Web page: www.shaftsoftware.com

Figure 3 Example of calculation results Slika 3 Primjer dijagramskog prikaza rezultata proračuna

228 59(2008)3, 228-238

I. SENJANOVIĆ, T. SENJANOVIĆ, S. TOMAŠEVIĆ, S. RUDAN CONTRIBUTION OF TRANSVERSE BULKHEADS...UDC: 629.544:629.5.023.463

Ivo SENJANOVIĆ

Tanja SENJANOVIĆ

Stipe TOMAŠEVIĆ

Smiljko RUDAN

Contribution of Transverse Bulkheads to Hull Stiffness of Large Container Ships

Preliminary communication

Ultra large container ships are rather fl exible and exposed to signifi cant wave deformations. Therefore, the hydroelastic strength analysis is required for these types of ships. The coupling of a beam structural model and a 3D hydrodynamic model is preferable for reasons of simplicity. In this paper, the contribution of large number of transverse bulkheads to general hull stiffness is analysed. The prismatic pontoon with the cross-section of a large container vessel is considered for this purpose. The 3D FEM torsional analysis is performed with transverse bulkheads included and excluded. The correlation analysis of the obtained deformations indicated the infl uence of transverse bulkheads on the ship hull stiffness. The analysis is done by employing the torsional theory of thin-walled girders.

Keywords: container ship, fi nite element method, stiffness, thin-walled girder, torsion

Doprinos poprečnih pregrada krutosti trupa velikih kontejnerskih bro-dova

Prethodno priopćenje

Vrlo veliki kontejnerski brodovi prilično su elastični i stoga podložni velikim valnim defor-macijama. Zato se danas njihova čvrstoća istražuje metodama hidroelastičnosti. Pritom se radi jednostavnosti sprežu gredni strukturni model i 3D hidrodinamički model. U ovom članku istražen je doprinos velikoga broja poprečnih pregrada kontejnerskih brodova općoj krutosti trupa. Za te potrebe razmatran je prizmatični ponton s poprečnim presjekom velikoga kontejnerskog broda. Konstruiran je 3D model konačnih elemenata i provedena je analiza uvijanja s uključenim i isključenim poprečnim pregradama. Korelacijskom analizom deformacija za ova dva slučaja ustanovljen je utjecaj poprečnih pregrada na krutost brodskoga trupa. Pritom se koristila teorija uvijanja tankostjenih nosača.

Ključne riječi: kontejnerski brod, metoda konačnih ele menata, krutost, tankostjeni nosač, uvijanje

Authors’ address:

Faculty of Mechanical Engineering

and Naval Architecture, University of

Zagreb,

Zagreb, Croatia

E-mail: [email protected]

Received (Primljeno): 2007-04-04

Accepted (Prihvaćeno): 2007-04-27

Open for discussion (Otvoreno za

raspravu): 2009-09-30

1 Introduction

Nowadays sea transport is rapidly increasing and ultra large container ships are built [1]. Since they are rather fl exible, their hydroelastic response becomes an imperative subject of inves-tigation. In the early design stage, the coupling of a FEM beam structural model with a 3D hydrodynamic model based on the radiation-diffraction theory is reasonable [2], [3].

The 1D FEM structural model is quite sophisticated since it takes into account the bending and shear stiffness, as well as the torsional and warping stiffness [4], [5]. The general hull stiffness is increased due to the large number of transverse bulkheads in holds. There are two types of bulkheads, i.e. ordinary watertight bulkheads and grillage ones. The distance between them is de-termined by the container length.

Transverse bulkheads stretch within one web frame spacing and are quite stiff. They can be directly included in the 1D FEM model as a short beam element with a closed cross-section [6], [7].

However, due to the large number of transverse bulkheads and to the model discontinuity, it is more practical and reasonable to take into account their continuous contribution to the general hull stiffness.

Different attempts to take the infl uence of transverse bulk-heads into account have been made. One of the fi rst approaches was to increase the deck thickness based on the equivalence of the deformation energy of transverse bulkhead girders and the increased deck energy [8]. Today, the usual way is to model transverse bulkheads by axial elastic springs at their joints to the ship hull. The spring effect is condensed in lumped bimoments [9]. Furthermore, in the case of a large number of transverse bulkheads, the lumped bimoments might be distributed along the hull girder [10]. The distributed bimoments are manifested as additional torque load, which depends on the variation of the twist angle as pure torsional torque. Therefore, only the torsional stiffness of the ship hull is increased due to the bulk-head infl uence.

