DNV RULES FOR RUDDER

CLASSIFICATION NOTESNo. 32.1

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STRENGTH ANALYSIS OFRUDDER ARRANGEMENTS

JANUARY 1984

FOREWORD

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Re-issue November 2002This is a re-issue of the version published in January 1984. No modifications have been made other than editorial ones, i. e.changes according to the current layout standard.

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Classification Notes - No. 32.1 3

January 1984

CONTENTS

1. GENERAL................................................................. 51.1 Introduction..................................................................51.2 Definitions ...................................................................52. CALCULATION FORMULAE.............................. 52.1 Rudder with neck bearing and heel pintle ...................5

2.2 Semi-spade rudder....................................................... 72.3 Spade rudder................................................................ 93. COMPLETE STRENGTH ANALYSIS................. 93.1 Two-dimensional grillage analysis.............................. 93.2 Two-dimensional framework analysis ...................... 10

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4 Classification Notes - No. 32.1

January 1984

DET NORSKE VERITAS


January 1984

1. General

1.1 Introduction

1.1.1 For scantlings of rudder, rudder stock and supportingstructures, the Rules for Classification of Steel Ships (hence-forth referred to as the Rules) specify requirements based oncalculated shear forces, bending moments and torsional mo-ments. Alternatively, a complete structural analysis may becarried out to demonstrate that the stresses are in compliancewith allowable stresses.

1.1.2 The force and moment distribution formulae given inSection 2 for various rudder types will normally be accepted asan improvement of the simplified formulae given in the Rules.

Note:Computer program on rudder calculations are offered by VER-ITAS and are available for external users as desktop computerversion (PILOT) and at the head office computer centre and con-nected terminals.

On request VERITAS may carry out and use the calculations asbasis for approval.

---e-n-d---of---N-o-t-e---

1.1.3 Strength analysis carried out in accordance with Sec-tion 3 will normally be accepted as basis for class approval.

1.2 Definitions

1.2.1 The following SI-units (International System of Units)are used in this Note:

— Mass unit: tonne (t) (= megagram (Mg))— Length unit: millimetre (mm) (metre (m) and centimetre

(cm) are used in some quantities and stated in each case)— Time unit: second (s)— Force unit: kilonewton (kN) (newton (N) is used in some

quantities and stated in each case)

1.2.2 Symbols not mentioned in the following list are givenin connection with relevant formulae. The general symbolsmay be repeated when additional definition is found necessaryin connection with specific formulae.

Fr = design rudder force in kN. See the Rules Pt.3 Ch.3Sec.2 D.

A = total rudder area in m2.Ab1 = rudder area in m2 above pintle bearing.Ab2 = rudder area in m2 below pintle bearing.Ia = mean moment of inertia in cm4 of rudder stock.Ib = mean moment of inertia in cm4 of rudder (above pintle

bearing, if any).Id = mean moment of inertia in cm4 of sole piece.Ih = mean moment of inertia in cm4 of rudder horn.Jh = mean polar moment of inertia in cm4 of rudder horn.Ah = side view area in m2 of rudder horn.E = modulus of elasticity in N/mm2.ωb = load on rudder blade in kN/m height of rudder.ωh = load on rudder horn in kN/m height of horn.

2. Calculation Formulae

2.1 Rudder with neck bearing and heel pintle

2.1.1 The design, mathematical model and resulting bendingmoment and shear force distribution are shown in Fig. 2-1.

2.1.2 The rudder is considered simply supported at the bear-ings. The sole piece is considered to be completely fixed at adistance forward of the after end of the propeller post depend-ing on the fineness of the ship’s lines, in the stern area. Theelasticity of sole piece, rudder and rudder stock is taken intoaccount.

2.1.3 The bending moment at neck bearing may be taken as:

ωb ≅

α =

kd =

hd, hr, hn, la, lb and ld = lengths in m as shown in Fig. 2-1.ldc = ld + c (m)

c = addition estimated on the basis of afterbody lines. Maybe taken as for ships with full lines (L = Rulelength of ship). For ships with narrow structure for-ward of propeller post, c is to be specially considered.

If the rudder is designed with the stern pintle instead of neckbearing (see Fig. 2-2), and the location hn < 0,1 hr, the aboveformula may be used if the term – hn

2 within brackets in the nu-merator is substituted by + hn

2.

For drawing of bending moment diagram:

2.1.4 The reaction forces at various bearings may be taken as:

— At rudder carrier:

— At neck bearing:

— At heel bearing:

Mn

ωb l( b2

hn2 ) 1

2--- α

8---+� �

� �–

1 α 1la Ib

lb Ia------------+� �

� �+

----------------------------------------------------- (kN m)=

Fr

hr----- (kN/m)

kd lb3

3 E Ib-------------- or

lb3Id

ldc3Ib

--------------

3 E Id

ldc3

-------------- is the sole piece spring constant.

