Ship stability analysis during launching from longitudinal sloping slipway by pneumatic airbags

Abstract

The changing forces acting on a ship during launching from a longitudinally sloping slipway can be divided into four stages: the 1st stage – from beginning of ship movement until the launching runners or the ship’s aft end contacts the water; 2nd stage – from the end of 1st stage until the ship begins to float off the ways; 3rd stage – from the end of 2nd stage until the ship paces the slipway threshold and is fully afloat; 4th stage – from the end of 3rd stage until comes to a full stop. The biggest danger for a ship during the launch is the end of stage 2 and the beginning of stage 3, when tipping is possible as the ship rotates around the slipway threshold; also as the ship pivots on the slipway’s threshold there are large concentrated forces imparted on the forward section of the ship due to pivoting pressures (the difference between the mass of the ship and launching devices and the rising buoyancy). This paper presents, for the first time, a new analysis method of taking into account the possibility of a tanker tipping during launching on pneumatic airbags. Launching calculations and diagrams for both the traditional and pneumatic airbags launching are presented along with a comparative analysis of the two launching methods. The analysis showed that during launching using pneumatic airbags, float off begins earlier than during a traditional launch. This earlier float off does not increase the risk of tipping but does increase the pivoting pressure due to the increased buoyancy from the airbags.

Nomenclature

the way passed by ship at the beginning of stage 2

W_{C}

launching weight of the ship and launching device

the volume of submerged hull part

specific gravity of the water

x_{G}

z_{G}

center of gravity coordinates

L_{1}

the length of forward runners part (from plane $y^{'} O^{'} z^{'}$ to aft perpendicular)

L_{2}

the length of back runners part (from plane $y^{'} O^{'} z^{'}$ to fore perpendicular)

the elevation of ship keel above slipway runners in plane $y^{'} O^{'} z^{'}$

the angel of runners inclined (slipway incline)

the angle of ship base plane inclined to horizon

T_{0}

the calculated water depth on slipway threshold

the length of underwater runners part

M_{W}

the moment of gravity force about back part of runners

M_{V}

the moment of buoyancy force about back part of runners

M_{W}^{'}

the moment of weight force about slipway threshold

M_{V}^{'}

the moment of buoyancy force about slipway threshold

x O^{'} y

the plane parallel to base plane of the ship

x O z^{'}

central plane of the ship

y^{'} O^{'} z^{'}

transverse plane, that passing through center of gravity – G-point (Fig. 1)

y O z

middle frame transversal plane

the length of each pneumatic airbag

K_{1}

safety factor

acceleration of gravity

load capacity per unit length of airbag

L_{d}

the length of contact between ship bottom and airbag at middle plane

N_{1}

the numbers of pneumatic airbags between slipway threshold and aft perpendicular

T_{av}

mean draft of the ship

x_{C}

z_{C}

center of buoyancy coordinates

x_{f}

x-coordinate of center of floatation waterline area

R_{1}

longitudinal metacentric radius

{GM}_{L}

longitudinal metacentric height

x_{f}^{'}

distance between center of floatation of waterline and plane $y^{'} O^{'} z^{'}$

T_{f}

draft at the fore perpendicular

T_{a}

draft at the aft perpendicular

M^{'}

static moment about plane $y^{'} O^{'} z^{'}$

distance, that airbag pass on slipway surface at the beginning 2nd stage

distance between airbags

distance between center of gravity and slipway threshold

N_{p}

pivoting pressure

C_{B}

the block coefficient of launching ship

L_{p p}

the length of the launching ship

the number of airbags

D_{1}

the external diameter of airbag

1. Introduction

Several methods of launching ships can be used. The principal methods are: floating off, gravity (end or side) launching, and mechanized methods. A longitudinal sloping slipway (end launching) is the most complicated method of launching a ship as it has a number of elements in the process which are largely uncontrollable. To launch a ship on a longitudinal sloping slipway requires a ship specific launching cradle to be mounted on hull above the slipway. The main disadvantages of this launching method are: the impossibility to control all aspects of the launch; the necessity of using a large number of steel slipway runners; attachment of the launching cradle to the ship’s hull; the large amount of wood required for both the slipway runners and the launching cradle; the complexity of ship transfer from the keel blocks to the launching cradle; the preparation and application of grease on to the slipway runners; pollution of shipyard’s water area by grease components; difficulty of dismounting and retrieval of the launching cradle after the ship’s launching; the long lead time to prepare for the launch; and the large number of man hours required for the entire process [2,4].

This new method of ship launching using pneumatic airbags eliminates the above mentioned disadvantages [1,3,5]. The research which began in [6] are continued in the present paper.