22959(2008)3, 228-238

CONTRIBUTION OF TRANSVERSE BULKHEADS... I. SENJANOVIĆ, T. SENJANOVIĆ, S. TOMAŠEVIĆ, S. RUDAN

The effect of transverse structure on the deformation of thin-walled girders is a challenging subject of contemporary investigations [11], [12], [13]. Recent literature shows that the problem is rather complex and the complicated solutions offered there reduce the applicative advantages of the combined beam theory and the thin-walled girder theory (1D + 2D) with respect to the direct 3D FEM analysis. In any case, the reliability of the 1D + 2D theory has to be checked by the correlation analysis with 3D FEM solutions.

In the light of the above circumstances, especially of the needs of the ship hydroelasticity analyses, where results of dry natural vibrations of the ship hull are required (modes, frequencies, mo-dal stiffness, modal mass), a simpler solution is preferable and more convenient. That was the motivation for the investigation of this challenging problem. Thus, the 3D FEM analysis of the prismatic hold structure with and without transverse bulkheads is performed. The equivalence of the maximum twist angle in the 1D and 3D models is used as a condition for determining the change of torsional beam stiffness. The reliability of approach is checked by the correlation for 1D and 3D warping functions and stress distribution.

2 Outline of the thin-walled girder theory

The thin-walled girder torsional theory is developed under assumptions that a considered structure is of membrane type (only in-plane deformation occurs) and that there is no distortion of the cross-section (twist angle is constant along the cross-section contour).

Figure 1 Beam torsionSlika 1 Uvijanje grede

A prismatic girder exposed to torsion is shown in Figure 1. The equilibrium of sectional torque, T, and the distributed external torsional load, μ

x, yields

(1)

According to the theory of thin-walled girders, the sectional torque consists of a pure torsional part and a warping contribu-tion [14]

(2)

where

E, G – Young’s modulus and shear modulusI

t, I

w – torsional and warping modulus

ψ – twist angle

Substitution of (2) into (1) leads to the ordinary differential equation of the fourth order

(3)

Its solution reads

(4)

where

(5)

and Ai are integration constants, while ψ

p represents a particular

solution which depends on μx.

Let us consider the twisting of the girder shown in Figure 1, which is loaded by torque M

t at the ends, while μ

x = 0. The warp-

ing of the girder ends is suspended. In this case the twist angle ψ is an anti-symmetric function and therefore A

0 = A

2 = 0. The

remaining constants A1 and A

3 are determined by satisfying the

boundary conditions

(6)

where u is the warping function (axial displacement) and u– is the relative sectional warping due to the unit beam deformation, dψ/

dx, defi ned according to the theory of thin-walled girders [15],

[16]. The fi nal expressions for the twist angle reads

(7)

Now, it is possible to determine sectional forces, i.e. pure torsional and warping torques (2)

(8)

and warping (sectorial) bimoment

(9)

Furthermore, the warping function (6) takes the form of

(10)

Torques Tt and T

w are the result of shear stresses τ

t and τ

w due

to pure torsion and suspended warping, respectively. The warping bimoment B

w represents the work of axial normal stress σ on the

displacement u– at a cross-section, i.e.

d dT xx= −μ .

T T T GIx

EIxt w t w= + = −d

d

d

d

3ψ ψ3 ,

EIx

GIxw t x

d

d

d

d

4

4

2

2

ψ ψ μ− = .

ψ β β ψ= + + + +A A x A x A x p0 1 2 3ch sh ,

β = GI

EIt

w

x l T M ux

ut= = = =: , ,d

d

ψ0

ψ ββ β

= −⋅

⎡

⎣⎢

⎤

⎦⎥

M l

GI

x

l

x

l lt

t

sh

ch.

T Mx

lT M

x

lt t w t= −⎛⎝⎜

⎞⎠⎟

=1ch

ch

ch

ch

ββ

ββ

,

B EIx

Mx

lw w t= = −d

d

sh

ch

2

2

ψ ββ β

.

uM

GI

x

lut

t

= −⎛⎝⎜

⎞⎠⎟

1ch

ch

ββ

.

230 59(2008)3, 228-238

I. SENJANOVIĆ, T. SENJANOVIĆ, S. TOMAŠEVIĆ, S. RUDAN CONTRIBUTION OF TRANSVERSE BULKHEADS...

(11)

where

(12)

Thus, by substituting (12) into (11) one fi nds the expression for warping modulus in (9)

(13)

3 Modelling of transverse bulkheads

The length of transverse bulkheads in large container ships is equal to one web frame spacing. They are of grillage type and therefore quite stiff. As a result, the bulkhead infl uence on the hull warping reduction is signifi cant.

