Dnv - Calculation of Shafts in Marine Applications

CLASSIFICATION NOTESNo. 41.4

DET NORSKE VERITASVeritasveien 1, NO-1322 Høvik, Norway Tel.: +47 67 57 99 00 Fax: +47 67 57 99 11

CALCULATION OF SHAFTS IN MARINE APPLICATIONS

FEBRUARY 2007

FOREWORDDET NORSKE VERITAS (DNV) is an autonomous and independent foundation with the objectives of safeguarding life, prop-erty and the environment, at sea and onshore. DNV undertakes classification, certification, and other verification and consultancyservices relating to quality of ships, offshore units and installations, and onshore industries worldwide, and carries out researchin relation to these functions.Classification NotesClassification Notes are publications that give practical information on classification of ships and other objects. Examples of de-sign solutions, calculation methods, specifications of test procedures, as well as acceptable repair methods for some componentsare given as interpretations of the more general rule requirements.A list of Classification Notes is found in the latest edition of Pt.0 Ch.1 of the �”Rules for Classification of Ships�” and the �”Rulesfor Classification of High Speed, Light Craft and Naval Surface Craft�”. The list of Classification Notes is also included in the current �“Classification Services �– Publications�” issued by the Society,which is available on request. All publications may be ordered from the Society�’s Web site http://exchange.dnv.com.The Society reserves the exclusive right to interpret, decide equivalence or make exemptions to this Classification Note.

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Main Changes�— A description of a method for calculating surface hardened/peened shafts has been added.�— Geometrical stress concentration factors for holes and slots have been aligned with IACS UR M68.�— An update of calculation examples, due to rule changes in Rules for Classification of Ships Pt.4 Ch.4 Sec.1 (i.e. reduction

of safety factor for low cycle criterion, has been made.�— Calculations of stainless steel in sea water have been amended for martensitic and duplex steel.�— A number of detected editorial errors have been corrected, and complementary explanations of calculation methods have

been made throughout the text.

For further, more detailed information, related to the changes in this new edition of Classification Note 41.4, please contact:[email protected]

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Classification Notes - No. 41.4 3

February 2007

CONTENTS

1. BASIC PRINCIPLES............................................... 41.1 Scope............................................................................41.2 Description of Method.................................................41.3 Limits of Application...................................................52. NOMENCLATURE ................................................. 62.1 Symbols .......................................................................63. THE LOW CYCLE FATIGUE CRITERION

AND TORQUE REVERSAL CRITERION .......... 63.1 Scope and General Remarks........................................63.2 Basic Equations ...........................................................73.3 Repetitive Nominal Peak Torsional Stress, max .........73.4 Repetitive Nominal Torsional Stress Range, ..............73.5 Component Influence Factor for Low Cycle Fatigue,

KL ..............................................................................123.5.1 Notch influence term for low cycle fatigue, fL( t, y) ....... 123.5.2 Surface condition influence term for low cycle fatigue,

fL(Ry, B) ............................................................................. 123.6 Surface Hardening/Peening .......................................123.6.1 General remarks .................................................................. 123.6.2 Calculation procedure ......................................................... 124. THE HIGH CYCLE FATIGUE CRITERION.... 134.1 Scope and General Remarks......................................134.2 Basic Equation ...........................................................134.3 High Cycle Fatigue Strengths, f and f....................134.4 Component Influence Factor for High Cycle

Fatigue, KH and KH ................................................14

4.4.1 Notch influence term for high cycle fatigue, fH ( t, mt) and fH ( b, mb) ................................................................... 14

4.4.2 Size (statistical) influence term for high cycle fatigue, fH(r) ..................................................................................... 14

4.4.3 Surface condition influence term for high cycle fatigue, fH (Ry, B) and fH (Ry, B).................................................. 15

4.5 Surface Hardening/Peening....................................... 154.5.1 General remarks .................................................................. 154.5.2 Calculation procedure ......................................................... 155. THE TRANSIENT VIBRATION CRITERION. 155.1 Scope and General Remarks ..................................... 155.2 Basic Equation........................................................... 155.2.1 The accumulated number of cycles, Nc ............................. 166. THE GEOMETRICAL STRESS

CONCENTRATION FACTORS.......................... 176.1 Definition and General Remarks............................... 176.2 Shoulder Fillets and Flange Fillets ........................... 176.3 U-Notch .................................................................... 186.4 Step with Undercut ................................................... 196.5 Shrink Fits ................................................................. 206.6 Keyways ................................................................... 216.7 Radial Holes ............................................................. 226.8 Longitudinal Slot....................................................... 226.9 Splines ....................................................................... 236.10 Square Groove (Circlip) ............................................ 24

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1. Basic Principles1.1 ScopeThis Classification Note consists of the procedure and the ba-sic equations for verification of the load carrying capacity forshafts. It is an S-N based methodology for fatigue life assess-ment mainly based on the DIN 743 Part 1-3: 2000-04Tragfähigkeits-berechnung von Wellen und Achsen andVDEH 1983 Bericht Nr. ABF 11 Berechnung von Wöhlerlin-ien für Bauteile aus Stahl, Stahlguss und Grauguss Syn-thetische Wöhlerlinien. However, it is �“adapted and simplified�” to fit typical shaft de-signs in marine applications, such as marine propulsion andauxiliaries onboard ships and mobile offshore units.

Examples of introduced simplifications are that axial stressesare considered negligible for marine shafting systems as theyare dominated by torsional- and bending stresses and direct useof mechanical strength from representative testing. Even though such shafts are exposed to a wide spectrum ofloads, just a few of the dominating load cases need to be con-sidered instead of applying e.g. Miner�’s & Palmgrens�’s cumu-lative approach. These typical few load cases are described in1.2 and illustrated in Figure 1-1.The permissible stresses in the different shafts depend on thesafety factors as required by the respective rule sections. A recipe for how to assess the safety of surface hardened orpeened shafts is given in this Classification Note.

Figure 1-1Applicable Load Cases (stress) and associated Number of cycles

1.2 Description of MethodLoad case A The criterion for this load case must not be perceived as a de-sign requirement versus static fracture or permanent distortion.Local yield will normally not be a decisive criterion for marineshafts. Much more relevant is the risk for Low Cycle Fatigue(LCF) failure. Cycling from stop (or idle speed) to a high op-erational speed, stresses the shaft material from zero stress toits maximum peak stress. A cycle is the completion of one rep-etition from zero (or idle speed) to a high operational speed andback to stop (or idle speed) again. This is often referred to as�“primary cycles�” comparable to the �“Ground-Air-Ground cy-cles�” (GAG) in the aircraft industry.Peak loads that accumulate 103 �– 104 load cycles during thelife time of the ship should be considered for the LCF. For cer-tain applications as e.g. short range ferries, higher number ofcycles may have to be considered. For the �“Low Cycle Fatiguecriterion�” presented in 3.2 a), 104 load cycles are used. Notethat the considered maximum peak stress may not necessarilybe associated with the maximum shaft speed, but could be anintermittent shock load, e.g. caused by a rapid clutching in orpassing a main resonance, see Figures 3-1 to 3-8. Load case B

The criterion for this load case is introduced to prevent fatiguefailure, caused by cyclic stresses, during normal and continu-ous operation (see also Figure 3-1 to 3-8). The number of loadcycles is associated with the total number of revolutions of theshaft throughout the vessels lifetime, which means up to oreven more than 1010 load cycles, deserving its name; �“HighCycle Fatigue (HCF) criterion�”. The criterion is presented in 4. Load case C This represents regular transient operations that are not cov-ered by A, i.e. accumulates more than 104 load cycles duringthe life time of the ship. In practice for marine purposes, thismeans shafts in direct coupled propulsion plants driven by typ-ically 5 to 8 cylinders diesel engine. The reason is that for suchplants the engine�’s main excitation order coincides with thefirst torsional natural frequency of the shafting system, wherethe �“steady state�” torsional vibration stress amplitudes normal-ly exceeds the level determined by Load case B, see Figure 3-2, 3-4 and 3-8. A speed range around this resonance rpm has tobe barred for continuous operation, and should only be passedthrough as quickly as possible, see Figure 3-5 to 3-8. Still, cer-tain plants may accumulate up to 1 million such load cyclesduring the life time of the ship. This is either caused by ratherslow acceleration or deceleration, or frequent passing (e.g. ma-

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noeuvring speed below the barred speed range). On the otherhand, optimized plants may accumulate as few cycles as 104(e.g. plants with barred speed range in the lower region of theoperational speed range or controllable pitch propeller (CPP)plants running through the barred speed range with zero or lowpitch).Load case D This criterion serves the purpose of avoiding repeated yield re-versals in highly loaded parts of the shaft. A yield reversal isdefined as yield in tension followed by yield in compression orvice versa, Figure 1-2. All paper-clip bending mechanical en-gineers are well aware of that, after just a few cycles, their�“workout�” is terminated by an early failure of the clip. Suchextreme loading regime is only applicable to shafts in plantswith considerable negative torque, e.g. �“crash stop�” manoeu-vres of reversible plants, see Figure 3-3 to 3-8. These torque re-versals are assumed to happen much less than 103 times.For optimisation of a shafting system, in particular when tran-sient operations are concerned, it is advised to use an iterativeapproach between dynamic analyses (as described in the Rulesfor Classification of Ships/High Speed, Light Craft and NavalSurface Craft Pt.4 Ch.3 Sec.1 G) and shaft design as presentedin this Classification Note.

Figure 1-2Hysteresis loops of plastification in the notch

Simplified diameter formulae are presented in the Rules forClassification of Ships/High Speed, Light Craft and Naval Sur-face Craft Pt.4 Ch.4 Sec.1 B206-B208 for various commonshaft designs. However, since the simplifications are made �“tothe safe side�”, these formulae will result in somewhat larger di-mensions than the basic criteria presented here.For the purpose of demonstration, a few examples of the cal-culation methods are presented in Appendix A.

1.3 Limits of ApplicationThe criteria presented in this Classification Note apply toshafts with:

�— material of forged or hot rolled steels with minimum ten-sile strength of 400 MPa

�— material tensile strength, B up to 1200 MPa1) and yieldstrength (0,2 % proof stress), y up to 900 MPa.1)

�— no surface hardening2) �— no chrome plating, metal spraying, welds etc. (which will

require special considerations)�— protection against corrosion (through oil, oil based coat-

ing, paint, material selection or dry atmosphere).3)

1) For applications where it may be necessary to take the advantage of ten-sile strength above 800 MPa and yield strength above 600 MPa, materialcleanliness has an increasing importance. Higher cleanliness than speci-fied by material standards may be required. See also Pt.4 Ch.2 Sec3B.

2) However, some general guidelines are given in 3.6 and 4.5.3) For steels as those mentioned in footnote 1) special protection against

corrosion is required. Method of protection is to be approved.


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2. Nomenclature2.1 SymbolsThe symbols in Table 2-1 are used. Only SI units are used.

