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DNV GL AS

RULES FOR CLASSIFICATION

ShipsEdition October 2015

Part 3 Hull

Chapter 14 Rudders and steering

FOREWORD

DNV GL rules for classification contain procedural and technical requirements related to obtainingand retaining a class certificate. The rules represent all requirements adopted by the Society asbasis for classification.

© DNV GL AS October 2015

Any comments may be sent by e-mail to [email protected]

If any person suffers loss or damage which is proved to have been caused by any negligent act or omission of DNV GL, then DNV GL shallpay compensation to such person for his proved direct loss or damage. However, the compensation shall not exceed an amount equal to tentimes the fee charged for the service in question, provided that the maximum compensation shall never exceed USD 2 million.

In this provision "DNV GL" shall mean DNV GL AS, its direct and indirect owners as well as all its affiliates, subsidiaries, directors, officers,employees, agents and any other acting on behalf of DNV GL.

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Rules for classification: Ships — DNVGL-RU-SHIP-Pt3Ch14. Edition October 2015 Rudders and steering

DNV GL AS

CHANGES – CURRENT

This is a new document.

The rules enter into force 1 January 2016.

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CONTENTS

Changes – current...................................................................................................... 3

Section 1 Rudders, sole pieces and rudder horns.......................................................61 General................................................................................................... 6

1.1 Basic assumptions................................................................................61.2 Definitions...........................................................................................61.3 Documentation requirements................................................................. 71.4 Certification requirements..................................................................... 91.5 Design considerations........................................................................... 91.6 Materials........................................................................................... 101.7 Equivalence....................................................................................... 11

2 Rudder force and rudder torque........................................................... 112.1 Rudder blades without cut-outs............................................................112.2 Rudder blades with cut-outs (semi-spade rudders)................................. 14

3 Rudder strength....................................................................................153.1 Strength calculations......................................................................... 15

4 Rudder stock and rudder shaft scantlings............................................ 164.1 Rudder stock scantlings...................................................................... 164.2 Rudder shaft scantlings.......................................................................17

5 Rudder blade........................................................................................ 185.1 Permissible stresses............................................................................185.2 Rudder plating................................................................................... 195.3 Connections of rudder blade structure with solid parts.............................205.4 Single plate rudders........................................................................... 22

6 Rudder stock and shaft couplings.........................................................236.1 Connection to steering gear.................................................................236.2 Horizontal flange couplings.................................................................. 236.3 Vertical flange couplings......................................................................256.4 Cone couplings with key..................................................................... 266.5 Cone couplings with special arrangements for mounting and dismounting

the couplings....................................................................................... 286.6 Rudder shaft couplings........................................................................31

7 Pintles...................................................................................................317.1 Scantlings..........................................................................................317.2 Couplings.......................................................................................... 317.3 Dimensions of threads and nuts...........................................................31

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7.4 Pintle housing.................................................................................... 328 Rudder stock bearing, rudder shaft bearing and pintle bearing.............32

8.1 Liners and bushes.............................................................................. 328.2 Minimum bearing surface.................................................................... 328.3 Bearing dimensions............................................................................ 338.4 Bearing clearances............................................................................. 34

9 Strength of sole pieces and of rudder horns.........................................349.1 Sole piece......................................................................................... 349.2 Rudder horn...................................................................................... 359.3 Rudder trunk..................................................................................... 37

Appendix A Guidelines for calculation of bending moment and shear forcedistribution........................................................................................................... 39

1 General................................................................................................. 392 Spade rudder........................................................................................ 39

2.1 Data for the analysis.......................................................................... 392.2 Moments and forces........................................................................... 39

3 Spade rudder with trunk...................................................................... 403.1 Data for the analysis.......................................................................... 403.2 Moments and forces........................................................................... 40

4 Rudder supported by sole piece........................................................... 404.1 Data for the analysis.......................................................................... 404.2 Moments and forces........................................................................... 41

5 Semi spade rudder with one elastic support.........................................415.1 Data for the analysis.......................................................................... 415.2 Moments and forces........................................................................... 425.3 Rudder horn...................................................................................... 42

6 Semi spade rudder with 2-conjugate elastic support............................ 436.1 Data for the analysis.......................................................................... 436.2 Moments and forces........................................................................... 446.3 Rudder horn bending moment............................................................. 446.4 Rudder horn shear force..................................................................... 456.5 Rudder horn shear stress calculation.................................................... 456.6 Rudder horn bending stress calculation................................................. 46

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SECTION 1 RUDDERS, SOLE PIECES AND RUDDER HORNS

1 General

1.1 Basic assumptions

1.1.1 Vessels shall be provided with means for steering (directional control) of adequate strength andsuitable design. The means for steering shall be capable of steering the ship at maximum ahead servicespeed, which shall be demonstrated.

1.1.2 Steering may be achieved by means of rudders, foils, flaps, steerable propellers or jets, yaw controlports or side thrusters, differential propulsive thrust, variable geometry of the vessel or its lift systemcomponents, or by any combination of these devices.

1.1.3 Requirements in this section are related to rudder and rudder design. For requirement to steering gearoperating the rudder, reference is made to Pt.4 Ch.10 Sec.1.If steering is achieved by means of waterjet or thrusters reference is made to Pt.4 Ch.5 Sec.2 and Pt.4 Ch.5Sec.3 respectively. Other means of steering is subject to special consideration.

1.1.4 All scantlings requirements given in this chapter are based on gross scantlings, hence the grossscantlings are to be equal or greater than these required gross scantlings.

1.1.5 This chapter applies to ordinary profile rudders, and to some enhanced profile rudders with specialarrangements for increasing the rudder force. Rudders not conforming to the profile types included in thischapter will be subject to special consideration.

1.1.6 This chapter applies to rudders made of steel. Rudders made of material different from steel will besubject to special consideration.

1.2 Definitions

1.2.1 Maximum ahead service speed and maximum astern speed shall be specified.

1.2.2 Maximum ahead service speed is the maximum service speed V. See Ch.1 Sec.4 [3.1.8].

1.2.3 Maximum astern speed is the speed which it is estimated the ship can attain at the designed maximumastern power at the deepest seagoing draught.

1.2.4 Some terms used for rudder, rudder stock and supporting structure are shown in Figure 1.

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Figure 1 Rudders

1.3 Documentation requirements

1.3.1 Documentation shall be submitted as required by Table 1.

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Table 1 Documentation requirements

Object Documentation type Additional description Info

Z030 – Arrangementplan

Covering rudders, propeller outlines, actuators,stocks, horns, stoppers and bearing lubricationsystem.Including specification of maximum speed ahead andaft, and Ice Class notation when applicable.

