#### Theory

1. Grounding is similar to dry-docking, but with certain modifications as follows :

a. Ship may have any displacement.

b. Trim may be by the stern or by the head or the ship may be on even keel.

c. The point of contact on grounding may be anywhere from forward to aft.

d. Ship may run aground on a sloping beach in which case the point of contact will be either the FP or the AP. As the tide falls, the trim will change, depending on the point of contact, and the ship may finally come to rest on the beach. The final trim may be by the head or by the stern, according to the slope of the beach, regardless of the original trim.

e. Ship may run aground on a rock at any point along its length, with deep waters at both ends. As the tide falls, the trim will change continuously, depending on the point of contact. There is no rest position.

2. After grounding and with the fall of tide, the underwater volume, hydrostatic draft, displacement and GM(F) will reduce, as in the case of dry docking, but the trim may increase or decrease or change in direction, depending on the point of contact and the initial trim.

3. Unlike dry-docking there are no side shores or bilge blocks and so the ship is never secured. Hence, even if the ship comes to rest on a beach and the tide continues to fall, the FSM will always apply. It also means that at some stage of fall of tide, the GM(F) may become negative and the ship is likely to capsize.

4. Because all the parameters continuously change with the fall of tide, there is no such thing as critical period or critical instant.

5. The drop in water level due to fall of tide will be measured at the point of contact, which may be anywhere along the length of the ship.

## Grounding Numericals

#### How to solve?

**ABBREVIATIONS**

**DF , DA & DR:- **Drafts forward, aft and at the rock or point of contact.

**d :-**Distance of the rock or the point of contact from the COF

**Fall in tide :-**Fall in tide after the first contact.

**Hyd :-**Hydrostatic draft.

**P :-**Loss of weight of the ship due to fall in tide after the first contact.

**T:-**Initial or final trim of the ship as stated.

**TF, TA & TR :-**Trim forward, aft and at the rock or point of contact.

**TC :-**Change of trim due to fall in tide after the first contact.

**FORMULAE APPLICABLE**

TC (cm.)= (P ´ d) / MCTC

Loss of GM = (P ´ KG) / (W – P)

GM(F) = KM – KG – Loss of GM – FSC

Drop (cm.) = (P / TPC) + (P ´ d2) / (MCTC ´ LBP)

*(This formula is to be used only if the other formulae cannot be used.)*

**CALCULATION OF INITIAL DATA REQUIRED FOR FURTHER CALCULATION**

**CASE 1**** — **

Data given : DF and DA .

Calculate Trim and mean draft. Using mean draft calculate *LCF* from hydrostatic tables.

Calculate Correction = (T ´ *LCF*) / LBP

*(This LCF will not be used again in any calculation.)*

Calculate Hyd = DA ± Correction *(Sign will depend on the trim.)*

Using Hyd calculate W, MCTC and LCF from the tables.

**CASE 2** —

Data given : W and LCG.

Using W calculate Hyd, MCTC, LCB and LCF from the tables.

Calculate the DF and DA as follows :

BG = LCB ~ LCG ; T = (W ´ BG) / MCTC

TA = (T ´ LCF) / LBP ; TF = T – TA

DF = Hyd ± TF ; DA = Hyd ± TA *(Sign will depend on the trim.)*

**CASE 3** —

Data given : W and T.

Using W calculate Hyd, MCTC and LCF from the tables.

Calculate the DF and DA as follows :

TA = (T ´ LCF) / LBP ; TF = T – TA

DF = Hyd ± TF ; DA = Hyd ± TA *(Sign will depend on the trim.)*

**In addition to the above data, following data will also be given :**

· KG and FSM

· The point of grounding, i.e. F.P. or A.P., and the slope of the beach, OR

· The point of contact in case of grounding on a rock.

**CALCULATION OF FINAL GM(F), DRAFTS AND FALL OF TIDE AFTER THE SHIP RUNS AGROUND AT F.P. OR A.P. AND COMES TO REST POSITION ON THE BEACH **

**STEP 1**** — CALCULATE THE LOSS OF WEIGHT (P) **

Final trim of the ship will be the slope of the beach as given in the data.