3. The Low Cycle Fatigue Criterion and Torque Reversal Criterion3.1 Scope and General RemarksThe low cycle fatigue criterion (LCF) and the torque reversalcriterion (load case A and D in 1.2, respectively) are applicablefor shafts subject to load conditions, which accumulate lessthan 104 load cycles. Typical such fatigue load conditions are:Load variations from

�— zero to full forward load�— zero to peak loads such as clutching-in shock loads, elec-

tric motor start-up with star-delta shift, ice shock loads(applicable for ships with ice class notations), etc.

�— full forward load to reversed load (<<103 load cycles).

Bending stress is normally disregarded since they are associat-ed with the number of shaft revolutions (rotary bending) andthus with High Cycle Fatigue; Load case B. Only stochasticbending stresses of relatively high amplitudes but few cycles

Table 2-1 SymbolsSymbol Term Unit

t Geometrical stress concentration factor, torsion

-

b Geometrical stress concentration factor, bending

-

Rotational speed ratio = n/n0 -Repetitive nominal torsional stress range

MPa (= N/mm2)

Nominal mean torsional stress at any load (or r.p.m.)

MPa (= N/mm2)

0 Nominal torsional stress at max-imum continuous power

MPa (= N/mm2)

ice rev Torsional stress due to ice shock while running astern

MPa (= N/mm2)

max Repetitive nominal peak torsion-al stress

MPa (= N/mm2)

vT Permissible torsional vibration stress amplitude for transient condition

MPa (= N/mm2)

max reversed Maximum reversed torsional stress

MPa (= N/mm2)

v Nominal vibratory torsional stress amplitude, ref. Pt.4 Ch.3 Sec.1 G100

MPa (= N/mm2)

f High cycle fatigue strength MPa (= N/mm2)vHC Permissible high cycle torsional

vibration stress amplitudeMPa (= N/mm2)

vLC Permissible low cycle torsional vibration stress amplitude

MPa (= N/mm2)

b Nominal reversed bending stress amplitude (rotating bending stress amplitude)

MPa (= N/mm2)

B Ultimate tensile strength 1) MPa (= N/mm2)f High cycle bending fatigue

strengthMPa (= N/mm2)

y Yield strength or 0.2% proof stress 1)

MPa (= N/mm2)

b Width of square groove (circlip) mmd Minimum shaft diameter at

notchmm

di Inner diameter of shaft at notch mmdh Diameter of hole mmD Bigger diameter in way of notch mme Slot width mmfL( t, y) Notch sensitivity term for low

cycle fatigue-

fL(Ry, B) Surface condition influence term for low cycle fatigue

-

fH ( t,mt) Notch influence term for high cycle torsional fatigue

-

fH ( b,mb) Notch influence term for high cycle bending fatigue

-

fH(r) Size (statistical) influence term for high cycle fatigue

-

fH (Ry, B) Surface condition influence term for high cycle torsional fatigue

-

fH (Ry, B) Surface condition influence term for high cycle bending fatigue

-

kec Eccentricity ratio -KA Application factor, torque range -

KA Application factor, repetitive cy-clic torques

-

KAP Application factor, temporary occasional peak torques

-

KAice Application factor, ice shock tor-ques

-

KH Component influence factor for high cycle bending fatigue

-

KH Component influence factor for high cycle torsional fatigue

-

KL Component influence factor for low cycle fatigue

-

l Total slot length mmmb Notch sensitivity coefficient for

high cycle fatigue (bending)-

mt Notch sensitivity coefficient for high cycle fatigue (torsion)

-

n Actual shaft rotational speed, r.p.m.

minutes-1

n0 Shaft rotational speed at maxi-mum continuous power, r.p.m.

minutes-1

NC Accumulated number of load cy-cles

-

Ne Accumulated number of load cy-cles during one passage up and down

-

P Maximum continuous power kWr Notch radius mmrec Radius to eccentric axial bore mmRa Surface roughness, arithmetical

mean deviation of the profilem

Ry Surface roughness, maximum height of the profile (peak to val-ley)

m

S Safety factor -t Thickness mmTV Vibratory torsional torque ampli-

tude, ref. Pt.4 Ch.3 Sec.1 G100kNm

T0 Torque at maximum continuous power

kNm

Wt Cross sectional modulus (first moment of area), torsion

mm3

1) For representative test pieces according to the Rules for Classification of Ships/High Speed, Light Craft and Naval Surface Craft Pt.2 Ch.2 Sec.5. If the mechanical properties are based on separately forged test pieces, the achieved properties must be reduced empirically in order to represent the properties of the real shaft.

Table 2-1 Symbols (Continued)Symbol Term Unit

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should be taken into account for the LCF (e.g. stochastic bend-ing stresses in water jet shafts due to aeration or cavitation, see3.3 and the Rules for Classification of Ships/High Speed, LightCraft and Naval Surface Craft Pt.4 Ch.4 Sec.1 F301 item 2).A component influence factor for LCF takes into account thedifference in LCF strength of the actual shaft section and a pol-ished plain test specimen push-pull loaded, see 3.5. The re-quired safety factor as given in the prevailing Rules forClassification of Ships/High Speed, Light Craft and Naval Sur-face Craft Pt.4 Ch.4 Sec.1 B203, is selected based on the un-certainties associated with the LCF criterion itself, the loadprediction, the non-cumulative approach and last but not least,the consequence of failure of a shaft disrupting an essentialfunction on board such as propulsion or power generation.

3.2 Basic EquationsThe peak stresses are limited to:For all shafts:

Additionally, for shafts in plants with considerable negativetorque:

max = repetitive nominal peak torsional stress according to3.3

= repetitive nominal torsional stress range accordingto 3.4

y = yield strength or 0.2 % proof stress limited to 0,7 B.This limitation is introduced for the calculation pur-pose only, since further �“irrational�” increase of theyield strength (by the steel heat treatment) increasesthe risk for brittle fracture.

S = required safety factor according to the Rules forClassification of Ships/ High Speed, Light Craftand Naval Surface Craft.

KL = component influence factor for low cycle fatigueaccording to 3.5.

t = Geometrical stress concentration factor in torsion,sec 6.

However, b) is only applicable when the principal stress is re-versed in connection with reversed torsion. I.e. not applicableto e.g. keyways and splines.

t is to be set to unity for shrink fits since the stress concentra-tion factors in Table 6-3 mainly consider the risk for fretting.

3.3 Repetitive Nominal Peak Torsional Stress, maxFor geared plants:

For direct coupled plants:

0 = nominal torsional stress at maximum continuouspower calculated as:

KA = application factor, repetitive cyclic torques definedas:

See Classification Note 41.2KAP = application factor, temporary occasional peak tor-

ques, see Classification Note 41.2KAice = application factor, ice shock torques, see Rules for

Classification of Ships Pt.5 Ch.1 and ClassificationNote 41.2

v = nominal vibratory torsional stress amplitude forcontinuous operation, alternatively, the representa-tive transient vibratory stress amplitude when pass-ing a barred speed range.

The influence of any stochastic bending moments of high am-plitudes (but few cycles) may be taken into account simply bymultiply 0 with:

b,max = nominal reversed bending stress amplitude due tomaximum stochastic bending moments (maximumrotating bending stress amplitude)

This simplification is justified because the estimation of thesestochastic bending moments is very uncertain and conserva-tive, see Rules for Classification of Ships/High Speed, LightCraft and Naval Surface Craft Pt.4 Ch.4 Sec.1 F301 item 2.

3.4 Repetitive Nominal Torsional Stress Range, Only applicable to plants with considerable negative torque.For reversible geared plants:

= Application factor, torque range defined as:

see Figure 3-3.As a safe simplification it may be assumed that

whichever is the highest.For direct coupled plants: = max + | max reversed|see Figure 3-4, 3-5, 3-6, 3-7 and 3-8.

max maximum value of ( + v) in the entirespeed range (for fwd running) or 0KAicewhichever is the highest

max reversed = maximum reversed torsional stress which isthe maximum value of ( + v) in the entirespeed range (for astern running)

= repetitive nominal torsional stress range isequal to the maximum forward torsionalstress plus the absolute value of the maxi-mum reversed torsional stress.As a safe simplification it may be assumedthat ( + or,( 0KAice + ( + v)astern) whichever is thehighest.

max = 0 KA or 0 KAP or 0 KAice whichev-er is the highest, see Figure 3-1 and 3-3.

max maximum value of ( + v) in the en-tire speed range (for fwd running) or

0·KAice whichever is the highest, see Figure 3-2.

a) L

ymax 2SK

b t2 y

S 3------------

T0 106

Wt-----------------

16 d T0 106

d4 di4�–

------------------------------=

T0 Tv+T0

------------------ 0 v+

0----------------=

2

0

maxb,

311

= 0 KA

0

reversedmax 0A(P)(ice)A

KK

KA = 2KA or 2KAP , or KAice + KA(P)


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Figure 3-1Typical torsional stresses in shafts for geared plants with unidirectional torque

Figure 3-2Typical torsional stresses in shafts for direct coupled plants, mean ( ) and upper stresses shown

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Figure 3-3Stress range for reversible geared plants

Figure 3-4Stress range for direct coupled plants with respect to shaft speed, r.p.m.


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Figure 3-5Stress range for direct coupled fixed pitch plants with respect to time (with high transient torsional stress amplitudes)

Figure 3-6Stress range for direct coupled fixed pitch plants with respect to time (with low transient torsional stress amplitudes)

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Figure 3-7Stress range for direct coupled CP-propeller plants with respect to time

Figure 3-8Stress range for direct coupled CP-propeller plants with respect to shaft speed, r.p.m.


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3.5 Component Influence Factor for Low Cycle Fatigue, KLThe component influence factor for low cycle fatigue takesinto account the difference in fatigue strength of the actualshaft component and a polished plain test specimen push �– pullloaded. The empirical formula is:

fL( t, y) = notch influence term for low cycle fatigue in-cluding notch sensitivity, see 3.5.1

fL(Ry, B) = surface condition influence term for low cyclefatigue, see 3.5.2.

3.5.1 Notch influence term for low cycle fatigue, fL( t, y)This notch influence term is a simplified approach that reflectsthe increasing influence of a notch with increasing materialstrength:

t = geometrical stress concentration factor, torsion ac-cording to 6

y = yield strength or 0.2% proof stress (not limited to0.7 B).

3.5.2 Surface condition influence term for low cycle fatigue, fL(Ry, B)This surface condition influence term is a simplified approachthat reflects the increasing influence of surface roughness forhigh material strength.For ordinary steels:

For stainless steel exposed to sea water:

Ry = surface roughness, maximum height of the profile(i.e. peak to valley) 6Ra. Minimum value to beused in the above formulae is RY = 1.0 m

3.6 Surface Hardening/Peening

3.6.1 General remarksSurface hardening is mainly applied in order to increase the

strength in areas with stress concentrations. The increase iscaused by higher hardness as well as compressive residualstresses. Peening (e.g. cold rolling, shot peening, laser peening) is ap-plied in order to induce compressive residual stresses (thework hardening effect is negligible). Both surface hardeningand peening have a transition to the shaft core area where thecompressive residual stresses shift into tensile stresses in orderto balance the outer compressive stresses.It is of vital importance that the applied working stresses do notalter the residual stresses in an unfavourable way, i.e. increas-ing residual tensile stresses. This is normally achieved bymeans of a minimum depth of hardening or peening. The basicprinciple is to calculate working stresses as a function of depthand to compare them with the assessed local strengths (hard-ness) and residual stresses along the same depth, see Figure 3-9.