FI

Z250 – Procedure Mounting and dismounting or rudder (including flapsas a detached component), rudder stock and pintles. FI

Z250 – Procedure Measurement of bearing clearances. FI

Z163 – Maintenancemanual

Flap rudders: Hinges, link systems and criteria forallowable bearing clearances. FI

Z110 – Data sheetNon-conventional rudder designs: Torquecharacteristics (torque versus rudder angle inhomogeneous water stream).

FI

Rudder arrangement

Z265 – Calculationreport1)

Expected life time of bearings subjected toextraordinary wear rate due to dynamic positioning. AP

Sole pieces and rudderhorns

H050 – Structuraldrawing AP

Rudder blades H050 – Structuraldrawing Including details of bearings, shafts and pintles. AP

Rudder stocks H030 – Detaileddrawing Including details of connections, bolts and keys. AP

Rudder and steering gearsupporting structures

H050 – Structuraldrawing

Including fastening arrangements (bolts, chockingand side stoppers). AP

1) Only for rudders included under DP-Control documentation, see Pt.6 Ch.3.

AP = For approval; FI = For information; ACO = As carried out; L = Local handling; R = On request; TA = Covered bytype approval; VS = Vessel specific

1.3.2 For general requirements to documentation, see Pt.1 Ch.3 Sec.2.

1.3.3 For a full definition of the documentation types, see Pt.1 Ch.3 Sec.3.

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1.4 Certification requirements

1.4.1 Components shall be certified as required by Table 2.

Table 2 Certification requirements

Object Certificate type Issued by Certificationstandard* Additional description

Structural parts

Shaft

Pintles

Stock

Carrier

MC Society

Bolts for flanged couplings

Stoppers

Rudder

MC ManufacturerBolts, except for flanged couplings

* Unless otherwise specified the certification standard is the Society's rules.

1.4.2 For a definition of the certificate types, see Pt.1 Ch.3 Sec.5.

1.5 Design considerations

1.5.1 Effective means are to be provided for supporting the weight of the rudder without excessive bearingpressure, e.g. by a rudder carrier attached to the upper part of the rudder stock. The hull structure in way ofthe rudder carrier is to be suitably strengthened.

1.5.2 All rudder bearings shall be accessible for measuring of wear without lifting or dismantling the rudder.Guidance note:In case cover plates are permanently welded to the side plating, it is recommended to arrange peep holes for inspection of securingof nuts and pintles.

---e-n-d---of---g-u-i-d-a-n-c-e---n-o-t-e---

1.5.3 Suitable arrangements are to be provided to prevent the rudder from lifting. The arrangement shalleffectively limit vertical movement of rudder in case of extreme (accidental) vertical load on rudder.

1.5.4 Means for draining the rudder completely after pressure testing or possible leakages shall be provided.Drain plugs shall be fitted with efficient packing.

1.5.5 In rudder trunks which are open to the sea, a seal or stuffing box is to be fitted above the deepestload waterline, to prevent water from entering the steering gear compartment and the lubricant from beingwashed away from the rudder carrier. If the top of the rudder trunk is below the deepest waterline, twoseparate stuffing boxes are to be provided.

Guidance note:One stuffing box with two separate lip seal rings in an acceptable alternative solution.


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1.5.6 Vibrations in the rudder structural elements are not considered in relation to the requirements forscantlings given in these rules.

Guidance note:Vibration analysis should be considered for semi-spade rudders.The lowest natural frequencies will normally fall in a frequency span which includes the blade passing frequency of a propeller.Particularly a coupled mode where torsion of rudder stock and bending of rudder horn are dominating may result in increased dynamicstresses in way of the lower pintle bearing.The natural frequencies will mainly depend on the torsion stiffness of the rudder stock, the bending stiffness of the rudder horn andthe distance between the centre of gravity of rudder and its rotational axis. The size of the rudder will also govern the frequencyrange in which these natural modes will fall. It is recommended to keep the lowest fundamental modes of a rudder away from theblade passing frequency in the full speed range. Normally it may not be possible to keep all the modes above the blade passingfrequency. Thus it is recommended to apply a method to determine the natural frequencies of a rudder either by means of FiniteElement Analyses or other reliable methods based on analytical approach/experience


1.5.7 Over-balanced rudders are subject to special consideration with respect to type of steering gear andrisk of an unexpected and uncontrolled sudden large movement of rudder causing severe change of ship'spre-set course. See Pt.4 Ch.10 Sec.1 [2.9].

Guidance note:A rudder shall be considered over-balanced, when balanced portion exceed 30% in any actual load condition. Special rudder types,such as flap rudders, are subject to special consideration.


1.6 Materials

1.6.1 Welded parts of rudders are to be made of approved rolled hull materials.

1.6.2 Material factor k for normal and high tensile steel plating may be taken into account when specified ineach individual rule requirement. The material factor k is to be taken as defined in Ch.3 Sec.1 [2.2], unlessotherwise specified.

1.6.3 Material grades for plates and sections for rudders, rudder trunks and rudder horns are in general tobe selected based on Class II in Ch.3 Sec.1 Table 9.The steel used for the rudder trunk is to be of weldable quality, with a carbon content not exceeding 0.23%on ladle analysis and a carbon equivalent Ceq not exceeding 0.41.Rudder trunks made of materials other than steel are to be specially considered by the Society.For rudder and rudder body plates subjected to stress concentrations, e.g. in way of lower support of semi-spade rudders or at upper part of spade rudders, Class III as given in Ch.3 Sec.1 Table 9 shall be applied.

1.6.4 Rudder stocks, pintles, coupling bolts, keys, rudder horns and rudder members shall be made of rolled,forged or cast carbon manganese or alloy steel in accordance with Pt.2 Ch.2.

Guidance note:It is recommended that rudder stocks and pintles are of weldable quality in order to obtain satisfactory weldability for any futurerepairs by welding in service.


For rudder stocks, pintles, keys and bolts the minimum yield stress shall not be less than 200 N/mm2.

1.6.5 Nodular cast iron may be accepted in certain parts after special considerations. Materials withminimum specified tensile strength lower than 400 N/mm2 or higher than 900 N/mm2 will normally not beaccepted in rudder stocks, shafts or pintles, keys and bolts.

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Nodular cast iron and cast steel parts for transmission of rudder torque by means of conical connections shallbe stress relieved.

1.6.6 For rudder stocks, pintles, keys and bolts the minimum yield stress is not to be less than 200 N/mm2.The requirements of this chapter are based on a material's yield stress of 235 N/mm2. If material is usedhaving a yield stress differing from 235 N/mm2 the material factor k is to be determined as follows:

where:

e = 0.75 for ReH > 235 N/mm2

e = 1.00 for ReH ≤ 235 N/mm2

ReH = specified minimum yield stress, in N/mm2, of material used, and is not to be taken greater than0.7Rm or 450 N/mm2, whichever is the smaller value

Rm = specified minimum tensile strength, N/mm2, of material used

1.7 Equivalence

1.7.1 The Society may accept alternative calculation methods to those shown in this chapter provided it isdemonstrated that the scantling and arrangements are of equivalent or better than those derived using therule calculation methods.

