S10S10S10(cont)(1986)(Rev.1
1990)(Corr.1
July 1999)(Corr.2
July 2003)(Rev.2
May 2010)(Rev.3
Mar 2012)(Corr.1
May 2015)(Rev.4
Apr 2015)(Corr.1
Dec 2015)(Rev.5
May 2018)(Rev.6
Sep 2019)(Rev.7
Feb 2023)

Rudders, Sole Pieces and Rudder Horns
S10.1 General
1.1

Basic assumptions

1.1.1 This UR applies to ordinary profile rudders, and to some enhanced profile rudders withspecial arrangements for increasing the rudder force.
1.1.2 This UR applies to rudders made of steel for ships with L ≥ 24m.
1.2

Design considerations

1.2.1 Effective means are to be provided for supporting the weight of the rudder without
excessive bearing pressure, e.g. by a rudder carrier attached to the upper part of the rudder
stock. The hull structure in way of the rudder carrier is to be suitably strengthened.
1.2.2 Suitable arrangements are to be provided to prevent the rudder from lifting.
1.2.3 In rudder trunks which are open to the sea, a seal or stuffing box is to be fitted above
the deepest load waterline, to prevent water from entering the steering gear compartment andthe lubricant from being washed away from the rudder carrier. If the top of the rudder trunk is
below the deepest waterline at scantling draught (without trim), two separate watertight seals /stuffing boxes are to be provided.
1.3

Materials

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

Note:
1.

Changes introduced in Rev.3 are to be uniformly implemented by IACS Members forships contracted for construction on or after 1 January 2013.

2.

The “contracted for construction” date means the date on which the contract to build thevessel is signed between the prospective owner and the shipbuilder. For further details
regarding the date of “contract for construction”, refer to IACS Procedural Requirement(PR) No. 29.

3.

Changes introduced in Rev.4 are to be uniformly implemented by IACS Members forships contracted for construction on or after 1 July 2016.

4.

Changes introduced in Rev.5 are to be uniformly implemented by IACS Members forships contracted for construction on or after 1 July 2019.

5.

Changes introduced in Rev.6 are to be uniformly implemented by IACS Members forships contracted for construction on or after 1 January 2021.

6.

Changes introduced in Rev.7 are to be uniformly implemented by IACS Members forships contracted for construction on or after 1 July 2024.

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S10
(cont)

1.3.2 Material factor k for normal and high tensile steel plating may be taken into accountwhen specified in each individual rule requirement. The material factor k is to be taken as
defined in UR S4, unless otherwise specified.
1.3.3 Steel grade of plating materials for rudders and rudder horns are to be in accordancewith UR S6.
1.3.4 Rudder stocks, pintles, coupling bolts, keys and cast parts of rudders are to be made ofrolled, forged or cast carbon manganese steel in accordance with UR W7, W8 and W11.
1.3.5 For rudder stocks, pintles, keys and bolts the specified minimum yield stress is not tobe less than 200 N/mm2. The requirements of this UR are based on a material's specified
minimum yield stress of 235 N/mm2. If material is used having a specified minimum yield stressdiffering from 235 N/mm2 the material factor k is to be determined as follows:

with

1.4

𝟐𝟐𝟐𝟐𝟐𝟐 𝒆𝒆�𝑹𝑹𝒆𝒆𝒆𝒆𝒌𝒌 = �
e= 0.75 for ReH > 235 N/mm2e= 1.00 for ReH ≤ 235 N/mm2

ReH= specified minimum yield stress, in N/mm2, of material used, and is not to betaken greater than 0.7σT or 450 N/mm2, whichever is the smaller value.

σT= tensile strength, in N/mm2, of material used.

Welding and design details

1.4.1 Slot-welding is to be limited as far as possible. Slot welding is not to be used in areas
with large in-plane stresses transversely to the slots or in way of cut-out areas of semi-spaderudders.
When slot welding is applied, the length of slots is to be minimum 75 mm with breadth of 2 t,
where t is the rudder plate thickness, in mm. The distance between ends of slots is not to be
more than 125 mm. The slots are to be fillet welded around the edges and filled with a suitablecompound, e.g. epoxy putty. Slots are not to be filled with weld.
Continuous slot welds are to be used in lieu of slot welds. When continuous slot welding isapplied, the root gap is to be between 6-10 mm. The bevel angle is to be at least 15°.
1.4.2 In way of the rudder horn recess of semi-spade rudders, the radii in the rudder plating
except in way of solid part in cast steel are not to be less than 5 times the plate thickness, but
in no case less than 100 mm. Welding in side plate is to be avoided in or at the end of the radii.
Edges of side plate and weld adjacent to radii are to be ground smooth.
1.4.3 Welds in the rudder side plating subjected to significant stresses from rudder bending
and welds between plates and heavy pieces (solid parts in forged or cast steel or very thickplating) are to be made as full penetration welds. In way of highly stressed areas e.g. cut-out
of semi-spade rudder and upper part of spade rudder, cast or welding on ribs is to be arranged.
Two sided full penetration welding is normally to be arranged. Where back welding is
impossible welding is to be performed against ceramic backing bars or equivalent. Steelbacking bars may be used and are to be fitted with continuously welded on one side to the
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S10
(cont)

heavy piecebevelled edge, see Figure 1. The bevel angle is to be at least 15° for one sided
welding.

Figure 1: Use of steel backing bar in way of full penetration welding of rudder side
plating
1.4.4 Requirements for welding and design details of rudder trunks are described in S10.9.3.
1.4.5 Requirements for welding and design details when the rudder stock is connected to therudder by horizontal flange coupling are described in S10.6.1.4.
1.4.6 Requirements for welding and design details of rudder horns are described in S10.9.2.3.
1.5
Equivalence
1.5.1 The Society may accept alternatives to requirements given in this UR, provided they
are deemed to be equivalent.
1.5.2 Direct analyses adopted to justify an alternative design are to take into consideration all
relevant modes 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.
1.5.3 If deemed necessary by the Society, lab tests, or full scale tests may be requested tovalidate the alternative design approach.
S10.2 Rudder force and rudder torque2.1

Rudder blades without cut-outs

2.1.1 The rudder force upon which the rudder scantlings are to be based is to be
determined from the following formula:

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S10
(cont)CR = K1 K2 K3 132 AV2[N]

where:
CR= rudder force[N]A= area of rudder blade [m2]V= maximum service speed, in knots, with the ship on summer load waterline.
When the speed is less than 10 knots, V is to be replaced by the expression:
Vmin = (V + 20) / 3
For the astern condition the maximum astern speed as defined in SOLAS
Regulation II-1/3.15 is to be used, however, in no case taken less than:
Vastern = 0.5 V

K1= factor depending on the aspect ratio λ of the rudder area.