In the torsional thin-walled girder theory, bulkheads can be modelled by axial elastic foundation at their joint to the hull struc-ture [9], [10]. In the case of a large number of bulkheads, the line foundation can be spread to the area foundation of the hull shell. The corresponding axial (tangential) surface load yields

(14)

where κ is the spread bulkhead stiffness and u is the warping function (6).

Axial load q causes an additional bimoment per unit length on the relative sectional warping u–

(15)

By substituting (14) into (15) one writes

(16)

where

(17)

is the sectional bulkhead stiffness.According to the theory presented in [9] and [10], the bulk-

head bimoment causes a distributed torque

(18)

where relation (16) is used for b. The torque μb is the transformed

bulkhead load and has to be equilibrated by sectional torques Tt

and Tw (2). Thus, by substituting (18) into (3), the differential

equation for girder torsion with bulkhead infl uence is obtained:

(19)

Generally, the value of k has to be calculated for a given bulkhead structure. It is a result of fl exural bulkhead stiffness.

4 Effect of transverse bulkheads

In order to take the infl uence of bulkheads into account, an-other approach can also be applied. A ship hull consists of a large number of open cross-section segments (holds) and of closed ones (bulkheads). For the open section, the torsional modulus I

t is quite

small, and therefore the warping modulus Iw plays the main role.

In a short bulkhead area, the torsional modulus of closed section It

0 is one order of magnitude higher than It, while Iw

0 is of the same order as I

w. For the reason of simplicity we can consider a

uniform girder with the equivalent torsional modulus It* , where

I I It t t< <* 0 , and the equivalent warping modulus Iw* equal to

Iw. In this case, the differential equation (3) takes the form of

(20)

where

(21)

GIb is the additional hull torsional stiffness due to bulkheads

as closed cross-section segments. Parameters k and GIb, in (19)

and (21) respectively, are equivalent quantities.Instead of bulkhead modelling by equivalent axial elastic

foundation, as it is usually done in literature, it is possible to determine the contribution of bulkheads by the 3D FEM analysis, as it is elaborated in Section 7. Let us assume, for the time being, that the end twist angles of a prismatic girder without and with transverse bulkheads are known, ψ(l) and ψ*(l) respectively. Referring to (7), one writes

(22)

(23)

where

(24)

(25)

Ratios of Eqs (23) and (22) lead to the transcendental equation for determining the unknown parameter y*

(26)

Now, the new value of torsional modulus can be determined by employing (24) and (25), i.e.

(27)

According to (21), the contribution of bulkheads to torsional stiffness is

(28)

B t u sw

s

= ∫ σ d ,

σ ψ= Ex

ud

d

2

2 .

I u t sw

s

= ∫ 2 d .

q u ux

= =κ κ ψd

d,

b qu s= ∫ ds

.

b kx

= d

d

ψ,

k u ss

= ∫κ 2 d

μ ψb

b

xk

x= =d

d

d

d

2

2 ,

EIx

GI kxw t

d

d

d

d

4

4

2

2 0ψ ψ− +( ) = .

EIx

GIxw t

d

d

d

d

4

4

2

2 0ψ ψ− =* ,

GI GI GIt t b* .= +

ψ ( )lM l

GI

y

yt

t

= −⎛⎝⎜

⎞⎠⎟

1th

ψ **

*

*( ) ,lM l

GI

y

yt

t

= −⎛⎝⎜

⎞⎠⎟

1th

y l lGI

EIt

w

= =β

y l lGI

EIt

w

* **

.= =β

11

112 2y

y

y y

y

y

l

l*

*

*

*( )

( )−

⎛⎝⎜

⎞⎠⎟

= −⎛⎝⎜

⎞⎠⎟

th th ψψ

..

I

I

y

yt

t

* *

.=⎛⎝⎜

⎞⎠⎟

2

I

I

y

yb

t

=⎛⎝⎜

⎞⎠⎟

−*

.2

1

23159(2008)3, 228-238


The twist angle (7) and the warping function (10) in non-dimensional form read respectively:

(29)

(30)

Referring to (8), the twisting and warping torques take the following form:

(31)

Furthermore, the twisting torque can be split into the hull part and the bulkhead contribution

(32)

where, proportionally to their torsional moduli (21),

(33)

Ratios I It t*

and I Ib t*

are defi ned by (27) and (28).Finally, the warping bimoment (9) takes the following non-

dimensional form:

(34)

The presented approach is based on the known ratio of end values of the twist angle for a girder without and with transverse bulkheads. Its reliability can be checked by the known ratio of warping functions in the middle of the girder. According to (10), it follows that

(35)

An additional way to check the obtained results is to compare the normal stress ratio at girder ends represented by the warping bimoments (9)

(36)

Actually, ratios (35) and (36) are related to the fi rst and second derivative of the twist angle, (6) and (9) respectively.