1.7.2 Direct analyses adopted to justify an alternative design are to take into consideration all relevantmodes of failure, on a case by case basis. These failure modes may include, amongst others: yielding,fatigue, buckling and fracture. Possible damages caused by cavitation are also to be considered.

1.7.3 If deemed necessary by the Society, lab tests, or full scale tests may be requested to validate thealternative design approach.

2 Rudder force and rudder torque

2.1 Rudder blades without cut-outs

2.1.1 The rudder force upon which the rudder scantlings are to be based, in N, is to be determined from thefollowing formula:

where:

CR = rudder forceA = area of rudder blade, in m2, including area of flap and rudder bulb, if any

= vertical projected area of nozzle rudderV = maximum service speed, in knots, as defined in Ch.1 Sec.4 [3.1.8]. When the speed is less than 10

knots, V is to be replaced by the expression:

Vmin = (V + 20) / 3

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For the astern condition the maximum astern speed is to be used, however, in no case less than:

Vastern = 0.5 VK1 = factor depending on the aspect ratio λ of the rudder areaK1 = (λ + 2) / 3, with λ not to be taken greater than 2λ = b2 / At,b = mean height of the rudder area in m. Mean breadth and mean height of rudder are calculated

according to the coordinate system in Figure 2At = sum of rudder blade area A and area of rudder post or rudder horn, if any, within the height b in m2

K2 = coefficient depending on the type of the rudder and the rudder profile according to Table 3K3 = 0.8 for rudders outside the propeller jet

= 1.15 for rudders behind a fixed propeller nozzle= 1.0 otherwise

Figure 2 Rudder dimensions

Table 3 Rudder profile type - coefficient

K2Profile Type

Ahead condition Astern condition

NACA-00 series Göttingen

1.10 0.80

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K2Profile Type

Ahead condition Astern condition

Flat side

1.10 0.90

Hollow

1.35 0.90

High lift rudders

1.70 to be specially considered; if notknown: 1.30

Fish tail

1.40 0.80

Single plate

1.00 1.00

Mixed profiles (e.g. HSVA) 1.21 0.90

2.1.2 The rudder torque, in Nm, is to be calculated for both the ahead and astern condition according to theformula:

where:

r = c (α – k), in mc = mean breadth of rudder area, in m, see Figure 2α = 0.33 for ahead condition

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α = 0.66 for astern conditionk = Af / A,Af = portion of the rudder blade area situated ahead of the centre line of the rudder stock, see Figure 3rmin = 0.1c, in m, for ahead condition

2.2 Rudder blades with cut-outs (semi-spade rudders)

2.2.1 The total rudder force CR is to be calculated according to [2.1.1]. The pressure distribution over therudder area, upon which the determination of rudder torque and rudder blade strength is to be based, is tobe derived as follows:

The rudder area may be divided into two rectangular or trapezoidal parts with areas A1 and A2, so that A =A1 + A2 (see Figure 3).

Figure 3 Rudder area distribution

The levers r1 and r2 are to be determined as follows:

r1 = c1 (α – k1) in m

r2 = c2 (α – k2) in m

where:

c1, c2 = mean breadth of partial areas A1, A2 determined, where applicable, in accordance with Figure 2k1 = A1f / A1,k2 = A2f / A2,A1f = portion of A1 situated ahead of the centre line of the rudder stockA2f = portion of A2 situated ahead of the centre line of the rudder stockα = 0.33 for ahead conditionα = 0.66 for astern condition

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For parts of a rudder behind a fixed structure such as the rudder hornα = 0.25 for ahead conditionα = 0.55 for astern condition

The resulting force of each part may be taken as:CR1 = CR · A1/A in NCR2 = CR · A2/A in N

The resulting torque of each part may be taken as:QR1 = CR1 r1 in NmQR2 = CR2 r2 in Nm

The total rudder torque is to be calculated for both the ahead and astern condition according tothe formula:

QR = QR1 + QR2 in NmFor ahead condition QR is not to be taken less than

3 Rudder strength

3.1 Strength calculations

3.1.1 The rudder force and resulting rudder torque as given in [2] cause bending moments and shear forcesin the rudder body, bending moments and torques in the rudder stock, supporting forces in pintle bearingsand rudder stock bearings and bending moments, shear forces and torques in rudder horns and heel pieces.The rudder body is to be stiffened by horizontal and vertical webs enabling it to act as a bending girder.

3.1.2 The bending moments, shear forces and torques as well as the reaction forces are to be determinedby a direct calculation or by an approximate simplified method considered appropriate by the Society. Forrudders supported by sole pieces or rudder horns these structures are to be included in the calculation modelin order to account for the elastic support of the rudder body. Guidelines for calculation of bending momentand shear force distribution are given in App.A.

3.1.3 At and above the upper carrier bearing above neck bearing the bending moment is to be taken as zero,except as follows:

— for rotary vane type actuators with two rotor bearings, calculation of bending moment influence may berequired if bending deflection in way of upper bearing, based on the design rudder force FR, exceeds twotimes the diametrical bearing clearances. In lieu of a direct calculation, the deflection of the rudder stock,in mm, between the rotor bearings, δub may be taken equal to:

Ia = moment of inertia of rudder stock in cm4.ℓ = ℓb - hf for arrangements with upper pintle bearing.ℓa = ℓb for arrangements with neck bearing.ℓb = distance in m from mid-height of neck bearing or upper pintle bearing, as applicable, to

mid-height of upper stock bearing.

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hf = distance in m from upper end of rudder to mid-height neck bearing.hub = centre distance of the rotor bearings in mm.

— the actuator force induced bending moment, in kNm, is to be taken as the greater of the following:

and

ha = vertical distance between force and bearing centre in m.FMTR = net radial force, in kN, on rudder stock in way of actuator corresponding to rudder torque

MTR, ref. Pt.4 Ch.10.M BU = bending moment, in kNm, at bearing above neck bearing.Fdes = radial force, in kN, induced by actuator at design pressure.