K1= (λ + 2) / 3, with λ not to be taken greater than 2.

λ= b2 / Atb= mean height of the rudder area, in m. Mean breadth and mean height of rudder
are calculated according to the coordinate system in Fig. 12.

At= 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 1.

K3= 0.8 for rudders outside the propeller jet.
= 1.15 for rudders behind a fixed propeller nozzle.
= 1.0 otherwise.

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S10
(cont)

Figure 12
Table 1
Profile Type
NACA-00 series Göttingen

Ahead condition

K2

Astern condition

1.10

0.80

1.10

0.90

1.35

0.90

1.70

1.30

1.40

0.80

1.00

1.00

1.21

0.90

Flat side

Hollow

High lift rudders

Fish tailSingle plate

Mixed profiles (e.g. HSVA)

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S10
(cont)

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

2.2

QR

= CR r[Nm]r= c (α – k1)[m]c= mean breadth of rudder area, in m, see Fig. 12.

α= 0.33 for ahead condition.

α= 0.66 for astern condition.

k1= Af / AAf= portion of the rudder blade area situated ahead of the centre line of the rudder
stock.

rmin= 0.1c for ahead condition, in m.

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

The total rudder force CR is to be calculated according to S10.2.1.1. The pressure distribution
over the rudder area, upon which the determination of rudder torque and rudder blade strength
is to be based, is to be 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 23).

Figure 23
The levers r1 and r2 are to be determined as follows:r1= c1 (α – k1)

[m]

r2

= c2 (α – k2)[m]c1, c2 = mean breadth of partial areas A1, A2 determined, where applicable, in
accordance with Figure 12.
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S10
(cont)k1= A1f / A1

k2

= A2f / A2

A1a

= portion of A1 situated aft of the centre line of the rudder stock.

A1f= portion of A1 situated ahead of the centre line of the rudder stock.

A2a= portion of A2 situated aft of the centre line of the rudder stock.

A2f= portion of A2 situated ahead of the centre line of the rudder stock.

α= 0.33 for ahead condition.

α= 0.66 for astern condition.

For parts of a rudder behind a fixed structure such as the rudder horn:α= 0.25 for ahead condition.

α= 0.55 for astern condition.

The resulting force of each part may be taken as:
𝑨𝑨
𝑨𝑨

CR1

= CR 𝟏𝟏 [N]CR2= CR 𝑨𝑨𝟐𝟐 [N]𝑨𝑨

The resulting torque of each part may be taken as:
QR1 = CR1 r1 [Nm]QR2 = CR2 r2 [Nm]The total rudder torque is to be calculated for both the ahead and astern condition according
to the formula:
QR= QR1 + QR2 [Nm]For ahead condition QR is not to be taken less than:

QR min = 0.1CR

A1c1 + A2c2
A

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S10
(cont)

S10.3 Rudder strength calculation
3.1
The rudder force and resulting rudder torque as given in S10.2 causes bending
moments and shear forces in the rudder body, bending moments and torques in the rudder
stock, supporting forces in pintle bearings and rudder stock bearings and bending moments,
shear forces and torques in rudder horns and heel pieces. The rudder body is to be stiffenedby horizontal and vertical webs enabling it to act as a bending girder.
3.2
The bending moments, shear forces and torques as well as the reaction forces are to
be determined by a direct calculation or by an approximate simplified method consideredappropriate by each individual society. For rudders supported by sole pieces or rudder horns
these structures are to be included in the calculation model in order to account for the elastic
support of the rudder body. Guidelines for calculation of bending moment and shear force
distribution are given in an annex to this UR.
S10.4 Rudder stock scantlings
4.1
The rudder stock diameter required for the transmission of the rudder torque is to bedimensioned such that the torsional stress is not exceeding the following value:𝜏𝜏 𝑇𝑇= 68 / k [N/mm2]

The rudder stock diameter for the transmission of the rudder torque is therefore not to be lessthan:
𝟑𝟑
𝒅𝒅𝒕𝒕 = 𝟒𝟒. 𝟐𝟐�𝑸𝑸
𝑹𝑹 𝒌𝒌 [mm]QR

= total rudder torque, in Nm, as calculated in S10.2.1.2 and/or S10.2.2.

k= material factor for the rudder stock as given in S10.1.3.5.

4.2

Rudder stock scantlings due to combined loads

If the rudder stock is subjected to combined torque and bending, the equivalent stress in therudder stock is not to exceed 118 / k, in N/mm2.
k= material factor for the rudder stock as given in S10.1.3.5.

The equivalent stress is to be determined by the formula:

Bending stress:
Torsional stress:

𝝈𝝈𝒄𝒄 = �𝝈𝝈𝟐𝟐𝒃𝒃 + 𝟑𝟑𝝉𝝉𝟐𝟐𝒕𝒕

[N/mm2]

𝝉𝝉𝒕𝒕 = 𝟓𝟓. 𝟏𝟏 × 𝟏𝟏𝟎𝟎𝟑𝟑 𝑸𝑸𝑹𝑹 ⁄𝒅𝒅𝟑𝟑𝒄𝒄[N/mm2]𝝈𝝈𝒃𝒃 = 𝟏𝟏𝟏𝟏. 𝟐𝟐 × 𝟏𝟏𝟎𝟎𝟑𝟑 𝑴𝑴⁄𝒅𝒅𝟑𝟑𝒄𝒄[N/mm2]

The rudder stock diameter is therefore not to be less than:
𝟔𝟔

M

𝒅𝒅𝒄𝒄 = 𝒅𝒅𝒕𝒕 �𝟏𝟏 +𝟒𝟒
𝟑𝟑(𝑴𝑴⁄𝑸𝑸𝑹𝑹 )𝟐𝟐[mm]= bending moment, in Nm, at the station of the rudder stock considered.