The application of such method can minimize the cost for production and mounting launching cradle, simplifies preparation work to launch a ship, and eliminates pollution of a shipyard’s water area. In order to launch a ship using airbags it is sufficient to build a flat inclined plane slipway made of concrete, crushed stone or coarse sand [8,9]. The most of modern vessels engine room and superstructure are located at the stern part of the ship. That increases the possibility of the ship tipping, bouncing and hitting the slipway threshold during the 3rd stage of launching on a longitudinal sloping slipway. Such researches have never been done before.

The object of the article is to explore a tanker’s launching on a longitudinal sloping slipway using pneumatic airbags in order to determine the possibility of tipping, to determine the pivoting pressure and to conduct the analysis of the airbags influence on the launching process.

2. Method of calculation

For these ship launching calculations on a longitudinal sloping slipway it is assumed that the ship speed is low as it moves along slipway thus, the influence of hydrodynamic forces are negligible. Therefore a static analysis has been carried out.

The main aim of the static analysis is to find the position of the vessel when the angular displacement occurs – tipping or floating off. Also of interest, are all the positions along the ways where high concentrated forces occur [10,11].

In order to determine the distance traveled by the ship before she begins to emerge and the value of pivoting pressure, as well as to determine the possibility of tipping, it is sufficient to consider the data from a static analysis of the ship launching at 2nd stage. The ship motion takes place parallel to the launching runners and the gravity forces, slipway reaction and water pressure are acting on her [2,4]. The buoyancy calculations are carried out by using Bonjean curves and the results are presented as a launching diagram – the complex of graphics presenting the dependence of values $W_{C}$ , $γ V$ , $M_{W}$ , $M_{V}$ , $M_{W}^{'}$ , $M_{V}^{'}$ on the distance S the ship moves at 2nd stage (Fig. 1) [1].

Fig. 1.

Forces acting on ship during launching.

It is admitted that ship launching is carried out astern. The coordinate system $x y^{'} z^{'}$ is determined for the ship during launching. The coordinate planes are main plane $x O^{'} y$ , central plane $x O z^{'}$ , transverse plane $y^{'} O^{'} z^{'}$ which is located at a distance $x_{G}$ from midship plane [4].

The abscissas of the ship’s center of buoyancy and center of gravity at calculated waterline area are measured from plane $y O z$ and $y^{'} O^{'} z^{'}$ . Let’s mark them as $x_{C}$ , $x_{f}$ and $x_{C}^{'}$ , $x_{f}^{'}$ respectively. Thus $x_{C}^{'} = x_{C} - x_{G}$ and $x_{f}^{'} = x_{f} - x_{G}$ .

The following symbols of launching device’s parts are used in the calculation: stationary part β, $T_{0}$ , λ and movable part α, $L_{1}$ , $L_{2}$ , b, c.

Angle α is slightly different from angle β and therefore it is accepted that $tg α \approx sin α \approx α$ and $cos α \approx 1$ .

As an example, the tanker with the below characteristics shown in Table 1 is used in the calculations.

Table 1

The tanker characteristics

Characteristics	Value
Length between perpendiculars $L_{p p}$ , m	174.6
Ship breadth B, m	31.1
Ship depth D, m	17.1
Ship draft T, m	12.2
Block coefficient $C_{B}$	0.79
Cargo density ρ, t/m³	0.75
Carrying cargo capacity W, t	59610
Launching weight of ship and launching device $W_{C}$ , t	12035.11
Lightship abscissa of center of gravity $x_{G}$ , m	−15.12
Light ship application of the center of gravity $z_{G}$ , m	10.93

Table 2

Ship launching airbags characteristics

Characteristics	Value
External diameter $D_{1}$ , m	1.8
Effective height (airbag height during the descent), m	0.9
Load capacity per unit length, t/m	11.54
Maximum length, m	18
Effective length, m	15
Weight, kg	581
Volume, m³	48.5

It is necessary to calculate the number of airbags and the distance between them for ship launching using pneumatic airbags [7,10]. The numbers of pneumatic airbags for ship launching could be taken from the following formula $\begin{matrix} (1) & N = K_{1} \frac{W_{C} \cdot g^{2}}{C_{B} \cdot R \cdot L_{d}} + N_{1}, \end{matrix}$ where $K_{1} = 1.2 \div 1.3$ ; $W_{C} = 12035.11 t$ ; $g = 9.81 {m/s}^{2}$ ; $C_{B} = 0.79$ ; $R = 11.54 t/m$ (see Table 2); $L_{d} = 15 m$ ; $N_{1} = 1$ .

According to (1) $N = 53 pcs$ are required.

The effective airbag length is 15 meters which is half the launched ship’s breadth. Therefore it is necessary to mount two rows of airbags under the ship (Fig. 2). Such airbag arrangement provides the required ship stability during launching. Thus, the total number of airbags is $N_{Σ} = 106 pcs$ .