TC = Initial T ± Final T

Calculate distance d as follows :

If point of grounding is at the A.P. : d = LCF

If point of grounding is at the F.P. : d = LBP – LCF

Formula **I : **P = (TC ´ MCTC) / d

**STEP 2**** — CALCULATE THE FINAL GM(F)**

W1 = W – P ; Using W1 calculate KM1 from the tables.

Formula **II :** Loss of GM = (P ´ KG) / W1 ; FSC = FSM / W1

Formula **III : **GM(F) = KM1 – KG – Loss of GM – FSC

**STEP 3**** — CALCULATE THE FINAL DRAFTS **

Using W1 calculate Hyd1 and LCF1 from the tables.

TA = (Final T ´ LCF1) / LBP ; TF = Final T – TA

DF = Hyd1 ± TF ; DA = Hyd1 ± TA *(Sign will depend on the final trim.)*

**STEP 4**** — FALL OF TIDE** is change in draft at the point of grounding, i.e. F.P. or A.P.

**CALCULATION OF FINAL DRAFTS AND FALL OF TIDE AFTER THE SHIP RUNS AGROUND ON THE BEACH AND THE GM(F) REDUCES TO ‘X’ m. **

**STEP 1**** — CALCULATE THE GM(F) AT THE REST POSITION ON THE BEACH, AS EXPLAINED IN PROBLEM 1**

There will be two possibilities as follows :

**Possibility 1 :**

If GM(F) @ Rest > X m. then final GM(F) will reduce to X m. after the rest position.

**Possibility 2 :**

If GM(F) @ Rest < X m. then final GM(F) will reduce to X m. before the rest position.

**STEP 2**** — CALCULATE LOSS OF WEIGHT (P1) WHEN FINAL GM(F) = X m.**

Using Hyd calculate KM from the tables.

Calculate P1 by combining Formulae **II & III : **

X = KM – KG – (P1 × KG) / (W – P1) – FSM / (W – P1)

**STEP 3**** — CALCULATE THE FINAL DRAFTS**

W2 = W – P1 ; Using W2 calculate Hyd2 , MCTC1 and LCF1 from the tables.

**Possibility 1 :**

Tide has fallen beyond the rest position, hence the final trim is the slope of the beach.

Calculate the final DF and DA as explained in Step 3 of Problem **1.**

**Possibility 2 — **

Calculate distance d1 as follows :

If point of grounding is at the A.P. : d1 = LCF1

If point of grounding is at the F.P. : d1 = LBP – LCF1

Formula **I :** TC = (P1 × d1) / MCTC1

Final T = Initial T ± TC

*If initial trim is by the stern and the point of contact is at AP — Subtract *

*If initial trim is by the stern and the point of contact is at FP — Add*

*If initial trim is by the head and the point of contact is at FP — Subtract*

*If initial trim is by the head and the point of contact is at AP — Add*

Calculate the final DF and DA as explained in Step 3 of Problem **1.**

**STEP 4**** —** **FALL OF TIDE** is change in draft at the point of contact i.e. at FP or AP.

**CALCULATION OF FINAL GM(F) AND DRAFTS AFTER THE SHIP RUNS AGROUND ON THE BEACH AND THE TIDE FALLS BY ‘X’ m.**

**STEP 1**** — CALCULATE THE FALL OF TIDE UPTO THE REST POSITION ON THE BEACH, AS EXPLAINED IN PROBLEM 1**

There will be two possibilities as follows —

**Possibility 1 : **

If fall of tide upto Rest < X m. then fall of tide of X m. will occur after the rest position.

**Possibility 2 :**

If fall of tide upto Rest > X m. then fall of tide of X m. will occur before the rest position.

**STEP 2**** — CALCULATE LOSS OF WEIGHT (P1) WHEN FALL OF TIDE = X m.**

Using Hyd calculate TPC from the tables.

Calculate distance d as explained in Step 1 of Problem **1.**

Calculate P1 using Formula **IV :** X = (P1 / TPC) + (P1 × d2) / (LBP × MCTC)

**STEP 3 ****— CALCULATE THE FINAL DRAFTS**

**Possibility 1 :**

Tide has fallen beyond the rest position, hence the final trim is the slope of the beach.