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S10
(cont)For a spade rudder with trunk extending inside the rudder, the rudder stock scantlings shall bechecked against the two cases defined in Annex S10.3.
4.3
Before significant reduction in rudder stock diameter are granted due to the applicationof steel with specified minimum yield stress exceeding 235 N/mm2, the Society may require theevaluation of the rudder stock deformations. Large deformations of the rudder stock are to be
avoided in order to avoid excessive edge pressures in way of bearings.
S10.5 Rudder blade
5.1

Permissible stresses

The section modulus and the web area of a horizontal section of the rudder blade are to besuch that the following stresses will not be exceeded:a)

In general, except in way of rudder recess sections where b) applies(i)

bending stress σb

110/k [N/mm2](ii) shear stress 𝜏𝜏

k
b)

50/k

(iii) equivalent stress 𝜎𝜎𝑒𝑒 = �𝜎𝜎𝑏𝑏2 + 3𝜏𝜏 2[N/mm2]120/k [N/mm2]= material factor for the rudder plating as given in S10.1.3.2.

In way of the recess for the rudder horn pintle on semi-spade rudders(i)bending stress σb(ii) shear stress 𝜏𝜏(iii) equivalent stress 𝜎𝜎𝑒𝑒 = �𝜎𝜎𝑏𝑏2 + 3𝜏𝜏 2

75[N/mm2]50[N/mm2]100[N/mm2]

Note: The stresses in b) apply equally to high tensile and ordinary steels.
5.2

Rudder plating

The thickness of the rudder side, top and bottom plating is not to be less than:𝑡𝑡 = 5.5𝑠𝑠𝑠𝑠√𝑘𝑘�𝑇𝑇𝑇𝑇𝑇𝑇𝑑𝑑 + 𝐶𝐶𝑅𝑅 10−4 ⁄𝐴𝐴 + 2.5[mm]

Tscd = summer scantling loadline draught, in m.
CR= rudder force, in N, according to S10.2.1.1.

A= rudder area, in m2.

𝛽𝛽 = �1.1 − 0.5(𝑠𝑠⁄𝑏𝑏)2

max. 1.00 if b/s ≥ 2.5.

s= smallest unsupported width of plating, in m.

b= greatest unsupported width of plating, in m.

k= material factor for the rudder plating as given in S10.1.3.2.
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S10
(cont)The thickness of the nose plates may be increased to the discretion of each Society. The
thickness of web plates 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 is to be of increased thickness per S10.5.3.4.
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 to beprovided with protrusions, except where not required as indicated below.
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-spaderudder is housed and 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 web plates and two vertical web plates.
5.3.3 Minimum section modulus of the connection with the rudder stock housing.
The section modulus of the cross-section of the structure of the rudder blade, in cm3, formedby vertical web plates and rudder plating, which is connected with the solid part where therudder stock is housed is to be not less than:
𝐻𝐻 −𝐻𝐻𝑥𝑥 2 𝑘𝑘
�
10−4 [cm3]
𝐻𝐻𝐸𝐸
𝑘𝑘𝑠𝑠

𝑊𝑊𝑠𝑠 = 𝑐𝑐𝑠𝑠 𝑑𝑑𝑐𝑐 3 � 𝐸𝐸

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 bya full penetration welded plate.
= 1.5 if there is an opening in the considered cross-section of the rudder.

dc= rudder stock diameter, in mm.

HE= vertical distance between the lower edge of the rudder blade and the upper edgeof the solid part, in m.

HX= vertical distance between the considered cross-section and the upper edge ofthe solid part, in m.

k= material factor for the rudder blade plating as given in S10.1.3.2.

ks= material factor for the rudder stock as given in S10.1.3.5.

The actual section modulus of the cross-section of the structure of the rudder blade is to be
calculated with respect to the symmetrical axis of the rudder.
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S10
(cont)The breadth of the rudder plating, in m, to be considered for the calculation of section modulus
is to be not greater than:b= sV + 2 Hx / 3[m]where:sV= spacing between the two vertical webs, in m, (see Figure 34).

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

Figure 3 4 Cross-section of the connection between rudder blade structure and rudder
stock housing, example with opening in only one side shown
5.3.4 The thickness of the horizontal web plates connected to the solid parts, in mm, as wellas that of the rudder blade plating between these webs, is to be not less than the greater of thefollowing values:tH= 1.2 t[mm]

tH

= 0.045 ds² / sH

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S10
(cont)

where:t= defined in S10.5.2.

dS= diameter, in mm, to be taken equal to:= dc, as per S10.4.2, for the solid part housing the rudder stock.
= dp, as per S10.7.1, 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 next vertical web.
5.3.5 The thickness of the vertical web plates welded to the solid part where the rudder stockis housed as well as the thickness of the rudder side plating under this solid part is to be notless than the values obtained, in mm, from Table 2.
Table 2 Thickness of side plating and vertical web plates

Type of rudder

Thickness of vertical web
plates, in mm

Thickness of rudder plating, inmm

Rudder blade
without
opening

Rudder bladewith opening

Rudder blade
without
opening

Area with
opening

1.2 t

1.6 t

1.2 t

1.4 t

1.4 t

2.0 t

1.3 t

1.6 t

Rudder supported bysole pieceSemi-spade and spaderudderst = thickness of the rudder plating, in mm, as defined in S10.5.2
The increased thickness is to extend below the solid piece at least to the next horizontal web.
5.4

Single plate rudders

5.4.1 Mainpiece diameter
The mainpiece diameter is calculated according to S10.4.1 and S10.4.2 respectively. For spaderudders the lower third may taper down to 0.75 times stock diameter.

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S10
(cont)

5.4.2 Blade thickness
The blade thickness is not to be less than:

𝑡𝑡𝑏𝑏 = 1.5𝑠𝑠𝑠𝑠√𝑘𝑘 + 2.5[mm]

where:s= spacing of stiffening arms, in m, not to exceed 1 m.

V

= speed, in knots, see S10.2.1.1.

k= material factor for the rudder plating as given in S10.1.3.2.

