Ship stability formule-4

Appendix I

Summary of stability formulae* Form coefficients Area of waterplane Area of amidships Volume of displacement Cb

L B Cw B d Cm L B d Cb Cm C p

Drafts When displacement is constant (for box shapes): Old density New draft Old draft New density When draft is constant: New density New displacement Old density Old displacement TPCSW FWA

Change of draft or dock water allowance

WPA 97.56 W 4 TPCSW FWA (1025 DW )

Homogeneous log: Relative density of log Draft Depth Relative density of water * See Note at end of the Appendix

25

486

Ship Stability for Masters and Mates

Variable immersion hydrometer: Density

My  M y M x   M y x    L

Trim W GM L 100 L Trimming moment or Change of trim MCTC W (LCB foap LCG foap ) MCTC

MCTC l Change of trim L Change of draft forward Change of trim Change of draft aft Change of draft aft

Effect of trim on tank soundings: Head when full Trim Length of tank Length of ship True mean draft: Correction Trim FY Length To keep the draft aft constant: d

MCTC L TPC l

To find GML: GM L L GG1 t

Simpson’s rules 1st rule: Area ⫽ h/3 (a ⫹ 4b ⫹ 2c ⫹ 4d ⫹ e)

or

1 3

CI 兺1

Summary of stability formulae

2nd rule: Area ⫽ 3h/8 (a ⫹ 3b ⫹ 3c ⫹ 2d ⫹ 3e ⫹ 3f ⫹ g)

or

3 8

CI 兺2

3rd rule: Area ⫽ h/12 (5a ⫹ 8b c)

or

1 12

CI 兺3

KB and BM Transverse stability For rectangular waterplanes: LB 3 12 BM ⫽ I/V I⫽

For box shapes: BM ⫽ B 2/12d KB ⫽ d/2 KM min ⫽ B/ 6 For triangular prisms: BM B 2/6d KB 2d/3 Depth of centre of buoyancy 1 d V  ⫹  ⫽ below the waterline 32 A  Longitudinal stability For rectangular waterplanes: BL3 12 I BM L L V IL

For box shapes: BM L

L2 12d

487

488


For triangular prisms: BM L

L2 6d

Transverse statical stability Moment of statical stability W GZ At small angles of heel: GZ GM sin By wall-sided formula: GZ (GM

1 2

BM tan 2 ) sin

By Attwood’s formula: GZ

v hh1 BG sin V

Stability curves: New GZ Old GZ GG1 sin heel or New GZ KN KG sin heel Dynamical stability W Area under stability curve  v(gh g1 h1 )  W BG(1 cos )    V lo

Total VHM SF W

l40 lo 0.8

Approx angle of heel

Actual HM

Total VHM SF

Actual HM 12° Maximum permissible HM Approx angle of due to grain shift

Reduction in GZ (GG H cos ) (GG V sin ) W ship displacement in tonnes GG H horiz movement of ‘G’ GG V vert movement of ‘G’ lo righting arm @ 0° (if upright ship) l40 righting arm @ 40° HM heeling moment SF stowage factor VHM volumetric heeling moment


489

List Final KG

Final moment Final displacement

wd Final W GG1 tan list GM GG1

Increase in draft due to list: New draft

1 2

b sin (d) cos . . . Rise of floor is zero put ‘r’ in if rise of floor exists. (measured at full Br. Mld)

Inclining experiment: Length of plumbline GM GG1 Deflection

Effect of free surface Virtual loss of GM

lb3 1 2 12 W n

Drydocking and grounding Upthrust at stern: P⫽

MCTC t l

or P ⫽ Old ⫺ New displacement Virtual loss of GM

P KM W

or

P KG WP

Pressure of liquids Pressure (P) Dwg Thrust P Area Depth of centre of pressure

I WL AZ

490


Bilging and permeability

S 100 per cent SF v Increase in draft A a Permeability

Strength of ships Stress

Load Area

Strain

Change in length y Original length R

Young’s modulus: E

Stress Strain

Bending moment: M Section modulus

E I R I Y

Shearing stress: q

F y It

Stress: f

E y R

Freeboard marks Distance Summer LL to Winter LL

1 48

Summer draft

Distance Summer LL to Tropical LL 418 Summer draft

Ship squat bT BH C S 0.81 Vk2.08 b 20

Blockage factor max


yo ⫽ H ⫺ T y 2 ⫽ y o ⫺ ␦max As b T Ac B H In open water: max

C b Vk2 100

In confined channel: C b Vk2 50 Width of influence ⫽ 7.7 ⫹ 20 (1 C b)2 max

Miscellaneous Angle of loll: 2 GM BM T

tan loll GM

2 initial GM cos loll

Heel due to turning: tan heel

v 2 BG g r GM

Rolling period: T 2

k

g GM

2k GM

Zero GM: tan list ⫽

3

2wd W BM

Theorem of parallel axes: I CG ⫽ I OZ ⫺ Ay 2

approx

491

492


or I NA I xx Ay 2 Permeability (␮) Permeability

BS 100 per cent SF

Increase in draft

Permeability ()