5 Ship particulars

A 7800 TEU container vessel of the following main particulars is considered, Figure 2.

GI

M l

x

l

x

y yt

t

**

*

* *ψ β= − sh

ch

GI

M

u

u

x

yt

t

* * *

* .= −1ch

ch

β

T

M

x

y

T

M

x

yt

t

w

t

* *

*

* *

*, .= − =1ch

ch

ch

ch

β β

T

M

T

M

T

Mt

t

h

t

b

t

* * *

,= +

T

M

I

I

T

M

T

M

I

I

T

Mh

t

t

t

t

t

b

t

b

t

t

t

*

*

* *

*

*

, .= =

B

M l

x

y yw

t

* *

* * .= − sh

ch

β

u

u

y

y

y

y

*

*

*( )

( ).

0

0

11

11

2

=⎛⎝⎜

⎞⎠⎟

−

−

ch

ch

B l

B l

y y

y yw

w

* *

*

( )

( ).= th

th

Figure 2 A 7800 TEU container vesselSlika 2 Kontejnerski brod nosivosti 7800 TEU

Figure 3 Midship cross-sectionSlika 3 Glavno rebro

232 59(2008)3, 228-238


Length overall Loa

= 334 mLength between perpendiculars L

pp = 319 m

Breadth B = 42.8 mDepth H = 24.6 mDraught T = 14.5 mDisplacement ∆ = 135530 t

Figure 4 Transverse bulkheadSlika 4 Poprečna pregrada

The midship cross-section is shown in Figure 3, while Figure 4 shows the transverse bulkhead. Properties of the open cross-section are determined by the STIFF program [17].

Cross-section area A = 6.394 m2

Horizontal shear area Ash

= 1.015 m2

Vertical shear area Asv = 1.314 m2

Vertical position of neutral line zNL

= 11.66 mVertical position of shear - - torsional centre z

D = –13.50 m

Horizontal moment of inertia Ibh

= 1899 m4

Vertical moment of inertia Ibv

= 676 m4

Torsional modulus It = 14.45 m4

Warping modulus Iw = 171400 m6

Position of deformation centre, zD, is rather low due to the

open cross-section. The relative warping of cross-section, u, is illustrated in Figure 5. Young’s modulus, shear modulus and Poisson’s ratio are: E = 2.06 · 108 kN/m2, G = 0.7923 · 108 kN/m2, ν= 0.3, respectively.

6 FEM models of a hull segment

Figure 6 Prismatic FEM model of a hull partSlika 6 Prizmatični model konačnih elemenata dijela trupa

The front holds of the ship as a prismatic thin-walled girder with the length of L = 2l = 174 m are considered. The FEM model is generated by the software [18]. It is constructed of four different types of superelements, and includes the total of 13 superelements, Figures 6 and 7. The shell fi nite elements are used. The model is clamped at the fore end and the only warp-ing is suspended at the aft end. The vertical distributed load is imposed at the aft cross-section, generating the total torque M

t = 40570 kNm, Figure 8, [19].There are two types of transverse bulkheads within the ship

hold space, i.e. the ordinary watertight bulkheads and bulkheads of grillage construction. Both types stretch within one web frame spacing. The bulkhead top ends with the stool. Such a bulkhead design makes them quite strong and therefore the general hull stiffness is increased.

Figure 5 Warping of cross-section, u–

Slika 5 Vitoperenje poprečnog presjeka, u–

23359(2008)3, 228-238


Figure 7 Superelement No. 3

Slika 7 Superelement br. 3

Figure 8 Load at the aft endSlika 8 Opterećenje kraja modela

Figure 9 Prismatic FEM model of a hull part with transverse bulkheads

Slika 9 Prizmatični model konačnih elemenata dijela trupa s poprečnim pregradama

Figure 10 Superelement No. 3 with a transverse bulkheadSlika 10 Superelement br. 3 s poprečnom pregradom

The FEM model of the ship segment with transverse bulk-heads and a typical superelement with the watertight bulkhead are shown in Figures 9 and 10 respectively. The boundary conditions and imposed load are the same as in the case of prismatic model without transverse bulkheads.