Fig. 2.

Airbag arrangement.

The longitudinal distance between the centers of two neighbor pneumatic airbags should meet the required strength of hull structure.

The distance between airbags centers l can be calculated by $\begin{matrix} (2) & l = \frac{L_{p p}}{N - 1}; l ⩽ 6 and l ⩾ \frac{π D}{2} + 0.5, \end{matrix}$ where $L_{p p} = 174.6 m$ ; $N = 106 pcs$ ; $D_{1} = 1.8 m$ .

The distance between airbags is taken as $l = 3.3 m$ .

Taking into account a contact length $L_{d}$ of ship’s bottom and airbags at midship the airbags with characteristics shown in Table 2 [3,5] are taken. Choosing the value of $L_{d}$ in the catalog (assortment) of airbags, depending on width of the ship, airbags circuit placement. The length of the working airbags body should correspond to the width of the vessel at midship.

In order to determine the possibility of the ship tipping during launching at 2nd stage the launching diagram has been built. The most suitable is English diagram.

Since the ship’s center of gravity moves parallel to slipway plane during the 2nd stage of longitudinal launching the beginning of emerge. tipping and maximal pivoting pressure can be determined. A static analysis of ship launching during the 2nd stage is made by using Bonjean curve. The point at which the aft perpendicular is over water boundary line is taken as a beginning of the 2nd stage [10].

The following are the input data for calculations: $W_{C}$ ; ship’s center of gravity coordinates $x_{G}$ and $z_{G}$ (Table 1); $L_{1} = 72.2 m$ ; $L_{2} = 102.4 m$ ; $c = 0.9 m$ ; $β = 0.05$ ; $α = β = 0.05$ ; $T_{0} = 3.6 m$ ; $λ = 68.7 m$ ; water specific gravity $γ = 1.025 {t/m}^{3}$ .

The approximate values of emergence waterline and pivoting pressure are determined during preliminary calculations for the ship. By using hydrostatic curves for displacement $W_{C}$ (Table 1) the following characteristics of the launched ship: $T = 2.98 m$ ; $x_{C} = 7.088 m$ ; $z_{C} = 1.561 m$ ; $x_{f} = 7.195 m$ ; $R_{1} = 646.3 m$ .

${GM}_{L} = R_{1} + z_{C} - z_{G} = 639.93 m$ and trim is $ψ = \frac{x_{g} - x_{C}}{{GM}_{L}} = - 0.0349$ are calculated and preliminary value of pivoting pressure is: $\begin{matrix} (3) & N_{p} = \frac{w_{C} \cdot {GM}_{L}}{L_{z} - x^{'} f} (α + ψ) = 1678.2 t, \end{matrix}$ where $x_{f}^{'} = x_{f} - x_{G} = 22.4 m$ .

By using hydrostatic curves the values of average draft $T_{av}^{'} = 2.5 m$ and $x_{f 2} = 7.28 m$ for volume $V_{2} = \frac{1}{γ} (W_{C} - N_{p}) = 10104.31 m^{3}$ are calculated.

The parameters of highest calculated waterline taken with 20% reserve: $\begin{matrix} (4) & h^{'} = 1.2 (T_{av}^{'} + x_{f}^{'} \cdot α), \end{matrix}$ where $h^{'} = 4.41 m$ .

The parameters of lower calculated waterline is determined assuming the ship’s center of gravity is at the slipway threshold $h^{″} = T_{0} - c = 2.7 m$ .

The lower calculated waterline is determined taken drafts at fore $T_{f}$ and aft $T_{a}$ perpendiculars: $\begin{matrix} (5) & \begin{array}{l} T_{f} = h^{″} - (\frac{L}{2} - x_{g}) α = - 2.67 m; \\ T_{A} = h^{″} + (\frac{L}{2} + x_{g}) \cdot α = 6.48 m . \end{array} \end{matrix}$ The position of lower calculated waterline is determined by $\begin{matrix} (6) & Δ h = (h^{'} - h^{″}) / n = 0.342 m . \end{matrix}$ where n is numbers of waterlines ( $n = 5$ ).

The distance 0.35 m between calculated waterlines is decided during calculations. The waterlines with step $Δ h = 0.35 m$ from the highest calculated waterline parallel to it downward on Bonjean curve are drawn.

The values of frames’ areas ω are taken from waterlines in Bonjean curve. The buoyancy force $γ V$ and its static moment M relatively to plane $y O z$ are calculated for each waterline. Thus static moment $M^{'}$ relatively to plane $y^{'} O^{'} z^{'}$ is $M^{'} = M - γ V x_{g}$ (Table 3).