4 Rudder stock and rudder shaft scantlings

4.1 Rudder stock scantlings4.1.1 Rudder stock diameter required for the transmission of the rudder torqueThe rudder stock diameter required for the transmission of the rudder torque is to be dimensioned such thatthe torsional stress is not exceeding the following value:

The rudder stock diameter for the transmission of the rudder torque, in mm, is therefore not to be less than:

where:

QR = total rudder torque, in Nm, as calculated in [2.1.2] and/or [2.2].k = material factor for the rudder stock as given in [1.6.6]

4.1.2 Rudder stock scantlings due to combined loadsIf the rudder stock is subjected to combined torque and bending, the von Mises stress in the rudder stock isnot to exceed 118 / k.

k = material factor for the rudder stock as given in [1.6.6]

The von Mises stress, in N/mm2, is to be determined by the formula:

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Bending stress, in N/mm2: σb = 10.2 · M/dc3

Torsional stress, in N/mm2: τt = 5.10 · QR/dC3

The rudder stock diameter, in mm, is therefore not to be less than:

M = bending moment, in Nm, at the station of the rudder stock considered, as described in [3].dt = rudder stock diameter for the transmission of the rudder torque, in mm, as defined [4.1.1].

4.1.3 High strength steelBefore significant reductions in rudder stock diameter due to the application of steels with yield stressesexceeding 235 N/mm2 are granted, the Society may require the evaluation of the rudder stock deformations.Large deformations of the rudder stock are to be avoided in order to avoid excessive edge pressures in wayof bearings. The slope of the stock is to be related to the bearing clearance, see [8.4].

4.1.4 In steering systems with more than one rudder where the torque from one actuator can be transferredto another, for instance by means of a connecting rod, the rudders stock shall not be permanently damagedwhen exposed to the sum of actuating loads.

4.2 Rudder shaft scantlings

4.2.1 At the lower bearing, the rudder shaft diameter, in mm, shall not be less than:

c =

ℓ, a and b are given in Figure 4, in mm.

The diameter, df in mm, below the coupling flange shall be 10% greater than dℓ. If, however, the rudder shaftis protected by a corrosion-resistant composition above the upper bearing, df may be equal to dℓ.

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Figure 4 Rudder shaft

5 Rudder blade

5.1 Permissible stresses

5.1.1 The section modulus and the web area of a horizontal section of the rudder blade are to be such thatthe following stresses, in N/mm2, will not be exceeded:

a) In general

(i) bending stress, σb 110 / k

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(ii) shear stress, 50 / k

(iii) von Mises stress

120 / k

k = material factor for the rudder plating as given in [1.6.2]

b) In way of the recess for the rudder horn pintle on semi-spade rudders

(i) bending stress, σb 75

(ii) shear stress, 50

(iii) von Mises stress

100

Guidance note:The stresses in b) apply equally to high tensile and ordinary steels.


Guidance note:The permissible stresses are to be understood as nominal stresses, i.e. stress concentrations are not considered.


5.2 Rudder plating

5.2.1 The thickness of the rudder side, top and bottom plating, in mm, is not to be less than:

where:

d = summer loadline draught, in mCR = rudder force, in N, according to [2.1.1]A = rudder area, in m2

β = ; max. 1.0 if b/s ≥ 2.5

s = smallest unsupported width of plating, in mb = greatest unsupported width of plating, in mk = material factor for the rudder plating as given in [1.6.2].

The thickness of the nose plates may be increased to the discretion of the Society. The thickness of webplates is not to be less than the greater of 70% of the rudder side plating thickness and 8 mm.

The rudder plating in way of the solid part, e.g. forged or cast steel, is to be of increased thickness per[5.3.4].

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5.3 Connections of rudder blade structure with solid parts

5.3.1 Solid parts in forged or cast steel, which house the rudder stock or the pintle, are normally to beprovided with protrusions.These protrusions are not required when the web plate thickness is less than:

— 10 mm for web plates welded to the solid part on which the lower pintle of a semi-spade rudder is housedand for vertical web plates welded to the solid part of the rudder stock coupling of spade rudders

— 20 mm for other web plates.

5.3.2 The solid parts are in general to be connected to the rudder structure by means of two horizontal webplates and two vertical web plates.

5.3.3 Minimum section modulus of the connection with the rudder stock housingThe section modulus of the cross-section of the structure of the rudder blade, in cm3, formed by vertical webplates and rudder plating, which is connected with the solid part where the rudder stock is housed is to benot less than:

where:

cS = coefficient, to be taken equal to:1.0 if there is no opening in the rudder plating or if such openings are closed by a full penetrationwelded plate1.5 if there is an opening in the considered cross-section of the rudder

dc = rudder stock diameter, in mmHE = vertical distance between the lower edge of the rudder blade and the upper edge of the solid part,

in mHX = vertical distance between the considered cross-section and the upper edge of the solid part, in mk = material factor for the rudder blade plating as given in [1.6.2].ks = material factor for the rudder stock as given in [1.6.6].

The actual section modulus of the cross-section of the structure of the rudder blade is to be calculated withrespect to the symmetrical axis of the rudder.

The breadth of the rudder plating, in m, to be considered for the calculation of section modulus is to be notgreater than:

where:

sV = spacing between the two vertical webs, in m, see Figure 5.

Where openings for access to the rudder stock nut are not closed by a full penetration welded plate, they areto be deducted.

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Figure 5 Cross-section of the connection between rudder blade structure and rudder stockhousing

5.3.4 The thickness of the horizontal web plates connected to the solid parts, in mm, as well as that of therudder blade plating between these webs, is to be not less than the greater of the following values:

tH = 1.2 ttH = 0.045 dS² / sH

where:

t = defined in [5.2]dS = diameter, in mm, to be taken equal to:

dc, as per [4.2], for the solid part housing the rudder stockdp, as per [7.1], for the solid part housing the pintle

sH = spacing between the two horizontal web plates, in mm

The increased thickness of the horizontal webs is to extend fore and aft of the solid part at least to the nextvertical web.

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5.3.5 The thickness of the vertical web plates welded to the solid part where the rudder stock is housedas well as the thickness of the rudder side plating under this solid part is to be not less than the valuesobtained, in mm, from Table 4.

Table 4 Thickness of side plating and vertical web plates

Thickness of verticalweb plates, in mm Thickness of rudder plating, in mm

Type of rudderRudder blade

without openingRudder bladewith opening

Rudder bladewithout opening Area with opening

Rudder supported by sole piece 1.2 t 1.6 t 1.2 t 1.4 t

Semi-spade and spade rudders 1.4 t 2.0 t 1.3 t 1.6 t

t = thickness of the rudder plating, in mm, as defined in [5.2]

The increased thickness is to extend below the solid piece at least to the next horizontal web.

5.4 Single plate rudders5.4.1 Mainpiece diameterThe mainpiece diameter is calculated according to [4.1] and [4.2] respectively. For spade rudders the lowerthird may taper down to 0.75 times stock diameter.