Figure 11 Deformation and stresses of the model without trans-verse bulkheads, σ

x [N/mm2]

Slika 11 Deformacije i naprezanja modela bez poprečnih pre-grada, σ

x [N/mm2]

Figure 12 Deformation and stresses of the model with transverse bulkheads, σ

x [N/mm2]

Slika 12 Deformacije i naprezanja modela s poprečnim pre-gradama, σ

x [N/mm2]

234 59(2008)3, 228-238


Deformed models, without and with transverse bulkheads, are shown in Figures 11 and 12 respectively. Distortion of the cross-section is negligible as a result of a double skin cross-sec-tion with very strong web frames. Due to the same reason, the bending stresses are negligible in comparison to the membrane stresses; therefore, the structure behaves as a membrane one. Dif-ferent colours in Figures 11 and 12 denote the levels of von Mises membrane stress. High stress concentration in the hatch coaming and the upper deck at the model ends confi rms the well-known fact caused by the suspended warping of the cross-section, [20].

7 Infl uence of transverse bulkheads

Since the 3D FEM model behaves as a membrane structure without distortion of the cross-section, the obtained results are comparable to those of the 1D analysis.

Figure 13 Rotation of the cross-sectionSlika 13 Zakret poprečnog presjeka

Rotation of the model free cross-section, determined analyti-cally by the beam theory (1D) and numerically by FEM (3D) for the pontoon with and without transverse bulkheads, is shown in Figure 13. The twist angle of the 3D analysis is somewhat higher than that of the 1D analysis.

(a)

Vertical position of the deformation centre in the 3D model is above that of the 1D model, points D

3D and D

1D in Figure 13,

respectively. The infl uence of transverse bulkheads on the position of the deformation centre is quite weak, point D

3D. The warping

of cross-section determined by the 3D analysis is rather close to that of the 1D analysis, Figure 5. Therefore, the warping correla-tion could be done only for one representative point of extreme displacement value. Let us chose the joint of the bilge and the inner bottom, Figure 5, where

(b)

Thus, the correlation of 3D and 1D analyses results is quite good concerning warping, while the 3D FEM model is more elastic than the 1D model from the twisting point of view. This could be caused by the shear infl uence on torsion which is not taken into account in the beam analysis, [10]. That fact might be the subject of further investigations. However, it does not have a signifi cant infl uence on the relative bulkhead contribution to the hull stiffness.

The twist angle ratio of the model with and without transverse bulkheads reads

(c)

The warping ratio in the bilge point is

(d)

The axial normal stress ratio in the hatch coaming, Figures 11 and 12, yields

(e)

In the considered numerical example, according to (24), y = 0.49541, while the solution of Eq. (26) gives y* = 0.7464. The variation of torsional stiffness, Eq. (27), is I It t

* .= 2 27 . It means that the bulkhead contribution is I Ib t = 1 27. .

Figure 14 Twist angleSlika 14 Kut uvijanja

ψψ

3

1

3

3

0 27690 10

0 22969 101 2055D

D

= ⋅⋅

=−

−

.

.. .

u

uD

D

3

1

1 02594

1 051920 9753= =.

.. .

mm

mm

ψψ

3

3

3

3

0 24876 10

0 27690 100 89837D

D

* .

.. .= ⋅

⋅=

−

−

u

uD

D

3

3

0 90013

1 025940 87737

* .

.. .= =mm

mm

σσ

3

3

5 93623

6 358050 93365D

D

* . /

. /. .= =N mm

N mm

2

2

23559(2008)3, 228-238


Figure 15 Deck warpingSlika 15 Vitoperenje palube

Figure 16 Twisting and warping torquesSlika 16 Torzijski momenti uvijanja i vitoperenja

Figure 17 Warping bimomentSlika 17 Bimoment vitoperenja

The girder displacements and sectional forces are determined for both cases, i.e. without and with transverse bulkheads, and are shown in Figures 14, 15, 16 and 17 in non-dimensional form. The corresponding formulae from Sections 2 and 4 are used. The twist angle ψ is reduced according to given values of the 3D FEM analysis, Figure 14. The warping of the cross-sec-tion u is also reduced, Figure 15. Its variation defi ned by the 1D analysis, Eq. (35) is

u

u

* ( )

( ). .

0

00 89388=

B l

B lw

w

* ( )

( ). .= 0 91633

ω π πn

t

t

w

t

n

L

GI

J

n

L

EI

GIn= + ⎛

⎝⎜⎞⎠⎟ =0

2

1 0 1 2, , , ...

ωω

π

π

n

n

y

n

y

n

*

*

.=+

⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

12

12

2

2

(f)

Discrepancy between the 1D analysis and the 3D FEM analy-sis, value (d), is only 1.9%.