Figure 3-9Local working- and residual stresses versus local strength (induc-tion hardening shown)

A hardness profile (HV versus depth) can be converted to ten-sile strength ( B) as:

B = 3.2HVIt is also of vital importance that the extension of hardening/peening in an area with stress concentration is duly considered.Any transition where the hardening/peening is ended is likelyto have considerable tensile residual stresses. This forms akind of �“soft spot�”.

3.6.2 Calculation procedureAs of this time subject to special consideration.

KL 1 fL t y+= fL Ry B+

fL t y t 1�– y900---------=

fL Ry B 10 4�–B 200�– Rylog=

yBy logR14.0),(RLf

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4. The High Cycle Fatigue Criterion4.1 Scope and General RemarksThe high cycle fatigue criterion (HCF) is applicable for shaftssubject to load conditions, which accumulate more than 3·106

load cycles, but typically 109 to 1010. It is based on the com-bined vibratory torsional and rotating bending stresses (axialstresses disregarded) relative to the respective component fa-tigue strengths. Typical HCF load conditions are:

�— Load variations due to torsional vibrations applicable forcontinuous operation (basically caused by engine firingpulses), and rotating bending moments due to forces fromgear mesh and forces as listed in the Rules for Classifica-tion of Ships/High Speed, Light Craft & Naval SurfaceCraft Pt.4 Ch.4 Sec.1 F300 1).

�— Load variations (torque and bending moment) due to iceimpacts on the propeller for ice classed vessels assumed toaccumulate ~106 load cycles, see the Rules for Classifica-tion of Ships Pt.5 Ch.1.

The fatigue strength is assumed to continuously drop beyond3·106, i.e. no fatigue limit or run-out is assumed for the trueshaft of ordinary grade for merchant steels. Since the HCF isassociated with higher number of cycles than ordinary fatiguetests, which are terminated for practical considerations at about108 cycles, it calls for a higher safety factor than for LCF. Theprevailing safety factor for HCF is, in addition to the reasonsmentioned in 3.1 for LCF, chosen due to:

�— The statistical nature of fatigue, realizing that the S-Ncurve is drawn as an average, best fit curve determinedwith only a few test specimens. Even for plain test speci-men the fatigue data contain considerable scatter and thestandard deviation increases with number of cycles

�— The inclusions in steel that have an important effect on thefatigue strength and its variability, but even vacuum re-melted steel shows significant scatter in fatigue strength

�— The shortcomings of the commonly used NDT-methods inthe marine industry (ultrasonic- and surface crack testing)and its workmanship on sized shafts (i.e. probability to de-tect defects)

�— Corrosion. Even though the shaft surface has some corro-sion protection (e.g. Dinol or Tectyl), there is always, afterlong time operation, a risk of corrosion on the surface ofinboard shafts, which considerably reduces the high cyclefatigue strength.

Simplified diameter formulae are given in the Rules for Clas-sification of Ships/High Speed, Light Craft and Naval SurfaceCraft Pt.4 Ch.4 Sec.1 B206 to B208 for various common shaftdesigns. However, since the simplifications are made �“to thesafe side�”, these formulae will result in somewhat larger di-mensions than the basic criteria presented here.

4.2 Basic EquationThe high cycle dynamic stresses are limited to:

v = nominal vibratory torsional stress for continuousoperation.

For geared plants: 0 (KA-1) unless a higher valueapplies below the full speed range (~90%-100% en-gine speed). KA is not to be taken lower than 1.1 inorder to cover for load fluctuations due to naviga-tional commands and �“rough sea hunting�”.

For ice class: As long as the natural frequency ofthe "propeller versus engine"-mode is much lowerthan the propeller blade passing frequency (ratio < 50%): 0.5 0 (KAice �–1), otherwise:

0 (KAice - 1)f = high cycle torsional fatigue strength, see 4.3 b = nominal reversed bending stress amplitude, see 4.1f = high cycle bending fatigue strength, see 4.3

S = required safety factor according to the Rules forClassification of Ships/High Speed, Light Craft andNaval Surface Craft.

4.3 High Cycle Fatigue Strengths, f and fThe correlation between fatigue strength and mechanicalstrength is a simple function of the yield strength (or 0.2%proof stress). For ordinary steels, the high cycle fatiguestrengths in bending and torsion are based on empirical formu-lae. For reversed stresses, the fatigue strength is presented as alinear function (e.g. 0.4 y+70) of the main parameter which isassumed to be the yield strength (see VDEH 1983). These�“weighted line of best fit�” equations are based on a systematiccollection of push-pull fatigue strength test results, each with50% survival probability, of different types of steel, see VDEH1983. Since these empirical formulas represent an �“average�” ofaverage fatigue values, they have been reduced by 10%.The mean stress influence is in principle the applied equivalent(von Mises) mean stress, but for marine shafting this means inpractice a function of the mean torsional stress. Finally, the fa-tigue strengths are reduced by the component influence factor.For stainless steel operating in seawater, the fatigue strengthsare dependent on the expected number of load cycles. Test re-sults are used as basis for the given figures, but are somewhaton the safe side in order to compensate for the influence of themean stress (thus the mean stress influence is not incorporatedas a separate parameter in these formulae).For stainless steel not exposed to corrosive environment (e.g.applicable to shaft section inside keyless couplings or keywaycouplings with additional sealing), the criteria for ordinarysteel apply.

2

2

f

b

2

f

v

S1


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y = yield strength or 0.2% proof stress limited to 0.7 B.This limitation is introduced for the calculation pur-pose only, since further �”irrational�” increase of theyield stress (by the steel heat treatment) increases therisk for brittle fracture.

KH = component influence factor for high cycle torsional fa-tigue, see 4.4

KH = component influence factors for high cycle bending fa-tigue, see 4.4

= nominal mean torsional stress at any load (or r.p.m.).

4.4 Component Influence Factor for High Cycle Fatigue, KH and KHThe component influence factor for high cycle fatigue takesinto account the difference in fatigue strength of the actualshaft component and a polished plain test specimen push �– pullloaded. The empirical formulae are:

fH ( t,mt) = notch influence term for high cycle torsionalfatigue, see 4.4.1

fH ( b,mb) = notch influence term for high cycle bendingfatigue, see 4.4.1

fH (r) = size (statistical) influence term for high cy-cle fatigue, see 4.4.2

fH (Ry, B) = surface condition influence term for high cy-cle torsional fatigue, see 4.4.3

fH (Ry, B) = surface condition influence term for high cy-cle bending fatigue, see 4.4.3.

4.4.1 Notch influence term for high cycle fatigue, fH ( t, mt) and fH ( b, mb)This term represents the influence of the geometrical stressconcentration, and the notch sensitivity, m, which represents the notch factor when they are combined as

For multi-radii transitions, see Table 6-1, resulting in low geo-metrical stress concentration factors ( t ~ 1.05 and b ~ 1.10), mt and mb are, as for a plain shaft, = 1.0.

If the calculated value of fH ( t,mt) or fH ( b,mb) is below uni-ty, the value 1.0 is to be used.Torsion:

t = geometrical stress concentration factor, torsion ac-cording to 6

mt = notch sensitivity coefficient for high cycle torsionalfatigue, calculated as:


Bending:

b = geometrical stress concentration factor, bending ac-cording to 6

mb = notch sensitivity coefficient for high cycle bendingfatigue, calculated as:


4.4.2 Size (statistical) influence term for high cycle fatigue, fH(r)This term represents a size factor, which qualitatively takesinto account the amount of material subject to high stresses. Itdoes not represent the stress gradient influence since that is in-cluded in the notch sensitivity coefficient, m, above. Further, itdoes not consider the reduced material strength due to sizesince that is covered by requirements of testing representativespecimen from forgings.

Table 4-1 Fatigue strengthsCycles1) 108 109 - 1010

Material

Ordinary steels

Stainless steel I2)

(in sea water)

Stainless steel II2) and III2)(in sea water)

1) The number of cycles refers to those accumulated in the full load range. 108 is a suitable estimate for e.g. pleasure craft. 109 to 1010 should be used for ferries, etc.

2) Stainless steel I = austenitic steels with 16-18 % Cr, 10-14 % Ni and > 2% Mo with B = 490�–600 MPa and y > 0,45 BStainless steel II = martensitic steels with 15-17 % Cr, 4-6 % Ni and > 1% Mo with B = 850-1000 MPa and y > 0,75 BStainless steel III �– ferritic-austenitic (duplex) steels with 25-27 % Cr, 4-7% Ni and 1-2% Mo with B = 600 to 750 MPa and y > 0,65 B.For other kinds of stainless steel, special consideration applies to fatigue values and pitting resistance, see the Rules for Classification of Ships Pt.4 Ch.4 Sec.1 B204

Torsion:

Bending:

f0.24 y 42 0.15 �–+

KH-----------------------------------------------------=

f0.4 y 70 0.4 �–+

KH-----------------------------------------------=

f105KH----------=

f90

KH----------=

f165KH-----------= f

140KH-----------=

f118KH----------= f

97KH----------=

f184KH-----------= f

150KH-----------=

KH fH t mt fH r fH Ry B+ +=

KH fH b mb fH r fH Ry B+ +=

m

t

ttt m)m,(Hf

r10.05601m

yt

b

bbbH m)m,(f

r20.05601m

yb

r0.01H rf

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February 2007

r = notch radius (use dh/2 or e/2 for radial holes or slots,respectively, see 6.7 or 6.8) or shaft radius whichev-er is smaller. If r >100, 100 is to be used.