Volume available for water 100 Volume available for cargo

Permeability ()

SF of cargo solid factor 100 SF of cargo

Solid factor Sinkage Tan

1 RD

A a

Effective length l

Volume of bilged compartment Intact waterplane area

BB H GM bilged

Drafts and trim considerations Correction to observed drafts

Midships draft corrected for deflection

l1 Trim L1

d FP (6 d m ) d AP 8

Correction of midships draft Distance of LCF from midships Trim to true mean draft when LBP LCF is not at amidships Second trim correction for position of LCF, if trimmed hydrostatics are not supplied a (form correction) True trim (MCTC2 MCTC1 ) 2 TPC LBP Alternative form correction

50 (True trim )2 (MCTC2 MCTC1 ) LBP


493

Note These formulae and symbols are for guidance only and other formulae which give equally valid results are acceptable.

Mass Volume

RD

substance FW

(L B d) Cb DWT light AW (L B) CW TPC

AW 100

Sinkage/rise

W TPC

FWA

Summer 4 TPCSW

DWA

(1025 dock ) FWA 25

MSS GZ GZ GM sin GZ [GM 1–2 BM tan2 ]sin GZ KN (KG sin ) Dynamic stability Area under GZ curve Area under curve (SR1)

1 h (y1 4y 2 y 3 ) 3

Area under curve (SR2)

3 h (y1 3y 2 3y 3 y 4 ) 8

lo

Total VHM SF

l40 lo 0.8 Actual HM

Total VHM SF

494


Approx angle of heel

Actual HM 12° Max permissible HM

Reduction in GZ (GGH cos ) (GGv sin ) Rolling period T (sec)

2 K g GM or

GG H/V

i T

FSC

l b3 T 12

FSC

FSM

tan

GG H GM

兺 Moments 兺 Weights

GG H

GM

兺 Moments 兺 Weights

w s length deflection

tan angle of loll

2 GM BM T

GM at angle of loll

tan

3

2ws BM T

2

K2 seconds GM T

ws

FSC

KG

2

or

2 initial GM cos

K2 gGM T


Draft when heeled (Upright draft cos ) ( 12 beam sin ) KM T KB BM T

Position of the metacentre

lT ∇ L B3 BM T ( box) 12 ∇ BM T

Distance Summer LL to Winter LL

1 Summer draft 48

Distance Summer LL to Tropical LL

1 Summer draft 488

KM L ⫽ KB ⫹ BM L BM L ⫽

IL ∇

BM L ( box) ⫽

MCTC

CoT

L3 B 12 ∇

GM L 100 LBP

兺 Trimming moment MCTC

Change of trim aft Change of trim

LCF LBP

LBP LCF LBP  LCF   Turn mean draft ⫽ Draft aft ⫾  trim LBP  

Change of trim fwd Change of trim

Trim

P

(LCG ⬃ LCB) MCTC

Trim MCTC LCF

P Reduction in TMD TPC Loss of GM

P KM T

or

P KG P

495

496


tan

v 2 BG g R GM

Permeability ()

Solid factor

Volume available for water 100 Volume available for cargo

1 RD

Permeability ()

SF of cargo solid factor 100 SF of cargo

Effective length l Sinkage

Volume of bilged compartment permea bility () Intact water plane area

I parallel axis I centroidal axis As 2 Tan

BB H GM bilged

Correction to observed drafts

l1 Trim L1

Midships draft corrected for deflection

d FP (6 d M ) d AP 8

Correction of midships draft to true mean drraft when CF not midships Distance of CF from midships trim (true trim at perp’s) LBP Second trim correction for position of CF iff trimmed hydrostatics are not supplied (form correction) True trim (MCTC2 MCTC1 ) 2 TPC LBP Alternative form correction

50 true trim 2 (M MCTC2 MCTC1 ) LBP

Summary Always write your formula first in letters. If you then make a mathematical error you will at least obtain some marks for a correct formula.

Ship stability formule-4

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