Initial DA / DF – Fall of X m. = Final DA / DF ; Final DA / DF ± Final T = Final DF / DA

**Note :***Sign will depend on the final trim. Value of P is not required.*

*Initial DA or DF will be used depending on whether the ship has run aground at A.P. or F.P. respectively.*

**Possibility 2 :**

W2 = W – P1 ; Using W2 calculate Hyd2 , MCTC1 and LCF1 from the tables.

Calculate distance d1 as explained in Step 3 of Problem **2.**

Formula **I : **TC = (P1 × d1) / MCTC1

Final T = Initial T ± TC

**Note : ***Sign will be as explained in Problem ***2***.*

Calculate the final DF and DA as explained in Step 3 of Problem **1.**

**STEP 4**** — CALCULATE THE FINAL GM(F)**

Using Hyd2 calculate KM2 from the tables.

Formula **II : **Loss of GM = (P1 × KG) / W2 ; FSC = FSM / W2

Formula **III :** GM(F) = KM2 – KG – Loss of GM – FSC

**CALCULATION OF FINAL DRAFTS AND FALL OF TIDE AFTER THE SHIP RUNS AGROUND ON A ROCK AND THE GM(F) REDUCES TO ‘X’ m.**

**(There is deep water at forward and aft ends)**

**STEP 1**** — CALCULATE LOSS OF WEIGHT (P) WHEN FINAL GM(F) = X m.**

Using Hyd calculate KM from the tables.

Calculate P by combining Formulae **II & III : **

X = KM – KG – (P × KG) / (W – P) – FSM / (W – P)

**STEP 2**** — CALCULATE THE FINAL DRAFTS**

W1 = W – P ; Using W1 calculate Hyd1 , MCTC1 and LCF1 from the tables.

Calculate distance d1 of the rock from COF1 using LCF1 and data about point of contact.

Formula **1 :** TC = (P × d1) / MCTC1

Final T = Initial T ± TC

**Note : ***Sign will be as explained in Problem ***2***.*

Calculate the final DF and DA as explained in Step 3 of Problem **1.**

**STEP 3**** — CALCULATE THE FALL OF TIDE**

Calculate initial distance d of the rock from the COF using the given data.

Initial trim at the rock (TR)= (Initial T ´ d) / LBP

Initial draft at the rock (DR)= Hyd ± Initial TR

*If initial trim is by the stern and the rock is forward of COF — Subtract *

*If initial trim is by the stern and the rock is aft of COF — Add*

*If initial trim is by the head and the rock is aft of COF — Subtract*

*If initial trim is by the head and the rock is forward of COF — Add*

Final TR = (Final T ´ d1) / LBP ; Final DR = Hyd1 ± Final TR

*(Apply sign as explained above)*

Fall of tide = Initial DR – Final DR

**CALCULATION OF FINAL GM(F) AND DRAFTS AFTER THE SHIP RUNS AGROUND ON A ROCK AND THE TIDE FALLS BY ‘X’ m.**

**(There is deep water at forward and aft ends)**

**STEP 1**** — CALCULATE THE LOSS OF WEIGHT (P) **

Using Hyd. calculate TPC from the tables.

Calculate distance d of the rock from the COF using the given data.

Calculate P using Formula **IV : **X = (P / TPC) + (P × d2) / (LBP × MCTC)

**STEP 2**** — CALCULATE THE GM(F) **

W1 = W – P

Formula **II :** Loss of GM = (P × KG) / W1 ; FSC = FSM / W1

Using W1 calculate KM1 from the tables.

Formula **III : **GM(F) = KM1 – KG – Loss of GM – FSC

**STEP 3**** — CALCULATE THE FINAL DRAFTS**

Using W1 calculate Hyd1 , MCTC1 and LCF1 from the tables.

Calculate distance d1 of the rock from COF1 using LCF1 and data about point of contact.

Formula **1 :** TC = (P × d1) / MCTC1

Final T = Initial T ± TC

**Note : ***Sign will be as explained in Problem ***2***.*

Calculate the final DF and DA as explained in Step 3 of Problem **1.**

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