4.4.3 Surface condition influence term for high cycle fatigue, fH (Ry, B) and fH (Ry, B)This term takes the surface roughness into account. The sur-face roughness due to the machining constitutes a kind of stressconcentration and reduced fatigue strength compared with apolished test specimen. The surface condition influence ishigher for more notch sensitive, high strength steels. For ordinary steels:Torsion:

Bending:

For stainless steel exposed to sea water:Torsion and Bending:

Ry = Surface roughness, maximum height of the profile(i.e. peak to valley) 6 Ra. Minimum value to beused in the above formulae is RY = 1.0 m

4.5 Surface Hardening/Peening

4.5.1 General remarksSurface hardening is mainly applied in order to increase the fa-tigue strength in areas with stress concentrations. The increaseis caused by higher hardness as well as compressive residualstresses. Peening (e.g. cold rolling, shot peening, laser peen-ing) is applied in order to induce compressive residual stresses(the work hardening effect is negligible). Both surface harden-ing and peening have a transition to the shaft core area wherethe compressive residual stresses shift into tensile stresses inorder to balance the outer compressive stresses.It is of vital importance that the extension of hardening/peen-ing in an area with stress concentration is duly considered. Anytransition where the hardening/peening is ended is likely tohave considerable tensile residual stresses. This forms a kindof �“soft spot�”.It is of vital importance that the applied dynamic workingstresses do not exceed the local fatigue strengths, neither at thesurface nor in the depth, especially at the transition to the core.This is may be achieved by means of a minimum depth of hard-ening or peening. The basic principle is to calculate both themean and dynamic working stresses as functions of depth, seeFigure 4-1. The residual stresses as a function of depth are thenadded to the mean working stresses. The local fatigue strength(for reversed stresses, i.e. R = -1) is determined as a functionof depth, depending on the local hardness and yield strength,respectively.

Figure 4-1Local dynamic working stresses and residual stresses versus localfatigue strength (induction hardening shown)

The criterion for safety against fatigue is then applied stepwisefrom surface to core.

4.5.2 Calculation procedureAs of this time subject to special consideration.

5. The Transient Vibration Criterion5.1 Scope and General RemarksAs mentioned in 1.2 load case C, the transient vibration crite-rion, which is applicable to vibration levels when passingthrough a barred speed range, applies in practice only to directcoupled plants. The permissible torsional vibration stress amplitudes, vT aredetermined by logarithmic interpolation between the low cyclecriterion, LCF ( vLC) and the high cycle criterion, HCF ( vHC)(adjusted down to 3·106 load cycles simply by reducing theHCF safety factor to 1.5) depending on the foreseen accumu-lated number of cycles, NC during the lifetime of the plant. Barred speed ranges are only permitted below a speed ratio

= n/n0 = 0.8 where n = actual r.p.m. and n0 = r.p.m. at fullpower.Simplified calculation method is given in the Rules for Classi-fication of Ships / High Speed, Light Craft and Naval SurfaceCraft Pt.4 Ch.4 Sec.1 B208 for some specific propeller and in-termediate shaft designs under the assumption that the barredspeed range is passed rapidly both upwards and downwards.

5.2 Basic Equation

vHC = permissible high cycle torsional vibration stress ampli-tude calculated according to the criteria in 4 but with a6.25% lower safety factor and without the bendingstress influence. For propeller shafts in way of and aftof the aft stern tube bearing the bending influence iscovered by an increase of S by 0.05.

vLC = permissible low cycle torsional vibration stress ampli-tude calculated according to the criteria in 3.2 a). Forpropeller shafts in way of and aft of the aft stern tubebearing the bending influence is covered by an in-crease of S by 0.05.

NC = accumulated number of load cycles (104<NC<3 106).For determination of the relevant NC, see the Rules for

yB4

ByH Rlog200103),(Rf

yB4

ByH logR200104),(Rf

yByHByH logR35.0),(R),(R ff

vLC

vHC

vHC

vLC log4.0

4C

vLC

0.4log

C

6

vHCvT 10N

N103


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DET NORSKE VERITAS

Classification of Ships/High Speed, Light Craft andNaval Surface Craft Pt.4 Ch.3 Sec.1 G401, or a de-tailed method in 5.2.1 and 5.2.2.

5.2.1 The accumulated number of cycles, Nc The accumulated number of cycles, Nc is to be based upon anequivalent number of cycles that results in the same accumu-lated partial damage (Miner�’s theory) as the real load spec-trum. This means that each cycle with somewhat lower stress

amplitudes will not be counted equally as one with the maxi-mum amplitude, but will be �“levelled�” in proportion to thenumber of cycles to failure for the maximum amplitude. Sincelow amplitudes do not contribute to the damage sum, this isdone in steps of 10% down to 60% of the maximum amplitude,see Figure 5-1 and Figure 5-2. The procedure is as follows:

1) Record torsional vibrations, see Figure 5-1 and Figure 5-2.

Figure 5-1v during stopping

Figure 5-2v during starting

2) Determine the maximum double amplitude 2 v consider-ing both start and stop and use this as reference for 100%.

3) Draw lines in both start and stop records with double am-plitude 100%, 90%, 80% and so on, see Figure 5-1 andFigure 5-2.

4) Count number of cycles N100 between 90% and 100%.This range will then be considered as amplitudes of 100%.Count number of cycles N90 between 80% and 90% (high-er amplitudes are not to be counted again!). This range willthen be considered as amplitudes of 90%, and so on.

5) Determine the double logarithmic -N curve by means ofvLC and vHC calculated according to 5.2, see Figure 5-3.

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February 2007

Figure 5-3Double logarithmic -N curve

With this characteristic the equivalent number of cycleswith 100% amplitude are calculated by means of divisionof the counted cycles with denominators determined as:

- 90% level with denominator



6) Calculate the equivalent number of cycles for one passageup and down as:

7) Determine accumulated number of cycles Nc as Ne timesnumber of passages, see 8.

8) The next question is; how many passages through thebarred speed range (up and down together) is to be fore-seen for the ship in question? DNV uses the followingguidelines, if not otherwise is substantiated:

�— Large carriers with fixed pitch propeller and manoeu-vring speed below the barred speed range respectivelywith controllable pitch propeller. Once every week,resulting in a total of 1 000 passages.

�— Large carriers with fixed pitch propeller and manoeu-vring speed above the barred speed range. Five timesevery week, resulting in 5 000 passages.

�— Vessels for short trade, once every day, resulting in7000 passages.

�— Ferries for short distances, 20 times per day, resultingin 150 000 passages.

Even though the above numbers are rough estimates oneshould bear in mind that logarithmic scales are used, sothat a mistake by a factor of 2 or 3 on the number passageswill have very little influence on the corresponding per-missible stresses. It is more important to put effort in try-ing to estimate the number of load cycles for each passagecorrectly by simulation technique or measurements fromsimilar plants in service.

6. The Geometrical Stress Concentration Factors6.1 Definition and General RemarksThe stress concentration factors, are defined:

�— for bending as the ratio between the maximum principalstress and the nominal bending stress

�— for torsion as the ratio between the maximum principalstress divided by or the maximum shear stress (which-ever is the highest) and the nominal torsional stress.

Stress concentration factors may be taken from well docu-mented measurements, finite element calculations, recognisedliterature or from the formulae summarised in 6.2 to 6.10. Only stress concentrations that are typical for shafting in ma-rine applications are described here.Note that these stress concentration factors are based on nom-inal stresses calculated by the minimum shaft diameter, d at thenotch, and considering the influence of a central axial holewith diameter di. The reduction of the first moment of area dueto keyway(s), longitudinal slot(s) or radial hole(s) is not con-sidered. It is recommended to design with generous fillet radii, in par-ticular in areas where the checking of the radius may be diffi-cult (e.g. keyways).

6.2 Shoulder Fillets and Flange Fillets The following stress concentration factors are valid for bothshoulder fillets and flange fillets provided that the inner diam-eter di is less than 0.5d:For shoulder fillets: Higher stress concentration factors apply if a part is shrunk onto the bigger diameter, D of the shoulder as indicated in the lastFigure in 6.3. This increase may be simplified by using a 10%larger value for D in the formulae in Table 6-1.

vHC

vLClog

1

3.1

vHC

vLClog

1

7.1

vHC

vLClog

1

4.2

N = N100 +

vHC

vLClog

1

90

1.3

N +

vHC

vLClog

1

80

1.7

N+

vHC

vLClog

1

70

2.4

N

3


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DET NORSKE VERITAS

6.3 U-Notch The stress concentration factors in Table 6-2 are valid for U-

notches (grooves), provided that the inner diameter, di is lessthan 0.5d.

Table 6-1 Stress concentration factors, shoulder fillets and flange fillets Design For low and high cycle fatigue

For combinations of small relative flange thicknesses (t/d) and small relative fillet radii (r/d) as e.g. (r + t)/d < 0.35 the t increases. This is to be taken into account by multiplying

t with 1 + (0.08 d/(r + t))2

Multiradii transitions with flange thickness t 0.2 d:

This may be obtained by e.g. starting with r1 = 2.5 d tangentially to the shaft over a 5 sector, followed by r2 = 0.65 d over 20 and finally r3 = 0.09 d over 65

D

r

d

D

t

r

db 1 1

1.24 rD d�–------------- 11.6 r

d--- 1 2 r

d---+

21.6 d

D---- r

D d�–-------------

3+ +

-----------------------------------------------------------------------------------------------------------------------+=

t 1 1

6.8 rD d�–------------- 38 r

d--- 1 2 r

d---+

24 d

D---- r

D d�–-------------

2+ +

------------------------------------------------------------------------------------------------------------+=

t

r3 r2 r1

20o5o

d/2

t 1.05

b 1.1

Table 6-2 Stress concentration factors, U-notch Design For low and high cycle fatigue

D d

r

b 1 1

0.4 rD d�–------------- 5.5 r

d--- 1 2 r

d---+

2+

-----------------------------------------------------------------------+=

t 1 1

1.4 rD d�–------------- 20.6 r

d--- 1 2 r

d---+

2+

--------------------------------------------------------------------------+=

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February 2007

6.4 Step with Undercut Stress concentration for a step with undercut may be deter-mined by interpolation between a shoulder fillet (6.2) and aU-notch (6.3), see Figure 6-1:

b(6.2) = stress concentration factor for a shoulder fillet,bending, see 6.2

b(6.3) = stress concentration factor for a U-notch, bend-ing, see 6.3

t(6.2) = stress concentration for a shoulder fillet, torsion,see 6.2.

Dimensions, see Figure 6-1.

Figure 6-1Stress concentration factors, step with undercut

b b 6.2 b 6.3 b 6.2�–d1 d�–D d�–--------------+=

t 1.04 t 6.2=

D d

r

b(6.2)

Shoulder fillet

with D d

rb(6.3)

U-notch

dD d 1

r

b

Step withundercut


February 2007

DET NORSKE VERITAS

6.5 Shrink FitsThe stress concentration factors in Table 6-3 is valid for shrinkfits. These factors mainly consider the risk for fretting and is

therefore not considered as geometrical stress consentrationfactors.

For hollow shafts with a relatively large inner diameter thenotch factors are higher. However, this can be compensated by

means of an expansion sleeve in the inner bore.

Table 6-3 Stress concentration factors, shrink fits

Design

For low cycle fatigue For high cycle fatigue

Keyway(s)

1.4Two keyways

Due to reduction in the cross section area, the above values are to be increased by 15%

The keyway(s) is also to be considered, see 6.6

Keyless

1.4

With relief

For designs with suitable relief grooves or shoulder steps at the end of the hub, the stress concentration may be considerably reduced, even close to unity if d = 1.1d1 and r = 2 (d-d1) and an axial overshoot near zero but not less than zero. This is valid only if no relative micro-movement between the 2 parts can be verified; see the Rules for Classification of Ships/High Speed, Light Craft and Naval Surface Craft Pt.4 Ch.4 Sec.1 B402 regarding surface pres-sure. However, the groove or shoulder itself is to be calculated as such.