The warping bimoment shown in Figure 17 is also reduced due to bulkheads. The 1D ratio, Eq. (36), yields

(g)

By comparing it to the 3D FEM stress ratio (e), a discrepancy of -1.9% is obtained. This fact confi rms quite good simulation of the bulkhead effect in the thin-walled girder theory.

The torques distributions are shown in Figure 16. The hull twisting torque in the case with bulkheads, T

h, is very close to

the pure twisting torque without bulkheads, Tt. The warping

torque Tw is now reduced in comparison to T

w due to the bulkhead

contribution, Tb.

The infl uence of the increased value of torsional stiffness on vibrations can, for instance, be analysed in the case of uncoupled natural vibration of a free thin-walled girder with suspended boundary cross-section warping. The corresponding formula for natural frequencies derived in Appendix reads, (A16)

(37)

The following relation between natural frequencies of a hull segment with and without transverse bulkheads exists:

(38)

For the fi rst natural frequencies of elastic modes one fi nds ω ω1 1

* = 1.05594. Thus, a 127% torsional stiffness increase due to bulkheads results in a 5.6% increase in the fi rst frequency [M3] in the considered case. It is evident from (38) that the variation of higher mode natural frequencies is decreased.

8 Bending stiffness analysis

8.1 Horizontal bending

Horizontal bending is analysed by the FEM model adapted for this purpose. The model aft end is entirely free and loaded by distributed loads, as shown in Figure 18. The vertical load generates a torque of M

t = 40570 kNm, while the total horizontal

force Fy, acting about the deformation centre, equilibrates it. In

this way, the girder is only exposed to horizontal bending.In the cases of the model without and with transverse

bulkheads, the horizontal force of pure bending takes values of F

y = 1500 kN and F

y* = 1565 kN, respectively. The cor-

responding maximum defl ections yield δy = 19.8654 mm and

δy* = 20.4069 mm, Figures 19 and 20. Thus, the moment of inertia

of the cross-section of the reinforced model can be expressed by that of the model without bulkheads:

236 59(2008)3, 228-238


Figure 18 Load at the model free aft end in the case of horizontal bending

Slika 18 Opterećenje horizontalnog savijanja na slobodnom stražnjem kraju modela

Figure 19 Horizontal bending of the model without transverse bulkheads, σ

vM [N/mm2]

Slika 19 Horizontalno savijanje modela bez poprečnih pregrada, σ

vM [N/mm2]

Figure 20 Horizontal bending of the model with transverse bulk-heads, σ

vM [N/mm2]

Slika 20 Horizontalno savijanje modela s poprečnim pregrad-ama, σ

vM [N/mm2]

(h)

The correction is rather small, approximately 1.56% and its infl uence on vibration is almost negligible. Stress concentration in the bilge area at the fi xed model end due to bending is evident, Figures 19 and 20.

Since pure torque Mt and horizontal forces F

y and F

y* for the

model without and with transverse bulkheads are known, it is possible to determine the vertical position of the deformation centre:

(i)

where h is the double bottom height. The obtained results are compared with those of pure torsion determined in Section 7, in Table 1. The 3D FEM analyses show that the torsional centre and the shear centre are not the same points. In the thin-walled girder theory these two centres are not distinguished, and the unique deformation centre is determined. Probably, the suspended warping in the 3D FEM torsional analysis has some infl uence on the vertical position of the torsional centre. We can see that the transverse bulkheads also infl uence the position of the torsional and shear centres.

Table 1 Vertical position of the deformation centre, zD [m]

Tablica 1 Vertikalni položaj središta deformacije, zD [m]

Without bulkheads With bulkheads

Torsional centre, 3D FEM twisting -10.60 -10.25

Shear centre, 3D FEM bending -12.52 -11.96

Deformation centre, 2D strip theory -13.50

8.2 Vertical bending

A similar FEM analysis is performed for the investigation of vertical bending stiffness. The total vertical force, imposed at the free model end, is F

v = 2000 kN. The corresponding defl ection at

the same place, in the case of the model without and with trans-verse bulkheads, yields δ

z = -28.3706 mm and δ

z* = -28.1697 mm,

respectively. Thus, for the corrected vertical moment of inertia of the cross-section one fi nds:

(j)

It is obvious that the infl uence of transverse bulkheads on the vertical stiffness is even lower than on the horizontal stiffness.