0 04 L,

Mf

ωb hr2

8---------------- (kN m)=

Pu

Mn

la-------=

n Pu 1la

lb----+

� ��

Fr

lb-----

hr

2---- hd+� �� (kN)+=

Pc Fr Pu Pn–+( ) (kN)=

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January 1984

Figure 2-1Rudder with neck bearing and heel pintle

Figure 2-2Rudder with stern pintle and heel pintle

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January 1984

2.2 Semi-spade rudder

2.2.1 The design, mathematical model and resulting bendingmoment and shear force distribution are shown in Fig. 2-3 and2.4.

2.2.2 The rudder is considered simply supported at the bear-ings. The rudder horn is considered to be completely fixed atthe counter, and the elasticity of rudder, rudder stock and rud-der horn is taken into account.

2.2.3 The bending moment at various positions in the rudderand stock may be taken as:

— At pintle bearing:


ωb1 ≅

ωb2 ≅

α =

k =

hn, la, lb, lc, lh and e = lengths in m as shown in Fig. 2-3.

Jh =

Ch = horizontal section area in cm2 enclosed by therudder horn.

= circumferential length of rudder horn in hori-zontal section divided by its mean wall thick-ness.




— At pintle bearing:

Pp = (Fr + Pu - Pn) (kN)

2.2.5 With reference to Fig. 2-4 the following design valuesmay be applied for the scantlings of rudder horn:

— Maximum bending moment:

Mh = Pp lh + 0,5 ωh lh2 (kN m)

— Maximum shear force:

Ph = Pp + ωh lh (kN)

— Torsional moment:

MhT = Pp e (kN m)

lh and e = length in m as shown in Fig. 2-4.

Mp

ωb2 lb22

2--------------------- (kN m)=

Mn

ωb1 lb12

hn2

–� �� 1

2--- α

8---+

� �� ωb2lb2

2 12---

lb1lb2-------- α

4---–+

� ��

+

1 α 143---

lalb1----------

IbIa-----+

� ��

+

----------------------------------------------------------------------------------------------------------------------------- (kN m)=

Fr Ab1

A lb1 hn–( )---------------------------- (kN/m)

FrAb2

Alb2--------------- (kN/m)

k lb13

3EIb--------------

E

lh2 6e

2,Jh

--------------lh

2

3Ih-------+

� ��

----------------------------------------

4Ch2

s/t�-------------

s/t�

Pu

Mnla

------- (kN)=

Pn Pu 1la

lb1-------+

� �� ωb1

2lb1---------- lb1 hn–( )2 Mp

lb1------- (kN)–+=

ωh

Fr A Ah+( )Ah

A2lh

----------------------------------- (kN/m)≡

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January 1984

Figure 2-3Semi-spade rudder

Figure 2-4Rudder horn

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January 1984

2.3 Spade rudder

2.3.1 The design and resulting bending moment and shearforce distribution are shown in Fig. 2-5.

2.3.2 The rudder is considered simply supported at the bear-ings.

2.3.3 The bending moment at the neck bearing may be takenas:

hn, la, lb, lu, ll = lengths in m as shown in Fig. 2-5.




Pn = (Fr + Pu) (kN)

Figure 2-5Spade rudder

3. Complete Strength Analysis

3.1 Two-dimensional grillage analysis

3.1.1 The rudder design may be regarded as a 2-dimensionalstructure with in-plane loading. The structural strength with re-spect to shear, bending and torsion is most conveniently ana-lysed by means of 2-dimensional grillage program.

3.1.2 If 2-dimensional grillage program is to be used, the fol-lowing features must be available :

— Linearly distributed loads on elements.— Arbitrary orientation of elements.— Arbitrary position of supports.— Hinged elements.— Solid circular elements or equivalent representation.— Built close section elements (box torsion) or equivalent

representation.

3.1.3 Fig. 3-1 shows a typical rudder design with heel supportand the corresponding structural model.

Mn

Fr

3 lu ll+( )---------------------- lb 2ll lu+( ) hn 2lu ll+( )+[ ] (kN m)=

Pu

Mn

la------- (kN)=

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January 1984

Figure 3-1Balance rudder and grillage model

3.2 Two-dimensional framework analysis

3.2.1 The calculations of shear forces, bending moments andtorsional moments (in rudder horn), and the correspondingstresses, may alternatively be carried out by means of 2-dimen-sional framework program.

In that case special modelling techniques will have to be ap-plied.

3.2.2 Fig. 3-2 shows a typical semi-spade rudder design andcorresponding structural model.

The rudder horn shear and bending stiffness is represented bya vertical element hinged to the “pintle arm” at the lower end.

The torsional effect due to the distance “e” between the centro-ids of pintle and horn may be represented by the horizontal fic-titious element, the “pintle arm”, hinged to the rudder blade. Ifthe length of this element is taken as “e” the cross-sectionalarea is given by:

Jh = as given in 2.2.3 in cm4

lh and e = lengths in m as given in Fig. 3-2.

3.2.3 If the feature “hinged element” is not available theabove model may still be used provided the bending stiffnessof the “pintle arm” is negligible.

Figure 3-2Semi-spade rudder and framework model

Apa

Jh

26000 lh e------------------------ (cm

2 )=

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