1) These notch factors also contain the influence of typical surface roughness and size influence. Hence the surface roughness terms and size influence term in the component influence factors are to be disregarded.

t 1 b

mb------- KH 1= t

mt------ KH 1=

d

1.4 B500---------+ 0.9 B

1000------------+

d

1.05 B500---------+ 0.71

1.2 B1000

----------------+

d 1

r

d

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February 2007

6.6 Keyways The stress concentration factors in the bottom of a keyway are

given in Table 6-4.

For hollow shafts with inner diameter greater than 0.5 d, the in-crease of nominal stresses due to the keyway(s) has to be con-sidered.Note that these stress concentration factors are to be used to-gether with the real nominal stresses applied at the end of thekeyway, see Figure 6-2, i.e. the influence of hub stiffening inbending and torque transmission due to friction may be consid-ered.The shrink fit between shaft and hub is also to be considered,see 6.5.

Figure 6-2Keyway Inside Hub

Table 6-4 Stress concentration factors, keywaysDesign For low and high cycle fatigueSemicircular ends

(for this formula gives too high stress concentration)


Sled runner ends


r

d

b 1.4 + 0.015 dr---=

rd--- 0.006

t 2.1 + 0.012 dr---=

rd--- 0.0075

r

d

b 1.4=

t 2.1 + 0.012 dr---=

rd--- 0.0075


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DET NORSKE VERITAS

6.7 Radial Holes For a radial hole with diameter dh < 0.2·d and a central bore di< 0.5·d (i.e. 2 outlet holes diametrically opposed), the stress

concentration factors at the surface in the radial hole outletsare:

The notch radius to be used in the formulae for component in-fluence factors in 4.4 is the radial hole radius = and the nominal stress is based on thefull section.

Note:The stress concentration in the intersection itself can be muchhigher and is subject to special consideration. For plants with

high torsional vibrations, such as direct coupled plants, no sharpedges are acceptable.

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

6.8 Longitudinal SlotFor longitudinal slots, the stress concentration factor is givenin Table 6-6.

The radius to be used in the formulae for component influencefactors in 4.4 is e/2.

Table 6-5 Stress concetration factors, radial holesDesign For low and high cycle fatigue

or simplified to:

As an approximation the same formulae may be used if only one outlet.With intersecting eccentric axial bore Multiply the above formulae with:

�— For kec > 0,85 the stress concentration will be specially considered.

di

dh

d

b 3 5.9 dhd-----�– 34.6

dhd-----

2+=

3.2t

dh

di d

rec

1 kec4+

kec eccentricity ratio�–2rec

d----------=

2d h

Table 6-6 Stress concetration factors, longitudinal slotsDesign For low and high cycle fatigue

= stress concentration factor for a radial hole with dh = e, in torsion, see 6.5.The formula is valid for:�— for outlets each 180 degrees and for outlets each 120 degrees�— slots with semicircular ends, i.e. r = e/2. A multi-radii slot end can reduce the

local stresses. �— no edge rounding (except chamfering), as any edge rounding increases the t

slightly.d di

l/2e

r

t

de

ddd

el

i157.0)5.6(tt

(6.5)t

2i

2h

2hh

t dd

dd10

dd15

dd32.3

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February 2007

6.9 SplinesThe stress concentration factors1) given in Table 6-7 may be

applied.

Surface hardened splines are subject to special consideration,see 3.6 and 4.5.Note that higher notch factors apply if the hub is very rigid, i.e.the torque transmission is concentrated at the hub edge, seeFigure 6.3.

Figure 6-3Spline connection with very rigid hub

Table 6-7 Stress concentration factors, splines

Design

For low cycle fatigue For high cycle fatigue

Involute splines:

1.15

Non-involute splines:

Higher notch factors apply (10%)

1) Note that the nominal stresses are based on the root diameter of the splines.2) These notch factors also contain the influence of typical surface roughness and size influence. Hence the surface roughness terms and size influence term

in the component influence factors are to be disregarded.

t 2

bmb------- KH 2=

tmt------ KH 2=

0.96 y1000------------+ 0.92 y

1500------------+


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DET NORSKE VERITAS

6.10 Square Groove (Circlip)The stress concentration factors given in Table 6-8 may be ap-plied.

Table 6-8 Stress concentration factors, square grove (circlip)Design For low cycle

fatigue For high cycle fatigue

3.0Other standards as e.g. DIN 743 indicate upper limits as 4 respectively 2.5. These rec-ommendations are probably based on fatigue test result, but without having checked the likely influence of compressive residual stresses from machining. Residual stresses in sharp notches can have a strong influence, and it is risky to rely on upper limits for such notches.

�— If , multiply the above formulae with:

tb

mb------- t

mt------

D d

b

r

1.03 0.4724 D d�–

r 2.9 100.514 0.00152 y+�–

+----------------------------------------------------------------------+ 1.48 0.1013 D d�–

r 100.514 0.00152 y+�–

+----------------------------------------------------------+

2bD d�–------------- 1.4 0.94 b

D d�–-------------

0.2�–

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February 2007

Appendix AExamplesThe examples presented in this Appendix are solely meant fordemonstration of the calculation method and are not to be used

for dimensioning purposes as fictitious loads and dimensionsare used.

Figure A-1Geared CP-propeller plant

A.1 Calculation example 1: Propeller shaft in a geared, controllable pitch plant with no ICE-classThe shaft arrangement is shown in Figure A-1.

A.1.1 In way of propeller flange

Figure A-2Propeller flange fillet

Relevant loads:

�— Torque at maximum continuous power, T0 = 62 kNm�— Application factors, repetitive cyclic torques (taken from

the torsional vibration calculations):In normal operation condition, KAnorm. = 1.2In misfiring condition, KAmisf. = 1.3

�— Rotating bending moment, Mb = 24.8 kNm (taken fromthe shaft alignment calculations including the bending mo-ment due to thrust eccentricity or as a simple and conserv-ative estimation by 0.4T0)

�— No significant clutching-in shock loads.

Relevant dimensions:

�— Outer diameter, d = 220 mm�— Inner diameter, di = 100 mm�— Flange diameter, D = 475 mm�— Radius of flange fillet, r = 30 mm�— Flange thickness, t = 65 mm�— Surface roughness in flange fillet, Ra = 0.8 m,

i.e. Ry 4.8 m.

Relevant material data:

�— EN 10083-1: C45E +N (1.1191) with the following me-chanical properties for representative test pieces1):Ultimate tensile strength, B 560 N/mm2Yield strength, y 275 N/mm2.

1) See the Rules for Classification of Ships Pt.2 Ch.2 Sec.5.

Calculations:The low cycle criterion (see 3.2):

�— Nominal torsional stress at maximum continuous power(see 3.3):

�— Application factor (maximum KA to be used in this crite-rion, see 3.3):KAmisf. = 1.3

�— Yield strength:y = 275 N/mm2

�— Safety factor (see the Rules for Classification of Ships/High Speed, Light Craft and Naval Surface Craft Pt.4 Ch.4Sec.1 B205):S = 1.25

�— Geometrical stress concentration factor, torsion (see 6.2):t = 1.33

�— Component influence factor for low cycle fatigue (see3.5):

A.1.2A.1.1

max 0 KAy

2 S KL------------------=

016 220 62 106

2204 1004�–------------------------------------------- N mm2 30.98 N mm2==


February 2007

DET NORSKE VERITAS

Consequently:

40.3 N/mm2 97.3 N/mm2

or actual safety factor, S = 3.0Conclusion: Criterion fulfilled.

The high cycle criterion (see 4.2):

�— Nominal vibratory torsional stress for continuous opera-tion (see 4.2):

v = (KAnorm. �– 1) 0 = (1.2 - 1) 30.98 N/mm2

= 6.20 N/mm2

�— Nominal reversed bending stress amplitude:

�— Geometrical stress concentration factors (see 6.2):Torsion: t = 1.33Bending: b = 1.61

�— Notch sensitivity coefficients (see 4.4.1):

�— Component influence factors for high cycle fatigue (see4.4):

�— High cycle fatigue strengths (see 4.3):

�— Safety factor (see the Rules for Classification of Ships/High Speed, Light Craft and Naval Surface Craft, Pt.4Ch.4 Sec.1 B205): S = 1.6

Consequently:

or actual safety factor, S = 3.8

Conclusion: Acceptable, and the dimensions could be reducedprovided that the stern tube bearing pressure remains withinthe acceptable rule limits in all operating conditions (full andidle speed respectively).

The transient vibration criterion (see 5):Not applicable since this is a geared plant with no barred speedrange.

A.1.2 In way of shaft coupling:

Figure A-3Propeller shaft end at shrink fit coupling

Relevant loads:

�— Torque at maximum continuous power, T0 = 62 kNm�— Application factors, repetitive cyclic torques (taken from

the torsional vibration calculations):In normal operation condition, KAnorm. = 1.2In misfiring condition, KAmisf. = 1.3

�— Negligible rotating bending moment (confirmed by shaftalignment analysis)

�— No significant clutching-in shock loads.


�— Outer diameter, d = 200 mm�— Inner diameter, di = 93 mm�— Surface roughness, Ra = 1.6 m, i.e. Ry 9.6 m.

Relevant material data:EN 10083-1: C45E +N (1.1191) with the following mechani-cal properties for representative test pieces1):Ultimate tensile strength, B 560 N/mm2.Yield strength, y 275 N/mm

1) See the Rules for Classification of Ships/High Speed, Light Craft and Na-val Surface Craft Pt.2 Ch.2 Sec.5

KL 1 + (1.33 -1)275900--------- 10 4�– 560 200�– 4.8 1.13=log+=

22 mmN13.125.12

275mmN30.981.3

v

f-----

2b

f------

21

S2-----+

b32 220 24.8 106

2204 1004�–----------------------------------------------- N/mm2 24.8 N/mm2= =

Torsion: mt 1 60275--------- 0.05�– 1

30------ 1.03=+=

Bending: mb 1 60275--------- 0.05�– 2

30------ 1.04=+=

Torsion: KH1.331.03---------- 0.01 30 3 10 4�– 560 200�– 4.8 1.42=log+ +=

Bending: KH1.611.04---------- 0.01 30 4 10 4�– 560 200�– 4.8 1.70=log+ +=

Torsion: f0.24 275 42+ 0.15�– 30.98

1.42----------------------------------------------------------------------= 72.78 N/mm2=

Bending: f0.4 275 70+ 0.4�– 30.98

1.70----------------------------------------------------------------= 98.59 N/mm2=

6.2072.78-------------

2 24.7898.59-------------

2+ 0.07 1

1.6-------

20.39==

Ø20

0

400

1.6

15

R3

-0.0

30-0

.061

93

DET NORSKE VERITAS


February 2007

Calculations:The low cycle criterion (see 3.2):

�— Nominal torsional stress at maximum continuous power(see 3.3):

�— Application factor (maximum KA to be used in this crite-rion, see 3.3):KAmisf. = 1.3

�— Yield strength:

y = 275 N/mm2

�— Safety factor (see the Rules for Classification of Ships/High Speed, Light Craft and Naval Surface Craft Pt.4Ch.4 Sec.1 B205):S = 1.25

�— Geometrical stress concentration factor, torsion (see 6.5):

t= 1.4�— Component influence factor for low cycle fatigue (see

3.5):

Consequently:

or actual safety factor, S = 2.3.