5.4.2 Blade thicknessThe blade thickness, in mm, is not to be less than:

where:

s = spacing of stiffening arms, in m, not to exceed 1 m;V = speed in knots, see [2.1.1];k = material factor for the rudder plating as given in [1.6.2].

5.4.3 ArmsThe thickness of the arms, in mm, is not to be less than the blade thickness:

The section modulus, in cm3, is not to be less than:

where:

C1 = horizontal distance from the aft edge of the rudder to the centreline of the rudder stock, in mk = material factor as given in [1.6.2] or [1.6.6], respectively.

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6 Rudder stock and shaft couplings

6.1 Connection to steering gear

6.1.1 The connection between rudder stock and steering gear is to be according to Pt.4 Ch.10.

dn

dg

d t

l t

hn

ds

Figure 6 Cone coupling

6.2 Horizontal flange couplings

6.2.1 The diameter of the coupling bolts, in mm, is not to be less than:

Where:

d = stock diameter, taken equal to the greater of the diameters dt or dc according to [4.1] and [4.2], inmm

n = total number of bolts, which is not to be less than 6

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em = mean distance [mm] of the bolt axes from the centre of the bolt systemks = material factor for the stock as given in [1.6.6]kb = material factor for the bolts as given in [1.6.6].

6.2.2 The thickness of the coupling flanges, in mm, is not to be less than the greater of the followingformulae:

Where:

kf = material factor for flange as given in [1.6.6]kb = material factor for the bolts as given in [1.6.6]db = bolt diameter, in mm, calculated for a number of bolts not exceeding 8.

6.2.3 The thickness of coupling flanges at the root section, in mm, shall not be less than:

kf = material factor for flange.M = bending moment in kNm at coupling.a = mean distance from centre of bolts to the longitudinal centre line of the coupling, in mm.d = diameter of rudder stock for stock flange, breadth of rudder for rudder flange, both in mm.β = factor to be taken from Table 5

Table 5 Table of β

d/a 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

β 1.8 1.5 1.25 1.0 0.8 0.6 0.45 0.35 0.25

β shall not be taken less than 0.25 when d/a is greater than 1.6.

kr is determined according to Table 6.

Table 6 Table of kr

kf 0.5 0.4 0.3

kr 70 75 84

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where:

rf = radius of fillet, in mm, not to be taken less than 0.3∙(a – 0.5d).

Guidance note:The mean distance, in mm, from centre of bolts to the longitudinal centreline of the coupling, a, may in general be taken as:

The mean distance, e, in mm, from the centre of bolts to the centre of the bolt system may in general be taken as:

n = Number of bolts.yi = Distance, in mm, from the longitudinal centreline of the rudder to the centre of bolt i.xi = Longitudinal distance, in mm, from, e.g. the rudder axis to the centre of bolt i.


6.2.4 The width of material between the perimeter of the bolt holes and the perimeter of the flange is not tobe less than 0.67 db.

6.2.5 Coupling bolts are to be fitted bolts and their nuts are to be locked effectively.

6.2.6 Requirements for welding and design details when the rudder stock is connected to the rudder byhorizontal flange coupling are described in Ch.13 Sec.1 [6.1.9].

6.3 Vertical flange couplings

6.3.1 The diameter of the coupling bolts, in mm, is not to be less than

where:

d = stock diameter in way of coupling flange, in mmn = total number of bolts, which is not to be less than 8kb = material factor for bolts as given in [1.6.6]ks = material factor for stock as given in [1.6.6].

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6.3.2 The first moment of area of the bolt assembly about the centre of the coupling, in cm3, is to be notless than:

6.3.3 The thickness of the coupling flanges is to be not less than the bolt diameter, and the width of theflange material between the perimeter of the bolt holes and the perimeter of the flange is to be not less than0.67 db.

6.3.4 Coupling bolts are to be fitted bolts and their nuts are to be locked effectively.

6.4 Cone couplings with key

6.4.1 A rudder stock cone coupling connection without hydraulic arrangement for mounting and dismountingshall not be applied for spade rudders.

6.4.2 An effective sealing shall be provided at each end of the cone coupling.

6.4.3 Tapered key-fitted (keyed) connections shall be designed to transmit rudder torque in all normaloperating conditions by means of friction in order to avoid mutual movements between rudder stock and hub.The key shall be regarded as a securing device.

6.4.4 Tapering and coupling lengthCone couplings without hydraulic arrangements for mounting and dismounting the coupling shall have a taperc on diameter of 1:8 - 1:12 where (see Figure 7):

The cone coupling is to be secured by a slugging nut. The nut is to be secured, e.g. by a securing plate.The cone shapes are to fit exactly. The coupling length ℓ is to be, in general, not less than 1.5d0.

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Figure 7 Cone coupling with key

6.4.5 Dimensions of keyThe shear area of the key, in cm2, is not to be less than:

where:

QF = design yield moment of rudder stock, in Nm:

dt = stock diameter, in mm, according to [4.1]k = material factor for stock as given in [1.6.6]dk = mean diameter of the conical part of the rudder stock, in mm, at the keyReH1 = specified minimum yield stress of the key material, in N/mm2

Where the actual diameter dta is greater than the calculated diameter dt, the diameter dta is to be used.However, dta applied to the above formula need not be taken greater than 1.145 dt.

The effective surface area, in cm2, of the key (without rounded edges) between key and rudder stock or conecoupling is not to be less than:

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where:

ReH2 = specified minimum yield stress of the key, stock or coupling material, in N/mm2, whichever is less.

6.4.6 The dimensions of the slugging nut, in mm, are to be as follows (see Figure 7):

external thread diameter: dg ≥ 0.65 d0

height: hn ≥ 0.6dg

outer diameter: dn ≥ 1.2 du or 1.5 dg, whichever is the greater.

6.4.7 Push upIt is to be proved that 50% of the design yield moment is solely transmitted by friction in the cone couplings.This can be done by calculating the required push-up pressure and push-up length according to [6.5.3] and[6.5.4] for a torsional moment Q'F = 0.5QF.

6.5 Cone couplings with special arrangements for mounting anddismounting the couplings

6.5.1 An effective sealing shall be provided at each end of the cone coupling.

6.5.2 Where the stock diameter exceeds 200 mm, the press fit is recommended to be effected by a hydraulicpressure connection. In such cases the cone is to be more slender, c ≈1:12 to ≈1:20.In case of hydraulic pressure connections the nut is to be effectively secured against the rudder stock or thepintle, see Figure 8.For the safe transmission of the torsional moment by the coupling between rudder stock and rudder bodythe push-up pressure and the push-up length are to be determined according to [6.5.3] and [6.5.4],respectively.