5.4.3 Arms
The thickness of the arms is not to be less than the blade thickness
ta

= tb

[mm]

The section modulus is not to be less than:Za= 0.5 s C12 V2 k [cm3]

C1= horizontal distance from the aft edge of the rudder to the centreline of the rudder
stock, in m.

k= material factor as given in S10.1.3.2 or S10.1.3.5 respectively.

S10.6 Rudder stock couplings
6.1

Horizontal flange couplings

6.1.1 The diameter of the coupling bolts is not to be less than:
𝒅𝒅𝒃𝒃 = 𝟎𝟎. 𝟔𝟔𝟔𝟔�𝒅𝒅𝟑𝟑 𝒌𝒌𝒃𝒃 ⁄𝒏𝒏 𝒆𝒆𝒎𝒎 𝒌𝒌𝒔𝒔 [mm]d= stock diameter, taken equal to the greater of the diameters dt or dc according toS10.4.1 and S10.4.2, in mm.

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

em= mean distance, in mm, of the bolt axes from the centre of the bolt system.

ks= material factor for the stock as given in S10.1.3.5.

kb= material factor for the bolts as given in S10.1.3.5.

6.1.2 The thickness of the coupling flanges, in mm, is not to be less than the greater of thefollowing formulae:𝑡𝑡𝑓𝑓 = 𝑑𝑑𝑏𝑏 �𝑘𝑘𝑓𝑓 ⁄𝑘𝑘𝑏𝑏
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S10

t f = 0.9d b(cont)

kf= material factor for flange as given in S10.1.3.5.

kb= material factor for the bolts as given in S10.1.3.5.

db= bolt diameter, in mm, calculated for a number of bolts not exceeding 8.

6.1.3 The width of material between the perimeter of the bolt holes and the perimeter of theflange is not to be less than 0.67 db.
6.1.4 The welded joint between the rudder stock and the flange is to be made in accordancewith Figure 4 5 or equivalent.

Figure 4 5 Welded joint between rudder stock and coupling flange
6.1.5 Coupling bolts are to be fitted bolts and their nuts are to be locked effectively.
6.2

Vertical flange couplings

6.2.1 The diameter of the coupling bolts, in mm, is not to be less than:
𝑑𝑑𝑏𝑏 = 0.81 𝑑𝑑⁄√𝑛𝑛 × �𝑘𝑘𝑏𝑏 ⁄𝑘𝑘𝑠𝑠

where:d= stock diameter, in mm, in way of coupling flange.

n= total number of bolts, which is not to be less than 8.

kb= material factor for bolts as given in S10.1.3.5.
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S10

ks

(cont)= material factor for stock as given in S10.1.3.5.

6.2.2 The first moment of area of the bolts about the centre of the coupling, m, is to be notless than:m= 0.00043 d3[cm3]

6.2.3 The thickness of the coupling flanges is to be not less than the bolt diameter, and thewidth of the flange material between the perimeter of the bolt holes and the perimeter of theflange is to be not less than 0.67 db.
6.2.4 Coupling bolts are to be fitted bolts and their nuts are to be locked effectively.
6.3

Cone couplings with key

6.3.1 Tapering and coupling length
Cone couplings without hydraulic arrangements for mounting and dismounting the couplingshould have a taper c on diameter of 1:8 - 1:12.
where:c= (d0 − du ) / ℓc(see Figure 5 6 and 5b7b)

The diameters d0, in mm, and du, in mm, are shown in Figure 5 6 and the cone length, ℓc, in
mm, is defined in Figure 5b7b.
The cone coupling is to be secured by a slugging nut. The nut is to be secured, e.g. by asecuring plate.
The cone shapes are to fit exactly. The coupling length ℓ, in mm, is to be, in general, not lessthan 1.5d0.

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S10

Figure 5 6 Cone coupling with key(cont)

Figure 5a 7a – Gudgeon outer diameter(da) measurement

Figure 5b 7b Cone length and coupling length
6.3.2 Dimensions of key
For couplings between stock and rudder a key is to be provided, the shear area of which, in
cm2, is not to be less than:𝑎𝑎𝑠𝑠 =

where:

17.55𝑄𝑄𝐹𝐹
𝑑𝑑𝑘𝑘 𝑅𝑅𝑒𝑒𝑒𝑒1

QF = design yield moment of rudder stock, in Nm.
𝑄𝑄𝐹𝐹 = 0.02664

𝑑𝑑𝑡𝑡 3𝑘𝑘

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.
dt= stock diameter, in mm, according to S10.4.1.

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S10

S10
(cont)k= material factor for stock as given in S10.1.3.5.

dk= mean diameter of the conical part of the rudder stock, in mm, at the key.

ReH1 = specified minimum yield stress of the key material, in N/mm2.
The effective surface area, in cm2, of the key (without rounded edges) between key and rudder
stock or cone coupling is not to be less than:𝑎𝑎𝑘𝑘 =

where:

5𝑄𝑄𝐹𝐹
𝑑𝑑𝑘𝑘 𝑅𝑅𝑒𝑒𝑒𝑒2

ReH2 = specified minimum yield stress of the key, stock or coupling material, in N/mm2,
whichever is less.
6.3.3 The dimensions of the slugging nut are to be as follows (see Figure 56):
external thread diameter:dg ≥ 0.65 doheight:hn ≥ 0.6 dg

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

6.3.4 Push up
It is to be proved that 50% of the design yield moment is solely transmitted by friction in thecone couplings. This can be done by calculating the required push-up pressure and push-up
length according to S10.6.4.2 and S10.6.4.3 for a torsional moment Q'F = 0.5QF.
6.3.5 Notwithstanding the requirements in S10.6.3.2 and S10.6.3.4, where a key is fitted tothe coupling between stock and rudder and it is considered that the entire rudder torque istransmitted by the key at the couplings, the scantlings of the key as well as the push-up force
and push-up length are to be at the discretion of the Society.
6.4
Cone couplings with special arrangements for mounting and dismounting thecouplings
6.4.1 Where the stock diameter exceeds 200 mm, the press fit is recommended to be effectedby a hydraulic pressure 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 the pintle.
For the safe transmission of the torsional moment by the coupling between rudder stock andrudder body the push-up pressure and the push-up length are to be determined according toS10.6.4.2 and S10.6.4.3 respectively.