Conclusion: Criterion fulfilled.


�— Nominal vibratory torsional stress for continuous opera-tion (see 4.2):

v = (KAnorm.�– 1) 0 = (1.2 - 1) 41.41 N/mm2 = 8.28 N/mm2

�— Geometrical stress concentration factor, torsion (see 6.5):

�— This notch factor contains also the influence of the surfaceroughness and size influence. Hence, the component influ-ence factor for high cycle torsional fatigue (see 4.4): KH = 1.38

�— High cycle torsional fatigue strength (see 4.3):

�— since rotating bending moment is negligible at

this section of the shaft�— Safety factor (see the Rules for Classification of Ships Pt.4

Ch.4 Sec.1 B205):S = 1.6

Consequently:

for actual safety factor, S = 8.8Conclusion: Acceptable, and it means that the shaft dimen-sions may be reduced, provided that acceptable friction torquetransmission through the shrink fit coupling is maintained.

The transient vibration criterion (see 5):Not applicable since this is a geared plant with no barred speedranges.

max 0 KAy

2 S KL------------------=

016 200 62 106

2004 934�–------------------------------------------- N/mm2 41.41 N/mm2= =

KL 1 1.4 1�– 275900---------+ 1.12= =

22

22

mmN2.98mmN8.53

mmN12.125.12

275mmN.41141.3

v

f----- b

f------ 1

S2-----+

tmt------ 0.71 1.2 560

1000---------------------+ 1.38= =

Torsion: f0.24 275 42+ 0.15�– 41.41

1.38---------------------------------------------------------------------- N mm2=

73.76 N/mm2=b

f------

20,

8.2873.76------------- 0.013 1

1.6-------

20.39= =


February 2007

DET NORSKE VERITAS

A.2 Calculation example 2: Oil distribution shaft in a direct coupled, controllable pitch propeller plant with no ICE-class:The shaft arrangement is shown in Figure A-4.

Main engine: 6 cylinder MAN B&W 6S42MC rated to 5300kW at 120 r.p.m.Control system: Only pitch control, i.e engine running at con-stant speed. No combinator mode.

Figure A-4Direct coupled CP-propeller plant

Figure A-5Design of slot in O.D.-shaft

A.2.1 In way of slot in the oil distribution shaft:Relevant loads:

�— Torque at maximum continuous power, T0 = 421.8 kNm atn0 = 120 r.p.m.

�— Negligible rotating bending moment (confirmed by shaftalignment analysis)

�— Calculated steady state torsional vibrations (full pitch) ac-cording to Figure A-6.

A.2.1

R30SEEN FROM

X

100 110 370

Ra 1.6Ra1.6

Y

Y

x

Ø 5

20

Ø 4

02

Ra 6.3

Ra 1.6 Ra 1.660

SECTION Y -Y

DET NORSKE VERITAS


February 2007

Figure A-6Torsional vibration calculations in normal condition with full pitch

In this case the calculated torsional vibration level for zeropitch reaches approximately the same level as for full pitch, seeFigure A-9. The reason for this is that both the engine excita-tion and propeller damping is lower for zero pitch than for fullpitch. The 6th order resonance speed is 72 r.p.m. (1. mode nat-

ural frequency = 433.6 vibr/min) for full pitch and 74 r.p.m. forzero pitch (1. mode natural frequency = 444 vibration/min.). Inmisfiring condition, the torsional vibration stresses due to the6th order excitation will be lower than for normal condition,due to one cylinder with no ignition, see Figure A-7.

Figure A-7Torsional vibration calculations in misfiring condition with reduced pitch corresponding to 3500 kW at 120 r.p.m.

Synthesis


February 2007

DET NORSKE VERITAS


�— Number of slots: 3 (each 120 ) �— Outer diameter, d = 520 mm�— Inner diameter, di = 402 mm�— Length of slot, l = 370 mm�— Width of slot, e = 60 mm�— Surface roughness at the edge of the slot, Ra = 1.6 m, i.e.

Ry 9.6 m


�— EN 10083-1: 42CrMo4 +QT (1.7225) with the followingmechanical properties as specified by the designer for rep-resentative test pieces1):Ultimate tensile strength, B 750 N/mm2Yield strength, y 450 N/mm2

1) See the Rules for Classification of Ships/High Speed, Light Craftand Naval Surface Craft Pt.2 Ch.2 Sec.5.

Calculations:The low cycle criteria (see 3.2):

a) Peak stress:

�— Nominal torsional stress at maximum continuouspower (see 3.3):

�— Maximum value of ( + v) in the entire speed range(see 3.3):In this case there are three potential maximum valuesof ( + v):

i) Running with full pitch at n0 = 120 r.p.m. in nor-mal condition:Nominal torsional stress: 0 = 23.77 N/mm2 andvibratory torsional stress (see Figure A-6):

v = 8.6 N/mm2. Thus, in this case ( + v) = 32.4 N/mm2.

ii) Running with full pitch at n0 = 120 r.p.m in mis-firing condition (one cylinder without combus-tion):Since the thermal load on the remaining 5 cylin-ders will be too high at 120 r.p.m, the power (bypitch reduction) has to be reduced to 3500 kW.Nominal torsional stress at 120 r.p.m is then:

Vibratory torsional stress at 120 r.p.m in misfiringcondition is according to the torsional vibrationcalculations: v misf. = 11.7 N/mm2. Thus, in thiscase ( + v) = 27.4 N/mm2.

iii) Running (accidentally*) at 6th order resonancespeed = 74 r.p.m with 0-pitch:Nominal torsional stress at 74 r.p.m with 0-pitchis assumed to be approximately 15% of the nom-inal torsional stress with full pitch:

Vibratory torsional stress at 74 r.p.m with 0-pitchis according to the torsional vibration calculations(see Fig.A-9):

Thus, in this case ( + v) = 41.36 N/mm2.

The maximum torsional stress occurs in case iii):

�— Yield strength:�— Safety factor (see the Rules for Classification of

Ships/High Speed, Light Craft and Naval SurfaceCraft Pt.4 Ch.4 Sec.1 B205): S = 1.25

�— Geometrical stress concentration factor, torsion (see6.8): t = 4.33

�— Component influence factor for low cycle fatigue (see3.5):

Consequently:

or actual safety factor, S = 2.0

Conclusion: Criterion fulfilled.b) Torque reversal:

�— The repetitive nominal stress range (see 3.4):Forward peak torsional stress, see criterion a):

Maximum reversed torsional stress:

Thus,

�— Safety factor (see the Rules for Classification ofShips/ High Speed, Light Craft and Naval SurfaceCraft Pt.4 Ch.4 Sec.1 B205):S = 1.25

Consequently:4.33 80 N/mm2 = 346.4 N/mm2 N/mm2 =415.6 N/mm2

or actual safety factor, S = 1.5.Conclusion: Criterion fulfilled even when running (acci-dentally) at 74 r.p.m. with 0-pitch as discussed in criteriona) (running steady state at 74 r.p.m. is not relevant). Theactual repetitive nominal stress range when passing quick-ly through the barred speed range will be lower, 33 N/mm2according to the measurements, see figure A-10. So the ac-tual safety factor is even higher than 1.5.

max v+ maxy

2 S KL------------------=

016 520 421.8 106

5204 4024�–--------------------------------------------------N mm2 23.77 N/mm2= =

0misf.35005300------------ 23.77 N/mm2 15.70 N/mm2= =

74r.p.m. 0.15 74120---------

223.77 N/mm2 1.36 N/mm2= =

* Accidentally, since running steady at 74 r.p.m is not relevantas the high cycle criterion below concludes with a barredspeed range around 74 r.p.m.

v74r.p.m. 40.0 N/mm2.=

v+ max v+ 74r.p.m 41.36 N/mm2= =

y 450 N/mm2=

KL 1 4.33 1�– 450900--------- 10 4�– 750 200�– 9.6log++ 2.72= =

41.36 N/mm2 4502 1.25 2.72---------------------------------N/mm2 66.2 N mm2=

t t= max max r eversed+2 y

S 3----------

max 41.3N mm2=

max r eversed 40 1�– .28 N mm2 38.7 N mm2= =

41.3 38.7 N mm2+ 80 N mm2= =

325.14502

DET NORSKE VERITAS


February 2007


�— Notch sensitivity coefficient for high cycle torsional fa-tigue (see 4.4.1):

�— Component influence factor for high cycle torsional fa-tigue (see 4.4):


is the speed ratio =


this section of the shaft�— Safety factor (see the Rules for Classification of Ships/

High Speed, Light Craft and Naval Surface Craft Pt.4 Ch.4Sec.1 B205): S = 1.6

Consequently:The permissible torsional vibratory stress as a function of shaftspeed is then:

This is then plotted as a function of engine speed in the calcu-lated torsional vibration diagram, see Figure A-8.

Figure A-8Calculated steady state torsional vibration stress and stress limit for continuous operation

Conclusion: The calculated stresses are higher than the al-lowable stresses for continuous operation around the 6th orderresonance at 74 r.p.m. This means that a barred speed range hasto be introduced from 67 to 79 r.p.m.. In order to take into ac-count some speed hunting and the inaccuracies of the calcula-tions and engine tachometer, the barred speed range should beextended some r.p.m. (~ 2%). In this case the barred speedrange should be set to 65 to 81 r.p.m.. However, this is to be

finally confirmed by torsional vibration measurements onboard during the sea trials.The transient vibration criterion (see 5):

v

f-----

2b

f------

21

S2-----+

mt 1 60450--------- 0.05�– 1

30------+ 1.02= =

KH4.331.02---------- 0.01 30 3 10 4�– 750 200�– 9.6log+ + 4.46= =

f0.24 450 42 0.15�–+ 0.15 2 23.77

4.46------------------------------------------------------------------------------------------------ N/mm2

33.6 0.12 2 N/mm2�–

=

=

nn0-----

b

f------

2 0

vHC33.6 0.12 2�–

1.6--------------------------------- N/mm2 21.0 0.075 2 N/mm2�–=

vT vHC3 106

NC----------------

0.4 log vLC

vHC-------------

vLCNC

104--------

0.4 log vHC

vLC-------------

=


February 2007

DET NORSKE VERITAS

�— Permissible high cycle torsional vibration stress, vHC, ascalculated above but with a safety factor of 1.5:


�— Accumulated number of load cycles when passing throughthe barred speed range, NC is assumed to be less than 105for this plant since the passage is with zero pitch.