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Figure 8 Cone coupling without key

6.5.3 Push up pressureThe push-up pressure, in N/mm2, is not to be less than the greater of the two following values:

where:

QF = design yield moment of rudder stock, as defined in [6.4.5], in Nmdm = mean cone diameter in mmℓ = cone length in mmµ0 = frictional coefficient, equal to 0.15Mb = bending moment in the cone coupling (e.g. in case of spade rudders), in Nm

It has to be proved by the designer that the push-up pressure does not exceed the permissible surfacepressure in the cone. The permissible surface pressure, in N/mm², is to be determined by the followingformula:

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where:

ReH = specified minimum yield stress of the material of the gudgeon in N/mm2

α = dm / dadm = diameter, in mmda = outer diameter of the gudgeon to be not less than 1.5 dm, in mm

6.5.4 Push-up lengthThe push-up length, in mm, is to comply with the following formula:

where:

Rtm = mean roughness, in mm, taken equal to 0.01c = taper on diameter according to [6.5.2].

Notwithstanding the above, the push up length is not to be less than 2 mm.Guidance note:Note: In case of hydraulic pressure connections the required push-up force, in N, for the cone may be determined by the followingformula:

The value 0.02 is a reference for the friction coefficient using oil pressure. It varies and depends on the mechanical treatment androughness of the details to be fixed. Where due to the fitting procedure a partial push-up effect caused by the rudder weight is given,this may be taken into account when fixing the required push-up length, subject to approval by the Society.


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6.6 Rudder shaft couplings

6.6.1 The cone coupling at the lower end of the rudder shaft shall be as required in [6.4] or [6.5], usingrelevant parameters for the shaft instead of the stock in the formulas.

6.6.2 The vertical coupling at the upper end of the rudder shaft shall be as required in [6.3], using relevantparameters for the shaft instead of the stock in the formulas.

7 Pintles

7.1 Scantlings

7.1.1 The pintle diameter, in mm, is not to be less than:

where:

B = relevant bearing force, in Nkp = material factor for pintle as given in [1.6.6].

7.2 Couplings7.2.1 TaperingPintles are to have a conical attachment to the gudgeons with a taper on diameter not greater than:1:8 - 1:12 for keyed and other manually assembled pintles applying locking by slugging nut,1:12 - 1:20 on diameter for pintles mounted with oil injection and hydraulic nut.

7.2.2 Push-up pressure for pintle bearingsThe required push-up pressure for pintle bearings, in N/mm², is to be determined by the following formula:

where:

B1 = supporting force in the pintle bearing, in Nd0 = pintle diameter, in mm.

The push up length is to be calculated similarly as in [6.5.4], using required push-up pressure and propertiesfor the pintle bearing.

7.3 Dimensions of threads and nuts

7.3.1 The minimum dimensions of threads and nuts are to be determined according to [6.4.6].

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7.4 Pintle housing7.4.1 LengthThe length of the pintle housing in the gudgeon is not to be less than the pintle diameter Dp. Dp is to bemeasured on the outside of liners.The thickness of the pintle housing is not to be less than 0.25 Dp.

7.4.2 ThicknessThe thickness of the pintle housing is not to be less than 0.25 Dp.

8 Rudder stock bearing, rudder shaft bearing and pintle bearing

8.1 Liners and bushes8.1.1 Rudder stock bearingLiners and bushes are to be fitted in way of bearings. The minimum thickness of liners and bushes is to beequal to:

— tmin = 8 mm for metallic materials and synthetic material— tmin = 22 mm for lignum material.

The difference in hardness of bushing and liners shall not be less than 65 Brinell. 13% Chromium steel shallbe avoided.The bushing shall be effectively secured to the bearing.

Guidance note:Bushing fitted by means of shrink fitting alone is not considered effectively secured. Additional physical stoppers need to be arrangedto prevent the bushing from accidentally rotating or shifting in vertical direction.


8.1.2 Pintle bearingThe thickness of any liner or bush, in mm, is neither to be less than:

where:

B = relevant bearing force, in N

nor than the minimum thickness defined in [8.1.1].

8.2 Minimum bearing surface

8.2.1 An adequate lubrication is to be provided.

The bearing surface Ab (defined as the projected area: length · outer diameter of liner), in mm2, is not to beless than:

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where:

F = reaction force, in N, in bearing as determined in [3.1]. For ram type actuator, F for the upperbearing shall be calculated considering reaction force induced by one ram

pa = allowable surface pressure, in N/mm2, according to Table 4.

The maximum surface pressure pa for the various combinations is to be taken as listed in Table 4. Highervalues than given in the table may be taken in accordance with makers’ specifications if they are verified bytests:

Table 7 Maximum surface pressure pa

Bearing material pa [N/mm2]

Lignum vitae 2.5

White metal, oil lubricated 4.5

Synthetic material with hardness between 60 and 70 Shore D1) 5.52)

Steel3) and bronze and hot-pressed bronze-graphite materials 7.0

Notes:

1) Indentation hardness test at 23°C and with 50 % moisture, are to be carried out according to a recognizedstandard. Synthetic bearing materials are to be of an approved type.

2) Surface pressures exceeding 5.5 N/mm2 may be accepted in accordance with bearing manufacturer's specificationand tests, but in no case more than 10 N/mm2.

3) Stainless and wear-resistant steel in an approved combination with stock liner.

8.3 Bearing dimensions

8.3.1 The length/diameter ratio of the bearing surface is not to be greater than 1.2.

The bearing length ℓp of the pintle, in mm, is to be such that

where:

Dp = actual pintle diameter measured on the outside of liners, in mm.

Bearing arrangements with a height of the bearing greater than above, may be accepted based on directcalculations provided by the designer showing acceptable clearances at the upper and lower edges of thebearing.

8.3.2 The thickness, in mm, of bearing material outside of the bushing shall not be less than:For balanced rudder or semi-spade rudders:

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For spade rudders:

Smaller thickness may be accepted based on direct analysis.

8.4 Bearing clearances

8.4.1 With metal bearings, clearances in mm shall not be less than db / 1000 + 1.0 on the diameter. If non-metallic bearing material is applied, the bearing clearance is to be specially determined considering thematerial’s swelling and thermal expansion properties. This clearance is not to be taken less than 1.5 mm onbearing diameter unless a smaller clearance is supported by the manufacturer’s recommendation and there isdocumented evidence of satisfactory service history with a reduced clearance.

9 Strength of sole pieces and of rudder horns

9.1 Sole piece9.1.1 Section modulus and sectional area

Figure 9 Sole piece

Referring to Figure 9, the section modulus around the vertical (z)-axis, in cm3, is not to be less than:

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The section modulus around the transverse (y)-axis, in cm3, is not to be less than:

The sectional area, in mm2, is not to be less than:

where:

k = material factor as given in [1.6.2] or [1.6.6] respectively.