The buoyancy forces of pneumatic airbags and actuating momentum are determined in the same way as for additional pontoons which sometimes are fixed to the hull for increasing of buoyancy force.

During the calculations of the distance the airbags pass the slipway it’s necessary to take into account that when the ship passes distance S, the airbags pass distance s. In order to determine this distance it’s considered that airbags have inextensible shell [6]. Thus $s = 1 / 2 S$ . Based on this statement the number of airbags that act on the ship launching process is determined as $\begin{matrix} (7) & N^{'} = \frac{\frac{S}{2} + l}{c + \frac{π}{2} (D_{1} - c) + l} . \end{matrix}$

The distance the ship passes from the beginning of the 3rd stage is calculated by formula (8) $\begin{matrix} (8) & S = L_{1} + \frac{h + c}{β} \end{matrix}$

The abscissa of buoyancy forces is calculated by formula (9) $\begin{matrix} (9) & x_{C}^{'} = \frac{M^{'}}{γ V} \end{matrix}$

The buoyancy force moments relatively to the runner’s aft end (10) and relatively to slipway threshold (11) are calculated: $\begin{array}{l} (10) & M_{V} = γ V (L_{2} - x_{C}^{'}) \\ (11) & M_{V}^{'} = - γ V (a + x_{C}^{'}), \end{array}$ where $a = L_{1} + λ - S$ (Fig. 1).

The calculation is performed for 1st waterline and presented in Table 3. The same calculation is used for each waterline. The calculation results are presented in English diagram (Fig. 3).

Table 3

The calculation of displacement and static moment for 1st waterline

1	2	3	4	5	6
No. frame	multiplier	Frame areas, sq.m		(3) − (4)	(2) × (5)

		fore	aft
10	0	50.838		50.838	0
9.11	1	37.897	63.799	−25.903	−25.903
8.12	2	25.088	76.135	−51.047	−102.094
7.13	3	12.631	87.803	−75.172	−225.515
6.14	4	0.869	97.904	−97.0356	−388.142
5.15	5	0	101.812	−101.814	−509.068
4.16	6	0	96.116	−96.116	−576.696
3.17	7	0	76.746	−76.746	−537.219
2.18	8	0	45.363	−45.363	−362.904
1.19	9	0	11.605	−11.605	−104.441
0.2	5	0	0	0	0
		127.322	657.283		−2831.98

$Σ 1$	784.6
$Σ 2$	−2831.98
$γ V$	7020.83
M	−221229.2
$M^{'}$	−115074.2

Based on the calculation taking into account the distance S which the ship passes at 2nd stage, next lines are built in the diagram: $W_{C}$ ; $γ V$ ; $M_{V}$ ; $M_{W} = W_{C} \cdot L_{2} = 123.24 \cdot 10^{4} t*m$ ; $M_{V}^{'}$ ; $M_{W}^{'}$ is a line between two points: point with abscissa $S = 140.9 m$ and ordinate $M_{W}^{'} = W_{C} \cdot a = 0 t*m$ and point with abscissa $S = 173.54 m$ and ordinate $M_{W}^{'} = W_{C} \cdot a = - 32.16 \cdot 10^{4} t*m$ [10].

Fig. 3.

British diagram of ship launching: —— – with airbags; – · – · – – without airbags.

3. Conclusions

Based on the above drawn diagram the following conclusions can be made:

the ship’s tipping during her launching using airbags will not occur at 2nd stage because $M_{V}^{'} > M_{W}^{'}$ . The critical position can occur at $S_{emer .} = 152.75 m$ when ship is launched using airbags and at $S_{emer .} = 159.12 m$ when ship is launched using standard launching device;

the ship floating off starts at distance $S = 152.75 m$ from the beginning of the 2nd stage ( $M_{V} = M_{W}$ ) when the ship is launched using airbags and at distance $S = 165.22 m$ when ship is launched using standard launching device;

the largest value of pivoting pressure is at this time (the moment of $M_{V} = M_{W}$ ) $N_{p} = W_{C} - γ V = 4507.1 t$ during launching using airbags and $N_{p} = 2028.1 t$ during launching using standard launching device;

the highest depth of launching device’s front end is determined by formula $\begin{matrix} (12) & T_{1} = S \cdot β = (λ + L_{1} + L_{2}) \cdot β = 12.75 m . \end{matrix}$

Based on the analysis of the ship stability during launching from longitudinal sloping slipway as explained above the following conclusions have been made:

Based on obtained results a comparison has been made, which showed that during ship launching using pneumatic airbags the ship floating off earlier, tipping and leap does not happen. However, additional buoyancy of airbags increases the pivoting pressure, but within a tolerable range.

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