1 AMERICAN BUREAU OF SHIPPING (ABS) The following longitudinal strength example calculations have been retrieved from ABS Steel Vessels Part 3: Hull construction and equipment. The calculations are for vessels to be classed for unrestricted service and follow the criteria defined below. i) Proportions: ii) Length: iii) Block Coefficient: 1.1 WAVE LOADING The maximum wave bending moment in (kNm) is obtained using the following formulae. Sagging Moment Hogging moment Where; The wave loading coefficient C 1 is dependent on the length of the vessel and can be calculated using equation 1
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1 AMERICAN BUREAU OF SHIPPING (ABS)
The following longitudinal strength example calculations have been retrieved from ABS Steel Vessels Part 3: Hull construction and equipment. The calculations are for vessels to be classed for unrestricted service and follow the criteria defined below.
i) Proportions:
ii) Length:
iii) Block Coefficient:
1.1 WAVE LOADING
The maximum wave bending moment in (kNm) is obtained using the following formulae.
Sagging Moment
Hogging moment
Where;
The wave loading coefficient C1 is dependent on the length of the vessel and can be calculated using equation
Once a maximum value for wave loading bending moment has been obtained a distribution factor must be applied to get an accurate value of the moment along the vessels length. The distribution factors can be obtained using Figure 1-1 below.
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Figure 1-1: Moment Distribution Factor
1.2 BENDING STRENGTH STANDARD
To find the total bending moment and still water bending moment the section modulus (cm2-m) amidships is to be calculated.
Where;
Once the section modulus is calculated the total bending moment can be obtained by rearranging equation
is defined as the nominal permissible bending stress of 17.5kN/cm2.
The total bending moment is considered to be the maximum sum of the still water bending moment and the wave induced bending moment, as below:
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1.3 EXAMPLE CALCULATIONS
Using the vessel particulars below obtained from…. Sample calculations are able to be carried out for a panamax bulk carrier.
Using equation the wave loading coefficient can be calculated. As the vessel is 230m long it falls into the first category.
Therefore:
The constants k1 and k2 are defined above. Using equations & the maximum hogging and sagging moments can be calculated.
Sagging Moment
Hogging moment
Once maximums have been calculated the longitudinal distribution factors must be applied to the moments. With the use of Matlab this was achieved and plotted to produce Figure 1-2.
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Figure 1-2: Wave loading bending moment distribution through length of vessel
To calculate the still water bending moment the section modulus for amidships is calculated with equation .
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Using the section modulus above the total bending moment can be calculated along the length of the vessel.
As both the wave loading bending moment and the total bending moment of the vessel have been calculated the still water bending moment is calculated using equation.
Once the still water bending moment was calculated it was plotted using Matlab. Figure 0-3 shows both hogging and sagging conditions for still water bending moment.
Figure 0-3: Still water bending moment distribution through length of vessel
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2 DET NORSKE VERITAS (DNV)
2.1 APPLICATION
The rules defined by the DNV for the classification of ships involve the following parameters:
L > 100m
CB > 0.6
L/B ≥ 5
B/D ≤ 2.5
Special considerations must be given for vessels that do not apply to the above. Those vessels that do however apply to these conditions are further subjected to calculations defining the global loading case of the vessel. In particular this involves the calculation of the longitudinal still water and wave loading bending moments and their distribution along the vessel’s length.
Additionally it should be noted that the wave bending moments calculated from DNV rules are assumed with a probability of 10-8 or less.
2.2 STILL WATER BENDING MOMENTS
The calculation of the still water bending moment using the DNV guidelines involves the consideration of the geometric parameters of the vessel. Initially the wave loading coefficient (CW) must be calculated dependent on the length of the vessel.
For L < 300m:
()
For 300m ≤ L ≤ 350m: ()
For L < 300m:
()
Now using the length, beam and block coefficient of the vessel, the maximum still water moment caused on the longitudinal axis may be determined.
For the maximum still water bending moment in a sagging direction: ()
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For the maximum still water bending moment in a hogging direction: ()
The loading at a distance x from the forward port of the vessel is found by the following:
()
From Equation (6), ksm is a distribution vector varied linearly as seen in Figure 2-4.
Figure 2-4: Still water moment loading along the length (L) of the vessel (Det Norske Veritas, 2013)
From Figure 2-4 it can be seen that the loading for the still water moment varies linearly with a changing gradient at points 0.1 L, 0.3 L, 0.7 L and 0.9 L. The maximum loading is seen to be present from 0.4 L to 0.7 L.
For still water conditions, the sign convention used can be seen in Figure 2-5.