9.1.2 von Mises stressAt no section within the length ℓ50 is the von Mises stress to exceed 115 / k. The von Mises stress, in N/mm2,is to be determined by the following formula:

where:

σb = Mb / Zz(x) in N/mm2

τ = B1 / As in N/mm2

Mb = bending moment at the section considered in NmMb = B1 x in NmMbmax = B1 ℓ50 in NmB1 = supporting force in the pintle bearing in N, normally B1 = CR / 2k = material factor as given in [1.6.2] or [1.6.6], respectively.

9.1.3 The sole piece shall be sloped in order to avoid pressure from keel blocks when docking. The solepiece shall extend forward of the after edge of the propeller boss, for sufficient number of frame spaces toprovide adequate fixation at the connection with deep floors of the aft ship structure. The cross section ofthis extended part may be gradually reduced to the cross section necessary for an efficient connection to theplate keel.

9.2 Rudder horn9.2.1 Rudder hornThe bending moments and shear forces are to be determined by a direct calculation or in line with theguidelines given in App.A [5] and App.A [6] for semi spade rudder with one elastic support and semi spaderudder with 2-conjugate elastic support respectively.

The section modulus around the horizontal x-axis, in cm3, is not to be less than:

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where:

Mb = bending moment at the section considered in Nm.

The shear stress, in N/mm2, is not to be larger than:

where:

k = material factor as given in [1.6.2] or [1.6.6], respectively.

9.2.2 von Mises stressAt no section within the height of the rudder horn is the von Mises stress to exceed 120 / k N/mm2. The vonMises stress, in N/mm2, is to be calculated by the following formula:

where:

σb = Mb / Zx in N/mm2

τ = B1 / Ah in N/mm2

B1 = supporting force in the pintle bearing in [N]Ah = effective shear area of rudder horn in y-direction in [mm2];

τT = MT 103 / (2 AT th) in N/mm2

MT = torsional moment in [Nm];AT = area in the horizontal section enclosed by the rudder horn in [mm2];th = plate thickness of rudder horn in [mm];

9.2.3 Rudder horn platingThe thickness of the rudder horn side plating, in mm, is not to be less than:

where:

L = rule length as defined in Ch.1 Sec.4 Table 2k = material factor as given in [1.6.2] or [1.6.6], respectively.

9.2.4 Connection to hull structure

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The rudder horn plating is to be effectively connected to the aft ship structure, e.g. by connecting the platingto side shell and transverse/longitudinal girders, in order to achieve a proper transmission of forces, seeFigure 10.When the connection between the rudder horn and the hull structure is designed as a curved transition intothe hull plating, special consideration is to be given to the effectiveness of the rudder horn plate in bendingand to the stresses in the transverse web plates.Brackets or stringer are to be fitted internally in horn, in line with outside shell plate, as shown in Figure 10.

Figure 10 Connection of rudder horn to aft ship structure

Transverse webs of the rudder horn are to be led into the hull up to the next deck in a sufficient number andmust be of adequate thickness.Strengthened plate floors are to be fitted in line with the transverse webs in order to achieve a sufficientconnection with the hull.The centre line bulkhead (wash-bulkhead) in the after peak is to be connected to the rudder horn.Scallops are to be avoided in way of the connection between transverse webs and shell plating.

9.2.5 Requirements for welding of rudder horns are described in Ch.13 Sec.1 [6.1.3].

9.3 Rudder trunk9.3.1 ScantlingsThe requirement applies to both trunk configurations, extending below stern frame or not.

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Where the rudder stock is arranged in a trunk in such a way that the trunk is stressed by forces due torudder action, the scantlings of the trunk are to be such that:

— the von Mises stress due to bending and shear does not exceed 0.35 ReH,— the bending stress, in N/mm2, in welded rudder trunk is to be in compliance with the following formula:

σ ≤ 80 / k

with:

σ = bending stress in the rudder trunk, as defined in [9.3.1]k = material factor for the rudder trunk as given in [1.6.2] or [1.6.6] respectively, not to be taken less

than 0.7ReH = specified minimum yield stress, in N/mm2, of the material used.

For calculation of bending stress, the span to be considered is the distance between the mid-height of thelower rudder stock bearing and the point where the trunk is clamped into the shell or the bottom of the skeg.

9.3.2 Requirements for welding of rudder trunks are described in Ch.13 Sec.1 [6.1.8].

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APPENDIX A GUIDELINES FOR CALCULATION OF BENDING MOMENTAND SHEAR FORCE DISTRIBUTION

1 GeneralThe evaluation of bending moments, shear forces and support forces for the system rudder – rudder stockmay be carried out for some basic rudder types as outlined in [2] – [6].

2 Spade rudder

2.1 Data for the analysisℓ10 - ℓ30 = lengths of the individual girders of the system in mI10 – I30 = moments of inertia of these girders in cm4

Load of rudder body, in kN/m:

PR = CR / (ℓ10 103)

2.2 Moments and forcesThe moments, in Nm, and forces, in N, may be determined by the following formulae:

Mb = CR (ℓ20 + (ℓ10 (2 c1 + c2) / 3 (c1 + c2)))B3 = Mb / ℓ30B2 = CR + B3

Figure 1 Spade rudder

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3 Spade rudder with trunk



PR = CR / ((ℓ10+ ℓ20)103)

3.2 Moments and forcesFor spade rudders with rudders trunks the moments, in Nm, and forces, in N, may be determined by thefollowing formulae:

MRis the greatest of the following values:

MR = CR2 (ℓ10 – CG2Z)MR = CR1 (CG1Z– ℓ10)

where:

CR1 = Rudder force over the rudder blade area A1CR2 = Rudder force over the rudder blade area A2CR1Z = Vertical position of the centre of gravity of the rudder blade area A1CG2Z = Vertical position of the centre of gravity of the rudder blade area A2MB = CR2 (ℓ10 – CG2Z)B3 = (MB+ MCR1) / (ℓ20 + ℓ30)B2 = CR+ B3

Figure 2 Spade rudder with trunk.

4 Rudder supported by sole piece

4.1 Data for the analysisℓ10 - ℓ50 = lengths of the individual girders of the system in m

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I10 – I50 = moments of inertia of these girders in cm4

For rudders supported by a sole piece the length ℓ20 is the distance between lower edge of rudder body andcentre of sole piece and I20 the moment of inertia of the pintle in the sole piece.

I50 = moment of inertia of sole piece around the z-axis in cm4

ℓ50 = effective length of sole piece in m


PR = CR / (ℓ10 103)

Z = spring constant of support in the sole piece, in kN/m:Z = 6.18 · I50 / ℓ50

3

4.2 Moments and forcesMoments and shear forces are indicated in Figure 3.

Figure 3 Rudder with neck bearing and sole piece.