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S10

S10
(cont)

Figure 6 8 Cone coupling without key
6.4.2 Push-up pressure
The push-up pressure is not to be less than the greater of the two following values:

preq1 =

2QF
10 3
d πμ0preq 2 =6M b 310
 2d m2
m[N/mm²]

𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟2= 6𝑀𝑀𝑐𝑐 103 [N/mm²]

where:
QF

𝑙𝑙2 𝑑𝑑𝑚𝑚= design yield moment of rudder stock, as defined in S10.6.3.2 in Nm.

dm= mean cone diameter, in mm, see Figure 56.

ℓ= coupling length, in mm.

µ0= frictional coefficient, equal to 0.15.

Mbc = bending moment in rudder stock at the top of the cone coupling (e.g. in case of
spade rudders), in Nm.
For spade rudder with trunk extending inside the rudder, the coupling shall be checked against
the two cases defined in Annex S10.3

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S10

S10
(cont)

It has to be proved by the designer that the push-up pressure does not exceed the permissible
surface pressure in the cone. The permissible surface pressure, in N/mm², is to be determinedby the following formula:𝑝𝑝perm =0.95R eH (1 − 𝛼𝛼 2 )√3 + 𝛼𝛼 4− 𝑝𝑝𝑏𝑏[N/mm²]

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S10

S10
(cont)

where:
3.5𝑀𝑀

𝑝𝑝𝑏𝑏 = 𝑑𝑑 ℓ𝑏𝑏𝑐𝑐
103
2
𝑚𝑚

[N/mm²]

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

α= dm /dadm= diameter, in mm, see Figure 56.

da= outer diameter of the gudgeon, in mm, see Figure 5 6 and Figure 5a7a. (Theleast diameter is to be considered).

The outer diameter of the gudgeon in mm shall not be less than 1.25 d0, with d0 defined in
Figure 56.
6.4.3 Push-up length
The push-up length𝛥𝛥ℓ, in mm, 𝛥𝛥ℓ is to comply with the following formula:
𝛥𝛥ℓ1 ≤ 𝛥𝛥ℓ ≤ 𝛥𝛥ℓ2

where:

∆ℓ1 =

Δℓ2 =

𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟 𝑑𝑑𝑚𝑚
0.8𝑅𝑅𝑡𝑡𝑡𝑡+
1 − 𝛼𝛼 2𝑐𝑐
𝐸𝐸 � 2 � 𝑐𝑐
𝑝𝑝perm 𝑑𝑑𝑚𝑚
0.8𝑅𝑅𝑡𝑡𝑡𝑡+
2
1 − 𝛼𝛼𝑐𝑐
𝐸𝐸 � 2 � 𝑐𝑐

[mm]

[mm]

Rtm

= mean roughness, in mm, taken equal to 0.01.

c= taper on diameter defined in S10.6.3.1.

Note: In case of hydraulic pressure connections the required push-up force Pe, in N, for thecone may be determined by the following formula:𝑐𝑐
𝑃𝑃𝑒𝑒 = 𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟 𝑑𝑑𝑚𝑚 𝜋𝜋 ℓ � + 0.02�
2

The value 0.02 is a reference for the friction coefficient using oil pressure. It varies and depends
on the mechanical treatment and roughness 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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S10
(cont)

S10.7 Pintles
7.1

Scantlings

The pintle diameter, in mm, is not to be less than:
𝑑𝑑𝑝𝑝 = 0.35�𝐵𝐵𝑘𝑘𝑝𝑝

where:

B= relevant bearing force, in N.

kp= material factor for pintle as given in S10.1.3.5.

7.2

Couplings

7.2.1 Tapering
Pintles are to have a conical attachment to the gudgeons with a taper on diameter not greaterthan:
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
The required push-up pressure for pintle in case of dry fitting, in N/mm², is to be determined by
preq1 as given below.
The required push-up pressure for pintle in case of oil injection fitting, in N/mm², is to be
determined by the maximum pressure of preq1 and preq2 as given belowfollowing formula:
𝐵𝐵 𝐵𝐵𝑑𝑑

𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟1 = 0.4 1 2 0
6𝑀𝑀𝑏𝑏𝑏𝑏

𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟2 = 2

where:

B1
rudder.

ℓ 𝑑𝑑𝑚𝑚𝑑𝑑𝑚𝑚 ℓ103[N/mm²][N/mm²]= Supporting force in the pintle, in N, e.g. B1 as defined in figure A4 for semi-spaded0= Pintle diameter, in mm, see Figure 59.

Mbp= bending moment in the pintle cone coupling to be determined by:𝑀𝑀𝑏𝑏𝑏𝑏 = 𝐵𝐵ℓ𝑎𝑎 [Nm]ℓa= length between middle of pintle-bearing and top of contact surface between cone
coupling and pintle in m, see Figure 79)

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S10
(cont)The required push-up length, Δℓ1, is to be calculated similarly as in S10.6.4.3, using the required
push-up pressure as defined above, and properties for the pintle.

Figure 79 pintle cone coupling indicating ℓa
7.3
The minimum dimensions of threads and nuts are to be determined according toS10.6.3.3.
7.4

Pintle housing

The length of the pintle housing in the gudgeon is not to be less than the pintle diameter dp. dp
is to be measured on the outside of liners.
The thickness of the pintle housing is not to be less than 0.25 dp.
S10.8 Rudder stock bearing, rudder shaft bearing and pintle bearing
8.1

Liners and bushes

8.1.1 Rudder stock bearing
Liners and bushes are to be fitted in way of bearings. For rudder stocks and pintles having
diameter less than 200 mm, liners in way of bushes may be provided optionally. The minimum
thickness of liners and bushes is to be equal to:
•
•

tmin
tmin

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

8.1.2 Pintle bearing

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S10

S10

The thickness of any liner or bush, in mm, is neither to be less than:(cont)

𝑡𝑡 = 0.01√𝐵𝐵

where:

B= relevant bearing force, in N.

nor than the minimum thickness defined in S10.8.1.1.
8.2

Minimum bearing surface

An adequate lubrication is to be provided.
The bearing surface Ab (defined as the projected area: length x outer diameter of liner) is not
to be less than:Ab= P / qa [mm2]where:
P= reaction force, in N, in bearing as determined in S10.3.2.

qa= allowable surface pressure according to the table below.