9 Conclusion

Hydroelastic analysis of large container vessels becomes an actual problem. For the reason of simplicity, a beam model of hull girder is coupled with a 3D hydrodynamic model. Instead of calculating the transverse bulkhead stiffness, the contribution of bulkheads to the global stiffness of the ship hull is determined by the 3D FEM analysis of a prismatic ship-like pontoon. This

IF

FI Iz

y y

y yz z

** *

/

/. .= =

δδ

1 01565

zh M

Fz

h M

FDt

yD

t

y

= − = −2 2

, ,**

I I Iyz

zy y

** . .= =

δδ

1 00713

23759(2008)3, 228-238


is a simple and reliable engineering approach. It was found that the increase in torsional stiffness is considerable in the illustrated numerical example. The infl uence of this fact on the resonant ship hull response to wave excitation is signifi cant and therefore has to be taken into account. On the other hand, the infl uence of the transverse bulkheads on vertical and horizontal bending stiffness is rather small and may be neglected.

However, some discrepancies between the thin-walled girder theory and the 3D FEM still exist. In the analysed numerical example of a ship hull segment, the twist angle determined by the beam analysis is signifi cantly smaller than that obtained by the 3D FEM analysis. Even the twist angle of the beam without bulkheads is still lower than that of the 3D FEM model reinforced by bulkheads. Also, there are some discrepancies of the shear centre position between the 1D and 3D models without bulk-heads. On the other hand, agreement between the cross-section warping is excellent. This problem will be the subject of further investigation.

Most of present papers dealing with problems of thin-walled structures are concentrated on the investigation within the thin-walled girder theory. The validation of results should be based on the correlation analysis with 3D FEM models which simu-late the structure behaviour in a more realistic way. Also, some model tests and full scale measurements are very valuable for this purpose.

Acknowledgement

The authors would like to express their gratitude to Prof. Radoslav Pavazza from the Faculty of Electroengineering, Mechanical Engineering and Naval Architecture, University of Split, for his useful consulting during the investigation of this challenging problem.

References

[1] Proceedings of International Conference on Design & Operation of Container Ships, RINA, London, 2006.

[2] MALENICA, Š., SENJANOVIĆ, I., TOMAŠEVIĆ, S., STUMPF, E.: “Some aspects of hydroelastic issues in the design of ultra large container ships”, The 22nd International Workshop on Water Waves and Floating Bodies, IWWWFB, Plitvice Lakes, Croatia, 2007.

[3] TOMAŠEVIĆ, S.: “Hydroelastic model of dynamic response of container ships in waves”, Ph. D. Thesis, FSB, Zagreb, 2007. (in Croatian).

[4] SENJANOVIĆ, I.: “Ship Vibrations”, 2nd Part, University of Zagreb, 1990.

[5] SENJANOVIĆ, I., GRUBIŠIĆ, R.: “Coupled horizontal and torsional vibration of a ship hull with large hatch openings”, Computers & Structures, 41(1991)2, p. 213-226.

[6] HASLUM, K., TONNESSEN, A.: “An analysis of torsion in ship hulls”, European Shipbuilding, No. 5/6 (1972), p.67-89.

[7] PEDERSEN, P. T.: “Torsional response of containerships”, Journal of Ship Research, 29(1985), p.194-205.

[8] SENJANOVIĆ, I.: “A solution to the torsion problem on container ships”, 1st and 2nd Parts, Brodogradnja, No. 1 and 2, 1972, (in Croatian).

[9] PAVAZZA, R., PLAZIBAT, B., MATOKOVIĆ, A.: “Idealisation of ships with large hatch openings by a thin-walled rod of open section on many elastic supports”, Thin-Walled Structures 32 (1998), p. 305-325.

[10] PAVAZZA, R.: “Bending and torsion of thin-walled beams of open section on elastic foundation”, Ph.D. Thesis, University of Zagreb, 1991.

[11] BOSWELL, L. F., LI, Q.: “Consideration of relationship between torsion, distortion and warping of thin-walled beams”, International Journal of Solids and Structures 21(1995), p. 147-161.

[12] KIM, J. H., KIM, Y. Y.: “Thin-walled multi-cell beam analysis for coupled torsion, distortion and warping deformation”, ASME Journal of Applied Mechanics 68(2001), p. 260-269.

[13] RENDEK, S., BLAŽ, I.: “Distortion of thin-walled beams”, Thin-Walled Structures 42(2004), p. 255-277.

[14] SENJANOVIĆ, I., FAN, Y.: “A higher-order torsional beam theory”, Engineering Modelling, 10(1997)1-4, p. 25-40.

[15] SENJANOVIĆ, I., FAN, Y.: “A higher-order theory of thin-walled girders with application to ship structures”, Computers & Structures, 43(1992)1, p. 31-52,

[16] SENJANOVIĆ, I., FAN, Y.: “A fi nite element formulation of ship cross-sectional stiffness parameters”, Brodogradnja, 41(1993)1, p. 27-36.