5 Semi spade rudder with one elastic support


Z = spring constant of support in the rudder horn, in kN/m:Z = 1 / (fb + ft) for the support in the rudder horn, see Figure 4

= unit displacement of rudder horn, in m, due to a unit force of 1 kN acting in the centre ofsupport

fb = 1.3 d3 / (6.18 In) in m/kN (guidance value)In = moment of inertia of rudder horn around the x-axis, in cm4, (see also Figure 4)ft = unit displacement due to torsionft =

in m/kN

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FT = mean sectional area of rudder horn in m2]ui = breadth, in mm, of the individual plates forming the mean horn sectional areati = thickness within the individual breadth ui in mmd = Height of the rudder horn, in m, defined in Figure 4. This value is measured downwards from

the upper rudder horn end, at the point of curvature transition, to the mid-line of the lowerrudder horn pintle

e = distance, in m, as defined in Figure 5


PR10 = CR2 / (ℓ10 · 103)PR20 = CR1 / (ℓ10 · 103)

for CR, CR1, CR2, see [3].


5.3 Rudder hornReferring to Figure 5, the loads on the rudder horn are as follows:

Mb = bending moment in Nm= min(B1z; B1d)

Q = shear force in N= B1

MT(z) = torsional moment in Nm= B1 e(z)

An approximation for B1, in N, is

B1 = CRb / (ℓ20 + ℓ30).

Figure 4 Semi-spade rudder.

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Figure 5 Rudder horn.

6 Semi spade rudder with 2-conjugate elastic support

6.1 Data for the analysisK11, K22, K12 = rudder horn compliance constants calculated for rudder horn with 2-conjugate elastic

supports (Figure 6). The 2-conjugate elastic supports are defined in terms of horizontaldisplacements, yi, by the following equations:

— at the lower rudder horn bearing:y1 = - K12B2- K22B1

— at the upper rudder horn bearing:y2 = - K11B2- K12B1

where:

y1, y2 = horizontal displacements, in m, at the lower and upper rudder horn bearings, respectivelyB1, B2 = horizontal support forces, in kN, at the lower and upper rudder horn bearings, respectivelyK11, K22, K12 = obtained, in m/kN, from the following formulae:

d = height of the rudder horn, in m, defined in Figure 6. This value is measured downwards

from the upper rudder horn end, at the point of curvature transition, to the mid-line of thelower rudder horn pintle

λ = length, in m, as defined in Figure 6. This length is measured downwards from the upperrudder horn end, at the point of curvature transition, to the mid-line of the upper rudder

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horn bearing. For λ = 0, the above formulae converge to those of spring constant Z for arudder horn with 1-elastic support, and assuming a hollow cross section for this part

e = rudder-horn torsion lever, in m, as defined in Figure 6 (value taken at z = d/2)I1h = moment of inertia of rudder horn about the x axis, in m4, for the region above the upper

rudder horn bearing. Note that I1his an average value over the length ℓ (see Figure 6)I2h = moment of inertia of rudder horn about the x axis, in m4, for the region between the upper

and lower rudder horn bearings. Note that I2his an average value over the length d - λ(see Figure 6)

Ith = torsional stiffness factor of the rudder horn, in m4

For any thin wall closed section:

FT = mean of areas enclosed by outer and inner boundaries of the thin walled section of rudderhorn, in m2

ui = length, in mm, of the individual plates forming the mean horn sectional areati = thickness, in mm, of the individual plates mentioned above.

Note that the Ithvalue is taken as an average value, valid over the rudder horn height.


PR10 = CR2 / (ℓ10 · 103)PR20 = CR1 / (ℓ-10 · 103)

for CR, CR1, CR2, see [3.2].


6.3 Rudder horn bending momentThe bending moment acting on the generic section of the rudder horn is to be obtained, in Nm, from thefollowing formulae:

— between the lower and upper supports provided by the rudder horn:

MH= FA1z

— above the rudder horn upper-support:

MH= FA1z + FA2 (z - dlu)

where:

FA1 = support force at the rudder horn lower-support, in N, to be obtained according to Figure 6, andtaken equal to B1

FA2 = support force at the rudder horn upper-support, in N, to be obtained according to Figure 6, andtaken equal to B2

z = distance, in m, defined in Figure 7, to be taken less than the distance d, in m, defined in the samefigure

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dlu = distance, in m, between the rudder-horn lower and upper bearings (according to Figure 6, dlu= d -λ).

6.4 Rudder horn shear forceThe shear force QH acting on the generic section of the rudder horn is to be obtained, in N, from the followingformulae:

— between the lower and upper rudder horn bearings:

QH= FA1

— above the rudder horn upper-bearing:

QH= FA1 + FA2

where:

FA1, FA2 = support forces, in N.

The torque acting on the generic section of the rudder horn is to be obtained, in Nm, from the followingformulae:

— between the lower and upper rudder horn bearings:

MT= FA1e(z)

— above the rudder horn upper-bearing:

MT= FA1e(z) + FA2e(z)

where:

FA1, FA2 = support forces, in N.e(z) = torsion lever, in m, defined in Figure 7.

6.5 Rudder horn shear stress calculationFor a generic section of the rudder horn, located between its lower and upper bearings, the following stressesare to be calculated:

— τS = shear stress, in N/mm2, to be obtained from the following formula:

— τT = torsional stress, in N/mm2, to be obtained for hollow rudder horn from the following formula:

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For solid rudder horn, τT is to be considered by the Society on a case by case basis.

For a generic section of the rudder horn, located in the region above its upper bearing, the following stressesare to be calculated:

— τS = shear stress, in N/mm2, to be obtained from the following formula:

— τT = torsional stress, in N/mm2, to be obtained for hollow rudder horn from the following formula:

For solid rudder horn, τT is to be considered by the Society on a case by case basis where:

FA1, FA2 = support forces, in NAH = effective shear sectional area of the rudder horn, in mm2, in y-directionMT = torque, in NmFT = mean of areas enclosed by outer and inner boundaries of the thin walled section of rudder

horn, in m2

tH = plate thickness of rudder horn, in mm. For a given cross section of the rudder horn, themaximum value of τT is obtained at the minimum value of tH.

6.6 Rudder horn bending stress calculationFor the generic section of the rudder horn within the length d, the following stresses are to be calculated:

— σB = bending stress, in N/mm2, to be obtained from the following formula:

where:

MH = bending moment at the section considered, in NmWX = section modulus, in cm3, around the x-axis (see Figure 7).

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Figure 6 Semi-spade rudder with 2-conjugate elastic support.

Figure 7 Rudder horn.

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