The allowable surface pressure qa for the various combinations is to be taken as reported inTable 3. Higher values than given in the table may be taken in accordance with makers’
specifications if they are verified by tests:
Table 3 Allowable surface pressure qa
Bearing materialqa [N/mm2]

lignum vitae

2.5

white metal, oil lubricated4.5

synthetic material with hardness greater than
60 Shore D1)steel3) and bronze and hot-pressed bronzegraphite materials
Notes:

5.52)
7.0

1)

Indentation hardness test at 23°C and with 50% moisture, are to be carried out according
to a recognized standard. 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 specification and 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

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

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S10

The bearing length Lp of the pintle, in mm, is to be such that:(cont)

Dp ≤ Lp ≤ 1.2 Dp
where:
Dp
8.4= Actual pintle diameter, in mm, measured on the outside of liners.

Bearing clearances

With metal bearings, clearances should not be less than db / 1000 + 1.0, in mm, on the diameter.
If non-metallic bearing material is applied, the bearing clearance is to be specially determinedconsidering the material’s swelling and thermal expansion properties. This clearance is not tobe taken less than 1.5 mm on bearing diameter unless a smaller clearance is supported by the
manufacturer’s recommendation and there is documented evidence of satisfactory service
history with a reduced clearance.
S10.9 Strength of sole pieces and of rudder horns
9.1

Sole piece

Figure 7 810 Sole piece
The section modulus around the vertical z-axis is not to be less than:Zz= Mb k / 80

[cm3]

The section modulus around the transverse y-axis is not to be less than:Zy= 0.5 Zz

The sectional area is not to be less than:As= B1 k / 48

[mm2]

k

= material factor as given in S10.1.3.2 or S10.1.3.5 respectively.
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S10
(cont)

9.1.1 Equivalent stress
At no section within the length ℓ50 is the equivalent stress to exceed 115 / k, in N/mm2. The
equivalent stress is to be determined by the following formula:𝜎𝜎𝑒𝑒 = �𝜎𝜎𝑏𝑏2 + 3𝜏𝜏 2[N/mm2]σb= Mb / Zz(x)[N/mm2]

𝜏𝜏= B1 / As[N/mm2]Mb= bending moment at the section considered [Nm].

Mb= B1 x[Nm]

Mbmax = B1 ℓ50[Nm]where:

9.2

B1= supporting force in the pintle bearing, in N, (normally B1 = CR / 2).

k= material factor as given in S10.1.3.2 or S10.1.3.5 respectively.
Rudder horn

When the connection between the rudder horn and the hull structure is designed as a curved
transition into the hull plating, special consideration is to be given to the effectiveness of therudder horn plate in bending and to the stresses in the transverse web plates.
The bending moments and shear forces are to be determined by a direct calculation or in linewith the guidelines given in Annex S10.5 and Annex S10.6 for semi spade rudder with one
elastic support and semi spade rudder with 2-conjugate elastic support respectively.
The section modulus around the horizontal x-axis is not to be less than:Zx= Mb k / 67

[cm3]

Mb

= bending moment at the section considered, in Nm.

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S10
(cont)The shear stress is not to be larger than:𝜏𝜏 τk= 48 / k [N/mm2]= material factor as given in S10.1.3.2 or S10.1.3.5 respectively.

9.2.1 Equivalent stress
At no section within the height of the rudder horn is the equivalent stress to exceed 120 / k, in
N/mm2. The equivalent stress is to be calculated by the following formula:𝜎𝜎𝑒𝑒 = �𝜎𝜎𝑏𝑏2 + 3(𝜏𝜏 2 + 𝜏𝜏 𝑇𝑇2 )[N/mm2]σb= Mb / Zx[N/mm2]

𝜏𝜏= B1 / Ah[N/mm2]

B1= supporting force in the pintle bearing, in N.

Ah= effective shear area of rudder horn in y-direction, in mm2.

𝜏𝜏 𝑇𝑇

= MT 103 / 2 AT thMT= 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.

k= material factor as given in S10.1.3.2 or S10.1.3.5 respectively.

[N/mm2]9.2.2 Rudder horn plating
The thickness of the rudder horn side plating is not to be less than:t = 2.4 Lk[mm]

where:
L= Rule length as defined in UR S2, in m.

k= material factor as given in S10.1.3.2 or S10.1.3.5 respectively.

9.2.3 Welding and connection to hull structure
The rudder horn plating is to be effectively connected to the aft ship structure, e.g. byconnecting the plating to side shell and transverse/ longitudinal girders, in order to achieve aproper transmission of forces, see Figure 8911.
Brackets or stringer are to be fitted internally in horn, in line with outside shell plate, as shownin Figure 8911.

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S10
(cont)

Figure 8 911 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 sufficientnumber.
Strengthened plate floors are to be fitted in line with the transverse webs in order to achieve a
sufficient connection 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.
The weld at the connection between the rudder horn plating and the side shell is to be full
penetration. The welding radius is to be as large as practicable and may be obtained bygrinding.
9.3

Rudder trunk

The requirements in this section apply to trunk configurations which are extended below stern
frame and arranged in such a way that the trunk is stressed by forces due to rudder action.
9.3.1 Materials, welding and connection to hull
The steel used for the rudder trunk is to be of weldable quality, with a carbon content notexceeding 0.23% on ladle analysis or a carbon equivalent CEQ not exceeding 0.41%.
Plating materials for rudder trunks are in general not to be of lower grades than corresponding
to class II as defined in UR S6.
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S10
(cont)

The weld at the connection between the rudder trunk and the shell or the bottom of the skeg is
to be full penetration.
For rudder trunks extending below shell or skeg, tThe fillet shoulder radius r, in mm, (see Figure
91012) is to be as large as practicable and to comply with the following formulae:r = 0.1dc / kwithout being less than:rr= 60 [mm]= 30 [mm]when σ ≥ 40 / k [N/mm²]
when σ < 40 / k [N/mm²]

where:dc= rudder stock diameter axis defined in S10.4.2.