�— Permissible low cycle torsional vibratory stress, vLC, ascalculated above:

Consequently:The permissible torsional vibratory stress in transient condi-tion as a function of shaft speed is then:

This is then plotted as a function of engine speed in the calcu-lated torsional vibration diagram:, see Figure A-9.

Figure A-9Calculated steady state torsional vibration stress with 0-pitch and stress limit for transient operation

Conclusion:If we compare the stress limit for transient operation with thecalculated steady state stress, we see from Fig.A-9 that the lim-it is almost reached at 74 r.p.m. However, when quickly pass-ing the barred speed range with 0-pitch, the stresses will nothave sufficient time to build up to the calculated steady statelevel. Also the expected number of cycles of significant loadmay be lower than the assumed 105 when quickly passingthrough the barred speed range. This is to be verified by meas-

urements on board during sea trials or be based on experiencewith similar plants. The measured shaft stresses converted to the O.D. shaft duringthe sea trials when passing quickly through the barred speedrange with zero pitch are shown on Figure A-10. The enginewas taken up to 64 r.p.m with zero pitch and kept there someseconds. Then the engine was brought as quickly as possiblewith zero pitch through the barred speed range to 90 r.p.m, seeFigure A-10.

vHC33.6 0.12 2�–

1.5---------------------------------N/mm2 23.4 0.08 2 N/mm2�–= =

nn0-----

vLC 66.2 0.15 2�– 23.77 N/mm2

66.2 3.57 2 N/mm2�–

=

=

vT 22.4 0.08 2�– 300.4 log 66.2 3.57 2�–

22.4 0.08 2�–----------------------------------

N/mm2

DET NORSKE VERITAS


February 2007

Figure A-10Measured transient torsional stress

Measurements were also carried out when slowing downthrough the barred speed range, showing similar stress cycles.From these measurements one can observe the following:

a) The maximum torsional vibratory stress amplitude is16.5 N/mm2, which is 41% of the calculated steady stateamplitude.

b) The equivalent number of cycles Nc with 100% amplitudefor one passage up and down are calculated according toitem 5.2.1 6) assuming that the number of cycles for start-ing is the same as for stopping:

The accumulated number of cycles Nc is then calculated asNe times expected number of passages during the shipslife-time, see 5.2.2: Once every week, i.e. 1000 passages, gives Nc = 7000 loadcycles, which is far less than the assumed 105 load cycles.Consequently, also the transient vibration criterion isfulfilled. N = 2 (N100 +

vHC

vLClog

1

90

1,3

N +

vHC

vLClog

1

80

1,7

N+

vHC

vLClog

1

70

2,4

N)

= 2 (2 +

22.3764.84

log

1

1.3

2 +

22.3764.84

log

1

1.7

0 +

22.3764.84

log

1

4.2

1)=2 (2 + 1.1 + 0 + 0.15) 7


February 2007

DET NORSKE VERITAS

A.3 Calculation example 3: Intermediate shaft in a direct coupled, fixed pitch plant with no ICE-class:The shaft arrangement is shown in Figure A-11.

Main engine: 5 cylinder MAN B&W 5L70MCE rated to 9000kW at 105 r.p.m.A Viscous type torsional vibration damper is mounted and thediameter of the fixed pitch propeller is 7,0 m.

Figure A-11Direct coupled fixed pitch propeller plant

A.3.1 In way of flange fillet of the intermediate shaft, first design proposal:First proposal for intermediate shaft design is a 500 mm diam-eter shaft with a flange fillet of multiradii design made fromunalloyed carbon steel JIS G3201 SF 590 A. Other relevant di-mensions are shown in Figure A-12.

Figure A-12Intermediate shaft flange fillet for ø500 mm shaft

Relevant loads:

�— Torque at maximum continuous power, T0 = 818.5 kNm at105 r.p.m.

�— Negligible rotating bending moment (to be confirmed byshaft alignment analysis)

�— Calculated steady state torsional vibrations according toFigure A-13.

Figure A-13Torsional vibration calculations in normal condition with ø500mm shaft


�— Outer diameter, d = 500 mm�— Multiradii transition, see Figure A-12 and Table 6-1

MAIN ENGINE

M. F

. SEI

A.3

ø500

ø900140

r1250r325r45

Intermediate Shaft

0

20

40

60

80

100

120

0 20 40 60 80 100

Speed [RPM]

Vib

rato

ry st

ress

[MPa

]

DET NORSKE VERITAS


February 2007

�— Flange thickness, t = 140 mm�— Flange diameter, D = 900 mm�— Surface roughness in flange fillet, Ra = 1.6 m, i.e. Ry =

9.6 m


�— Steel JIS G3201 SF 590 A with the following minimummechanical properties for representative test pieces1):Ultimate tensile strength, B 590 N/mm2Yield strength, y 295 N/mm2

1) See the Rules for Classification of Ships/High Speed, Light Craft andNaval Surface Craft Pt.2 Ch.2 Sec.5

Calculations:The Low Cycle Criteria (see 3.2):

a) Peak stress:


�— Maximum value of ( + v) in the entire speed range(see 3.3):The peak torque in misfiring condition will be lowerthan in normal condition since the engine speed andengine load will be reduced to 86% and 63% of MCRrespectively. Therefore, in this case there are two potential maxi-mum values of ( + v):

i) Running at n0 = 105 r.p.m in normal condition:Nominal torsional stress: 0 = 33.35 N/mm2 andvibratory torsional stress (see Figure A-13): v =26.9 N/mm2. Thus, in this case ( + v) = 60.25 N/mm2

ii) Running (accidently*) at 5th order resonancespeed = 78 r.p.m: Nominal torsional stress at 78 r.p.m:

Vibratory torsional stress (steady state) at 78r.p.m is according to the torsional vibration calcu-lations (see Figure A-13): v78 r.p.m.=109 N/mm2.Thus, in this case ( + v) = 128.0 N/mm2.* Accidently, since running steady at 78 r.p.m isnot relevant as the high cycle criterion below con-cludes with a barred speed range around 78 r.p.m.However, passing through a barred speed range inthis l-range will result in transient torsional vibra-tions close to the steady state level.

The maximum torsional stress occurs in case ii):

�— Yield strength:

y = 295 N/mm2

�— Safety factor (see the Rules for Classification ofShips/High Speed, Light Craft and Naval SurfaceCraft Pt.4 Ch.4 Sec.1 B205): S = 1.25

�— Geometrical stress concentration factor, torsion ac-cording to Table 6-1 in 6.2:

t = 1.05 �— Component influence factor for low cycle fatigue (see

3.5):

Consequently:

or actual safety factor, S = 1.1Conclusion:The above calculation is somewhat on the conservativeside since the allowable stresses are compared with thepeak stresses taking the calculated steady state vibratorystresses at 78 r.p.m. into account. Since a certain speedrange around 78 r.p.m. will be barred for continuous run-ning due to the high cycle criterion below, the actual vibra-tory stresses will be slightly lower than the calculatedones. This must be confirmed by measurements. See alsoconclusion to the transient criterion below.

b) Torque reversal:

�— Repetitive nominal torsional stress range due to re-versing (see 3.4):

�— Safety factor (see the Rules Pt.4 Ch.4 Sec.1 B205):S = 1.25

Consequently:

or actual safety factor 1.27Conclusion:Criterion fulfilled.The high cycle criterion (see 4.2):

�— Notch sensitivity coefficient for high cycle torsional fa-tigue (see 4.4.1): mt=1.0 due to multiradii transition


max v+ maxy

2SKL--------------=

016 818.5 106

5003-------------------------------------N mm2 33.35 N mm2= =

78r.p.m.78105---------

233.35 N mm2 18.4 N mm2= =

v+ max v+ 78r.p.m. 128.0= = N mm2

KL 1 1.05 1�– 295900--------- 10 4�–+ 590 200�– 9.6log+ 1.05= =

128.0 N mm2 2952 1.25 1.05----------------------------------- N mm2 112.4 N mm2=

t2

S 3----------

2 v+ 2 128 N mm2 256.0 N mm2= =

1.05 256.0 N mm2 2 2951.25 3---------------------- N mm2

268.8 N mm2 272.5N mm2

v

f-----

2 b

f------

2 1

S2-----+


February 2007

DET NORSKE VERITAS




this section of the shaft�— Safety factor (see the Rules Pt.4 Ch.4 Sec.1 B205):

S = 1.6Consequently:The permissible torsional vibratory stress as a function of shaftspeed is then:

This is then plotted as a function of engine speed in the calcu-lated torsional vibration diagram, see Figure A-14:

Figure A-14Calculated steady state torsional vibrations stress and stress limitfor continuous operation with ø500 mm shaft.

Conclusion: The calculated stresses are higher than the allowable stressesfor continuous operation around the 5th order resonance at 78r.p.m. In this case it means that a relatively wide barred speedrange has to be introduced from 72 to 84 r.p.m. According tothe Rules, the barred speed range can not be introduced in theoperational range above 0.8 n0 = 84 r.p.m.

The Transient Vibration Criterion (see 5):



�— Accumulated number of load cycles when passing throughthe barred speed range, NC is assumed to be at least 105 forthis plant since the passage is with fixed pitch and thebarred speed range is in the upper region of the -range.


Consequently:The permissible torsional vibratory stress in transient condi-tion as a function of shaft speed is then:


Figure A-15Calculated steady state torsional vibrations stress and stress limitfor transient operation with ø500 mm shaft.

Conclusion:If we compare the limit for the transient operation with the cal-culated steady state stresses around 78 r.p.m, we see from Fig-ure A-15 that the limit is exceeded by 44%. Since the barredspeed range is in the upper speed range and this is a fixed pitchpropeller plant, it is not possible to "quickly" pass this barredspeed range. The stresses will have sufficient time to build upto the steady state level and the accumulated number of cycleswill be relatively high. In this case it is also unlikely that thepeak stress criterion above will be fulfilled. Therefore the in-termediate shaft design should be changed or alternatively avery much larger damper to be fitted. Of cost reason the last al-ternative is not further pursued.It is of course possible to only change the material to a materialwith better mechanical properties, but in order to move thebarred speed range down so that the accumulated load cycleswill be fewer, the shaft diameter should also be reduced. Thishas also another benefit with regards to shaft alignment sincethe shaft line will be more flexible.