[17] ...: “STIFF, User’s Manual”, FSB, Zagreb, 1990.[18] ...: “SESAM, User’s Manual”, Det Norske Veritas, Høvik, 2003.[19] SENJANOVIĆ, T.: “Utjecaj skladišne konstrukcije na krutost trupa

kontejnerskih brodova”, Diplomski rad, FSB, Zagreb, 2007.[20] SENJANOVIĆ, I., FAN, Y.: “Pontoon torsional strength analysis

related to ships with large deck openings”, Journal of Ship Research, 35(1991)4, p. 339-351.

[21] SENJANOVIĆ, I., ĆATIPOVIĆ, I., TOMAŠEVIĆ, S.: “Coupled fl exu-ral and torsional vibrations of ship-like structures”, (in preparation).

Appendix

Torsional beam vibrations

The differential equation of uncoupled torsional beam vibra-tions can be written as an extension of the static equation (3), [4], [21]

(A1)

where the twist angle ψ and the distributed torque μx are time

dependent quantities. The symbol Jt0 denotes the polar mass

moment of inertia. Natural vibrations are harmonic and Eq. (A1) is reduced to the homogeneous form

(A2)

where ψ and ω are the natural mode and the natural frequency, respectively.

Solution of (A2) is assumed in exponential form

(A3)

By substituting (A3) into (A2) one fi nds the following bi-quadratic characteristic equation:

(A4)

Its four roots yield

(A5)

where

(A6)

EIx

GIx

Jt

tw t t x

∂∂

− ∂∂

+ ∂∂

=4

4

2

20

2

2

ψ ψ ψ μ ( ),

EIx

GIx

Jw t t

d

d

d

d

4

4

2

22 0 0

ψ ψ ω ψ− − = ,

ψ α= e x .

α α ω4 22 0

0− − =GI

EI

J

EIt

w

t

w

.

α γ ηj i= ± ±, ,

γ ω= + +GI

EI

J EI

GIt

w

t w

t21 4 12

0

2( )

238 59(2008)3, 228-238


(A7)

Thus, the solution of (A2) takes the following form:

(A8)

Let us consider vibrations of a free beam with suspended warping at its ends. The corresponding boundary conditions read

(A9)

that leads to

(A10)

In the case of symmetric modes, A1 = A

3 = 0, while for anti-

symmetric modes A2 = A

4 = 0. The corresponding eigenvalue

problems yield

(A11)

(A12)

For a nontrivial solution, determinants of (A11) and (A12) have to be zero. That leads to the frequency equations

(A13)

(A14)

with the same eigenvalue formula for the symmetric (n = 0, 2…) and anti-symmetric (n = 1, 3…) modes

(A15)

Taking (A15) into account, one fi nds the following expression for natural frequencies of torsional vibrations from (A7)

(A16)

Integration constants A2 and A

4, and A

1 and A

3 are determined

from (A11) and (A12), respectively. Symmetric and anti-sym-metric natural modes according to (A8) yield

(A17)

(A18)

In case n = 0, the natural frequency ω0 =0, Eq. (A16), and

the natural mode ψ0 = 1, Eq. (A17), that is related to the rigid

body rotation.

η ω= + −GI

EI

J EI

GIt

w

t w

t21 4 12

0

2( ).

ψ γ γ η η= + + +A x A x A x A x1 2 3 4sh ch sin cos .

x l T u= ± = =: ,0 0

x lx x

= ± = =: , .d

d

d

d

3ψ ψ0 03

γ γ η ηγ γ η η

sh

sh

l l

l l

A

A

−⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎩

⎫⎬⎭

=sin

sin3 32

4

0{{ }

γ γ η ηγ γ η η

ch

ch

l l

l l

A

A

cos

cos3 31

3

0−

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎩

⎫⎬⎭

= {{ }.

γ η γ η γ η( ) sin2 2 0+ =sh l l

γ η γ η γ η( ) cos2 2 0+ =ch l l

η πl

nn= =

20 1 2, , , ...

ω π πn

t

t

w

t

n

l

GI

J

n

l

EI

GIn= + ⎛

⎝⎜⎞⎠⎟ =

21

20 1 20

2

, , , ...

ψ η η γ γ γ ηn n n n n n nl x l x n= ⋅ + ⋅ =sin ch sh cos , , ...0 2

ψ η η γ γ γ ηn n n n n n nl x l x n= ⋅ − ⋅ =cos sh ch sin , , ...1 3

1 2 modelling of propulsion shaft 3 line and basic

Documents