σ= bending stress in the rudder trunk, in N/mm².

k= material factor for the rudder trunk as given in S10.1.3.2 or S10.1.3.5
respectively.
The radius may be obtained by grinding. If disk grinding is carried out, score marks are to be
avoided in the direction of the weld. The radius is to be checked with a template for accuracy.
Four profiles at least are to be checked. A report is to be submitted to the Surveyor.
Rudder trunks comprising of materials other than steel are to be specially considered by theSociety.

Figure 9 1012 Fillet shoulder radius

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S10
(cont)

9.3.2 Scantlings
The scantlings of the trunk are to be such that:-the equivalent stress due to bending and shear does not exceed 0.35 ReH.

-

the bending stress on welded rudder trunk is to be in compliance with the following
formula:σ ≤ 80 / k[N/mm²]with:σ= bending stress in the rudder trunk, as defined in S10.9.3.1.

k= material factor for the rudder trunk as given in S10.1.3.2 or S10.1.3.5
respectively, not to be taken less than 0.7.

ReH

= 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 midheight of the lower rudder stock bearing and the point where the trunk is clamped into the shellor the bottom of the skeg.

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S10
(cont)

Annex
Guidelines for calculation of bending momentand shear force distribution
AnnexS10.1

GeneralThe evaluation of bending moments, shear forces and support forces for the system rudder–rudder stock may be carried out for some basic rudder types as outlined in AnnexS10.2AnnexS10.6.

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S10
(cont)

AnnexS10.2

Spade rudder

Data for the analysisℓ10 - ℓ30 = Lengths of the individual girders of the system, in m.
I10 – I30 = Moments of inertia of these girders, in cm4.
Load of rudder body:PR= CR / (ℓ10 103)

[kN/m]

Moments and forces
The moments and forces may be determined by the following formulae:
Mb= CR (ℓ20 + (ℓ10 (2 c1 + c2) / 3 (c1 + c2))) [Nm]B3

= Mb / ℓ30

[N]

B2

= CR + B3[N]

The maximum moment, MRC, in top of the cone coupling as shown in Figure A1 is applicablefor the connection between the rudder and the rudder stock.

Figure A 1

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S10

S10
(cont)

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S10

S10
(cont)

AnnexS10.3

Spade rudder with trunk

Data for the analysisℓ10 - ℓ30 = Lengths of the individual girders of the system, in m.
I10 – I30 = Moments of inertia of these girders, in cm4.
Load of rudder body:PR= CR / ((ℓ10 + ℓ20)103)[kN/m]

Moments and forces
For a spade rudder with trunk extending inside the rudder, the strength shall be checked
withagainst the following two cases:a) pressure applied on the entire rudder areab) pressure applied only on rudder area below the middle of neck bearing.
For spade rudders with rudders trunks The moments, in Nm, and forces, in N, for the twocases defined above may be determined according to Figure A2 a) and b), respectively.

Figure A2 a)
Full rudder force CR = CR1+CR2 and total rudder torque QR = QR1 + QR2 with rudders
stock bending moment Mb = MCR2 - MCR1Page 33 of 43

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S10
(cont)

Figure A2 b)
Rudder force CR2 corresponding to rudder torque QR2 acting at rudder blade area A2
with rudders stock bending moment Mb = MCR2
MR is the greatest of the following values:
MCR1 = CR1 (CG1Z – ℓ10)
MCR2 = CR2 (ℓ10 – CG2Z)
where:
CR1 : Rudder force over the rudder blade area A1.
CR2 : Rudder force over the rudder blade area A2.
CG1Z : Vertical position of the centre of gravity of the rudder blade area A1 from base.
CG2Z : Vertical position of the centre of gravity of the rudder blade area A2 from base.
CR= CR1 + CR2

B3= (MCR2 - MCR1) / (ℓ20 + ℓ30)

B2

= CR + B3

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S10
(cont)

Figure A 2
AnnexS10.4

Rudder supported by sole piece

Data for the analysisℓ10 - ℓ50 = Lengths of the individual girders of the system, in m.
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 and centre 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.

Load of rudder body:PR= CR / (ℓ10 103) [kN/m]

Z= spring constant of support in the sole piece.

Z= 6.18 x I50 / ℓ503 [kN/m]

Moments and forces
Moments and shear forces are indicated in Figure A 3

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S10

S10
(cont)

Figure A 3

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S10

S10
(cont)

AnnexS10.5

Semi spade rudder with one elastic support

Data for the analysisℓ10 - ℓ50 = Lengths of the individual girders of the system, in m.
I10 – I50 =Moments of inertia of these girders, in cm4.
Z

= spring constant of support in the rudder horn.

Z= 1 / (fb + ft) [kN/m] for the support in the rudder horn (Figure A 4).

fb= unit displacement of rudder horn, in m, due to a unit force of 1 kN acting in thecentre of support.

fb= 1.3 d3 / (6.18 In)In= moment of inertia of rudder horn around the x-axis, in cm4, (see also Figure A4).

ft= unit displacement due to torsion.

ft=de 2 ∑ u i t i 3.14 × 10 8 FT2

FT= mean sectional area of rudder horn, in m2.

ui= breadth, in mm, of the individual plates forming the mean horn sectional area.

ti= thickness within the individual breadth ui, in mm.

d= Height of the rudder horn, in m, defined in Figure A 4. This value is measured
downwards from the upper rudder horn end, at the point of curvaturetransition, to the mid-line of the lower rudder horn pintle.