A.3.2 In way of flange fillet of the intermediate shaft, sec-ond design proposal:Second proposal for intermediate shaft design for the same

KH1.051.0----------- 0 01 100 3 10 4�– 590 200�– 9.6log++ 1.26= =

f0.24 295 42 0.15 233.35�–+

1.26--------------------------------------------------------------------------- N mm2

89.5 3.97 2 N mm2�–

==

nn0-----

b

f------

20

vHC89.5 3.97 2�–

1.6----------------------------------N mm2 55.9 2.48 2 N mm2�–=

Intermediate Shaft

0

20

40

60

80

100

120

0 20 40 60 80 100

S pe e d [RPM]

Vib

rtor

y st

ress

[M

Pa]

Limit forcontinuous operation

vT vHC3 106

NC----------------

0.4 log vLC

vHC-------------

vLCNC

104--------

0.4 log vHC

vLC-------------

=

vHC89.5 3.97 2�–

1.5------------------------------------N mm2 59.7 2.65 2 N mm2�–= =

nn0-----

vLC 112.4 33.35 2 N mm2�–=

265.27.5935.334.112log4.0

2vT /3065.27.59

2

2

mmN

DET NORSKE VERITAS


February 2007

plant is a 380 mm diameter shaft with the same multiradiiflange transition as the first proposal, but made from quenchedand tempered alloyed steel EN10083-1 - 34CrNiMo6 +QT.Other relevant dimensions are shown in Figure A-16.

Figure A-16Intermediate shaft flange fillet for ø380 mm shaft

Relevant loads:

�— Since the shaft stiffness has been reduced, the calculatedsteady state torsional vibrations have been changed ac-cording to Figure A-17.

Figure A-17Torsional vibration calculations in normal condition with ø380mm shaft


�— Outer diameter, d = 380 mm�— Multiradii transition, see Figure A-16 and Table 6-1�— Flange thickness, t = 110 mm�— Flange diameter, D = 700 mm�— Surface roughness in flange fillet, Ra = 1.6 m, i.e. Ry =

9.6 m


�— Quenched and tempered alloyed steel EN10083-134CrNiMo6 +QT with the following minimum mechani-cal properties for representative test pieces1):Ultimate tensile strength, B 900 N/mm2

Yield strength, y 700 N/mm2

1) See the Rules for Classification of Ships/ High Speed, Light Craft andNaval Surface Craft Pt.2 Ch.2 Sec.5

Calculations:The Low Cycle Criteria (see 3.2):

a) Peak stress:


�— Maximum value of ( + v) in the entire speed range(see 3.3) will be when running (accidently*) at 5th or-der resonance speed = 51 r.p.m.:Nominal torsional stress at 51 r.p.m:

Vibratory torsional stress (steady state) at 51 r.p.m isaccording to the torsional vibration calculations (seeFigure A-17): v51r.p.m.= 145.7 N/mm2.

Thus, in this case ( + v)max = 163.6 N/mm2

* Accidently, since running steady at 51 r.p.m. is not relevant as thehigh cycle criterion below concludes with a barred speed rangearound 51 r.p.m. Using the steady state vibration level is a conserv-ative asumption in this case since passing through the barred speedrange with 0.5 will happen quickly and the transient vibrationlevel will normally be approximately 80% of the steady state level.

�— Yield strength:y=700 N/mm2, however limited to 0.7 B= 630

N/mm2 in the fatigue strength calculation.�— Safety factor (see the Rules for Classification of

Ships/ High Speed, Light Craft and Naval SurfaceCraft Pt.4 Ch.4 Sec.1 B205):S=1.25

�— Geometrical stress concentration factor, torsion ac-cording to Table 6-1 in 6.2:

�— t = 1.05�— Component influence factor for low cycle fatigue (see

3.5):

Consequently:

or actual safety factor, S = 1.7Conclusion:Criterion fulfilled.b) Torque reversal:

Ø380

Ø700110

r950r247r45

Intermediate Shaft

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100

Speed [RPM]

Vib

rtor

y st

ress

[MPa

]

max v+ maxy

2SKL--------------=

016 818.5 106

3803------------------------------------- N mm2 75.97 N mm2= =

51r.p.m.51

105---------

275.97 N mm2 17.9 N mm2= =

KL 1 1.05 1�– 700900--------- 10 4�–+ 900 200�– 9.6log+ 1.11= =

163.6 N mm2 6302 1.25 1.11----------------------------------N mm2 227 N mm2=

t2 y

S 3----------


February 2007

DET NORSKE VERITAS

�— Repetitive nominal torsional stress range due to reversing(see 3.4):

�— Safety factor (see the Rules for Classification of Ships andHigh Speed Light Crafts and Naval Surface Craft Pt.4Ch.4 Sec.1 B205):S = 1.25

Consequently:

or actual safety factor 2.12Conclusion:Criterion fulfilled.


�— Notch sensitivity coefficient for high cycle torsional fa-tigue (see 4.4.1): mt = 1.0 d due to multiradii transition




�— Safety factor (see the Rules for Classification of Ships/High Speed, Light Craft and Naval Surface Craft Pt.4Ch.4 Sec.1 B205): S = 1.6

Consequently:The permissible torsional vibratory stresses as a function ofshaft speed is then:


Figure A-18Calculated steady state torsional vibrations stress and stress limitfor continuous operation with ø380 mm shaft.

Conclusion:The calculated stresses are higher than the allowablestresses for continuous operation around the 5th order reso-nance at 51 r.p.m. A barred speed range has to be introducedfrom 48 to 56 r.p.m. However, the barred speed range shouldbe confirmed by torsional vibration measurements on boardduring the sea trials.The Transient Vibration Criterion (see 5):

where:



�— Accumulated number of load cycles when passing throughthe barred speed range, NC is assumed to be reduced to5·104 in this case since the barred speed range has beenmoved down in the speed range.


Consequently:The permissible torsional vibratory stresses in transient condi-tion as a function of shaft speed is then:

�— since rotating bending moment is negligible at this section of the shaft

2 v+ 2 163.6 N mm2 327.2 N mm2= =

1.05 327.2 N mm2 2 6301.25 3------------------ N mm2

343.6 N mm2 582 N mm2

v

f-----

2 b

f------

2 1

S2-----+

KH1.051.0----------- 0.01 100 3 10 4�– 900 200�– 9.6log++ 1.36= =

f0.24 630 42 0.15 275.97�–+

1.36----------------------------------------------------------------------------- N mm2=

142.1 8.38 2 N mm2�–=nn0-----

b

f------

20,

vHC142.1 8.38 2�–

1.6------------------------------------- 88.8 5.2 2 N mm2�–=

vT vHC3 106

NC----------------

0.4 vLC

vHC-------------log

vLCNC

104--------

0.4 vHC

vLC-------------log

=

vHC142.1 8.38�– 2

1.5-------------------------------------- N mm2 94.7 5.59 2 N mm2�–= =

nn0-----

vLC 227 75.97 2 N mm2�–=

259.594.797.75227log0.4

2vT N/mm3059.57.94

2

2

DET NORSKE VERITAS


February 2007


Figure A-19Calculated steady state torsional vibrations stress and stress limitfor transient operation with ø380 mm shaft.

Conclusion:If we compare the stress limit for transient operation with thecalculated steady state stress, we see from Fig. A-19 that thelimit is just reached at 51 r.p.m. However, when quickly pass-ing the barred speed range, the stresses will certanly not be ashigh as the calculated steady state stresses. This is to be veri-fied by measurements on board during sea trials or be based onexperience with similar plants.The measured shaft stresses during the sea trials for this plantare presented in Figure A-20 and A-21.

Figure A-20Measured stresses with respect to shaft speed.

Figure A-21Measured stresses with respect to time

From these measurements one can observe the following:a) The maximum torsional vibratory stress amplitude is 120N/mm2, which is 82% of the calculated steady state amplitude.

b) The equivalent number of cycles, Ne with 100% amplitudefor one passage up and down are calculated according to5.2.1.6) assuming that the number of cycles for starting is thesame as for stopping.

The accumulated number of cycles Nc is then calculated as Netimes expected number of passages during the ships life-time,see 5.2.2:Once every week, i.e. 1000 passages, gives Nc = 38000 loadcycles, which is slightly less than the assumed 5·104 load cy-cles.Consequently, also the transient vibration criterion is fulfilled.

A.3.3 In way of flange fillet of the intermediate shaft, using simplified method:Since this shaft has a flange transition with a multiradii design,the simplified diameter formulae in the Rules for Classifica-tion of Ships/ High Speed, Light Craft and Naval Surface CraftPt.4 Ch.4 Sec.1 B208 may be used:The minimum shaft diameter is:

However, the shaft diameter is also to fulfill the criteria for tor-sional vibration:

a) For continuous operation (high cycle fatigue):For 0.9:

For 0.9:

Intermediate Shaft (ø380 mm)

0

20

40

60

80

100

120

140

160

48 49 50 51 52 53 54 55 56

Speed [RPM]

Vib

rtor

y st

ress

[MPa

]

Calculated steady state torsional vibration stress

222

vLC N/mm1.209N/mm1055197.75227

222

vHC N/mm4.93N/mm1055159.57.94

Ne 2N100N90

1 3

1vLC

vHC-------------log

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

-------------------------------N80

1 7

1vLC

vHC-------------log

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

-------------------------------+ +=

= 2 (13 +

93.4209.1log

1

1,3

8 +

93.4209.1log

1

1,7

11 )=2 (13 + 3.8 + 2.4) 38.

d 100k Pn0------ 560

B 160+-----------------------3

100 1.0 9000105

------------ 560590 160+------------------------3 mm= 400 mm=

1B 160+

18----------------------CkCd1.38=


February 2007

DET NORSKE VERITAS

b) For transient operation (passing barred speed range):

These curves are plotted together with the torsional vibrationcalculations, see figure A-22.

Figure A-22Torsional vibration calculations in normal condition with ø400mm shaft.

For a 400 mm diameter shaft the 5th order resonance will bearound 56 r.p.m and the vibration level is found by interpola-tion between the levels with 380 mm and 500 mm diametershafts.Conclusion:Figure A-22 shows that for a 400 mm diameter shaft the calcu-lated vibration stress exceeds the permissible stress in transientcondition. For the simplified critera, the permissible transientstresses are to be compared with steady state vibrations and notthe actual transient vibrations as for the detailed criteria above.The shaft diameter can not be reduced because the criterion forminimum diameter must be met, and the diameter can not beincreased either since the permissible stresses will never ex-ceed the calculated steady state vibrations. The only possibili-ties left are to introduce a high strength material or to install alarger damper. The transient criterion will be fulfilled if thetensile strength is increased to:

This means that the material must be changed from carbonsteel to quenched and tempered alloyed steel.

590 160+18

------------------------1.0 0.35 0.93 400 0.2�– 1.38 N mm2+

36.3 N mm2=

=

1B 160+

18----------------------CkCd 3 2 2�– =

590 160+18

------------------------= 1.0 0.35 0.93 400 0.2�– 3 2 2�– N mm2+

78.8 52.5 2 N mm2�–=

2 1.7 1

ck--------- 1.7 78.8 52.5 2�–

1.0--------------------------------------- N mm2= =

134.0 88.7 2 N mm2�–=

Intermediate Shaft

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100

Speed [RPM]

Vib

rtor

y st

ress

[MPa

]

ø380 mm

ø500 mm

ø400 mm

2

1

B140105--------- 590 160+ 160 N mm2�– 840 N mm2=

Dnv - Calculation of Shafts in Marine Applications

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