[m/kN] (guidance value)()[m/kN]

e(z) = distance as defined in Figure A 5, in m.
Load of rudder body:
PR10 = CR2 / (ℓ10 x 103) [kN/m]
PR20 = CR1 / (ℓ20 x 103) [kN/m]for CR, CR1, CR2, see S10.2.
Moments and forces
Moments and shear forces are indicated in Figure A 4.
Rudder horn
The loads on the rudder horn are as follows:Mb= bending moment = B1 z [Nm],Page 37 of 43

Mbmax = B1 d

[Nm]

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S10
(cont)q= shear force = B1 [N]MT(z) = torsional moment = B1 e(z) [Nm]An estimate for B1 is:
B1= CR b / (ℓ20 + ℓ30)

[N]

Figure A 4

Figure A 5

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S10
(cont)

Annex S10.6

Semi spade rudder with 2-conjugate elastic support

Data for the analysis
K11, K22, K12 : Rudder horn compliance constants calculated for rudder horn with 2-conjugate
elastic supports (Figure A 6).The 2-conjugate elastic supports are defined in terms of horizontal
displacements, yi, by the following equations:at the lower rudder horn bearing:y1= - K12 B2 - K22 B1

at the upper rudder horn bearing:y2= - K11 B2 - K12 B1

where:
y1, y2 : Horizontal displacements, in m, at the lower and upper rudder horn bearings,
respectively.
B1, B2 : Horizontal support forces, in kN, at the lower and upper rudder horn bearings,
respectively.
K11, K22, K12 : Obtained, in m/kN, from the following formulae:

K11 = 1.3

λ3
e2 λ
+
3EJ1h GJ th λ3
λ2 (d λ ) e 2 λ+K 22 = 1.3 
+
EJ
EJ
GJ th
3
2
1h
1h 

2
 λ3
(d λ)3  + e 2d
λ2 (d λ ) λ(d λ )+++K12 = 1.3 

EJ1h
EJ1h
3EJ 2h  GJ th 3EJ1hd : Height of the rudder horn, in m, defined in Figure A 6. This value is measured
downwards from the upper rudder horn end, at the point of curvature transition, to the
mid-line of the lower rudder horn pintle.
λ : Length, in m, as defined in Figure A 6. This length is measured downwards from theupper rudder horn end, at the point of curvature transition, to the mid-line of the upper
rudder horn bearing. For λ = 0, the above formulae converge to those of spring constant
Z for a rudder horn with 1-elastic support, and assuming a hollow cross section for thispart.
e : Rudder-horn torsion lever, in m, as defined in Figure A 6 (value taken at z = d/2).
J1h : Moment of inertia of rudder horn about the x axis, in m4, for the region above theupper rudder horn bearing. Note that J1h is an average value over the length λ (see Figure
A 6).

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S10

J2h : Moment of inertia of rudder horn about the x axis, in m4, for the region between theupper and lower rudder horn bearings. Note that J2h is an average value over the lengthd - λ (see Figure A 6).

(cont)Jth : Torsional stiffness factor of the rudder horn, in m4.
For any thin wall closed section:

J th =

2

4FT∑ ut iii

FT : Mean of areas enclosed by outer and inner boundaries of the thin walled section of
rudder horn, in m2.
ui : Length, in mm, of the individual plates forming the mean horn sectional area.
ti : Thickness, in mm, of the individual plates mentioned above.
Note that the Jth value is taken as an average value, valid over the rudder horn height.
Load of rudder body:
PR10 = CR2 / (ℓ10 x 103) [kN/m]
PR20 = CR1 / (ℓ20 x 103) [kN/m]for CR, CR1, CR2, see S10.2.2.
Moments and forces
Moments and shear forces are indicated in Figure A 6.
Rudder horn bending moment
The bending moment acting on the generic section of the rudder horn is to be obtained, in Nm,from the following formulae:•

between the lower and upper supports provided by the rudder horn:
MH = FA1 z•

above the rudder horn upper-support:
MH = FA1 z + FA2 (z - dlu)

where:
FA1 : Support force at the rudder horn lower-support, in N, to be obtained according to
Figure A 6, and taken equal to B1.
FA2 : Support force at the rudder horn upper-support, in N, to be obtained according to
Figure A 6, and taken equal to B2.

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S10

z : Distance, in m, defined in Figure A 7, to be taken less than the distance d, in m, definedin the same figure.

(cont)

dlu : Distance, in m, between the rudder-horn lower and upper bearings (according to
Figure A 6, dlu = d - λ ).
Rudder horn shear force
The shear force QH acting on the generic section of the rudder horn is to be obtained, in N,
from the following formulae:•

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 thefollowing formulae:•

between the lower and upper rudder horn bearings:
MT = FA1 e(z)•above the rudder horn upper-bearing:
MT = FA1 e(z) + FA2 e(z)where:
FA1, FA2 : Support forces, in N.
e(z) : Torsion lever, in m, defined in Figure A 7.
Rudder horn shear stress calculation
For a generic section of the rudder horn, located between its lower and upper bearings, thefollowing stresses are to be calculated:𝜏𝜏𝑆𝑆 : Shear stress, in N/mm2, to be obtained from the following formula:𝜏𝜏𝑆𝑆 =𝐹𝐹𝐴𝐴1
𝐴𝐴𝐻𝐻𝜏𝜏 𝑇𝑇 : Torsional stress, in N/mm2, to be obtained for hollow rudder horn from the following
formula:𝜏𝜏 𝑇𝑇 =

𝑀𝑀𝑇𝑇 10−3
2F 𝑇𝑇 𝑡𝑡𝐻𝐻

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S10

S10
(cont)For solid rudder horn, 𝜏𝜏 𝑇𝑇 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, thefollowing stresses are to be calculated:𝜏𝜏𝑆𝑆 : Shear stress, in N/mm2, to be obtained from the following formula:𝜏𝜏𝑆𝑆 =

𝐹𝐹𝐴𝐴1 + 𝐹𝐹𝐴𝐴2
𝐴𝐴𝐻𝐻𝜏𝜏 𝑇𝑇 =

𝑀𝑀𝑇𝑇 10−3
2𝐹𝐹𝑇𝑇 𝑡𝑡𝐻𝐻

𝜏𝜏 𝑇𝑇 : Torsional stress, in N/mm2, to be obtained for hollow rudder horn from the following
formula:For solid rudder horn, 𝜏𝜏 𝑇𝑇 is to be considered by the Society on a case by case basis where:
FA1, FA2 : Support forces, in N.

AH : Effective shear sectional area of the rudder horn, in mm2, in y-direction.
MT : Torque, in Nm.
FT : 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,
the maximum value of 𝜏𝜏 𝑇𝑇 is obtained at the minimum value of tH.

Rudder horn bending stress calculation

For 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:

σB =

MH
WX

where:MH : Bending moment at the section considered, in Nm.
WX : Section modulus, in cm3, around the x-axis (see Figure A 7).

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S10
(cont)

Figure A 6

Figure A 7

End ofDocument

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