Figure 2-5: Sign convention of Ms (Det Norske Veritas, 2013)
2.3 WAVE LOADING BENDING MOMENTS
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For the calculation of wave loading bending moments using the DNV guidelines, the CW
value achieved in Equations (1) , (2) or (3) for still water conditions is used. Furthermore a new variable α is defined as follows:
α = 1.0 for seagoing conditions= 0.5 for harbour and sheltered water conditions
For the maximum wave loading bending moment in a sagging direction: ()
For the maximum wave loading bending moment in a hogging direction:
()
To apply this wave loading moment across the length of the vessel a distribution vector is used.
()
From Equation (9), kwm is a distribution vector varied linearly as seen in Figure 2-6.
Figure 2-6: Wave bending moment distribution (Det Norske Veritas, 2013)
In Figure 2-6 it can be seen that kwm may approach 1.2. This however will only occur for ships defined as having a “high speed and large flare” (Det Norske Veritas, 2013). Table 1-1 shows the adjustments to kwm for these vessels.
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Table 1-1: Adjustments to kwm for vessels with high speed or large flare
In Table 1-1 the coefficients CAV and CAF are defined as follows:
-- ( )
-- ( )
-- ( )
Where:
ADK = projected area in the horizontal plane of the upper deck forward of 0.2 L from F.P.
AWP= area of water plane forward of 0.2 L from F.P.
zf = vertical distance from summer load water line measured at F.P.
SAMPLE CALCULATION
From the list of vessels used in the similar vessel analysis, an average was taken across the data collected for both panama and cape size vessels. For the following sample calculation the following particulars were taken as an average for a panama vessel:
Length (L) Beam (B) Draught (T) Mass (m) Block Coefficient
(CB)
230m 33m 14m 74000t 0.68
Initially the maximum still water wave loading moment is calculated. Since L < 300m, Equation (1) is used:
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Now, from Equations (4), the maximum still water bending moment in a sagging direction:
From Equation (5), the maximum still water bending moment in a hogging direction:
Using Equation (6) the maximum bending moment in both the hogging and sagging direction is able to be plotted similarly to the distribution in Figure 2-4. The final result is seen in Figure 0-7.
Figure 0-7: Still water bending moment distribution through length of vessel
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The maximum wave loading bending moment for the vessel is calculated using Equations (7) and (8).
From Equation (7), the maximum wave loading moment in a sagging direction (assuming a seagoing condition):
From Equation (8), the maximum wave loading moment in a hogging direction (assuming a seagoing condition):
On average the vessels chosen in the similar vessel analysis were did not have a high speed and large flare and therefore Equations 10, 11 and 12 do not apply and the kwm value in Figure2-6 will have a maximum of 1. Now by using Equation (9) the maximum bending moment in both the hogging and sagging direction can be plotted similarly to the distribution in Figure2-6. The final result is seen in Figure 0-8.
Figure 0-8: Wave loading moment distribution through length of vessel
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3 LLOYDS REGISTER (LRS)
3.1 APPLICATION
The rules defined by the LRS are considered to be accurate for the following vessels:
If a vessel has particulars outside of these ranges special considerations must be applied in the calculations. Vessels that do however fall between the regions defining the particulars are able to be subjected to LRS calculations defining the global loading case of the vessel.
3.2 WAVE LOADING BENDING MOMENTS
Calculation of the wave loading bending moment using the LRS guidelines involves the consideration of the geometric parameters of the vessel. Initial calculations involve finding the coefficient, C1, as defined below:
For L < 90m: ()
For 90m ≤ L ≤ 300m:
()
For 300m ≤ L ≤ 350m:
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()
For 350m ≤ L ≤ 500m:
()
The longitudinal distribution factor, C2, is given as a vector following the points defined below, connected with linear interpolation:
0 at aft end of L
1 between 0.4 L and 0.65 L from aft
0 at the fwd end of L
A ship service factor, f1, is defined as follows:
f1 = 1 for seagoing conditions ()
0.5 ≤ f1 < 1 for service restriction ()
Dependent on the direction of the moment (hogging or sagging) another ship service factor f2
can be found.
For sagging:
f2 = -1 ()
For hogging:
()
Overall, the wave loading bending moment can be defined as follows:
()
3.3 STILL WATER BENDING MOMENTS
For the calculation of the still water bending moment using the LRS guidelines, a total bending moment of the vessel must first be calculated using the sectional modulus of the vessel.
()
In Equation 10, a value for kL can be determined from Table 3-3. Values for f1 and C1 can be taken from Equations (1) to (6).
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Table 3-3: Values for kL from yield stress (Lloyds Register, 2012)
The permissible hull vertical bending stress of the vessel is necessary in calculating the permissible still water bending moments using the LRS rules. For the continuous longitudinal structural members within 0.4 L amidships, the stress can be found as follows:
()
For members outside of 0.4 L amidships:
()
For simplicity in graphing this loading pattern however a linear progression is plotted for Equation (12). The permissible still water bending moment can now be calculated (for hogging and sagging respectively):
()
3.4 SAMPLE CALCULATION
From the list of vessels used in the similar vessel analysis, an average was taken across the data collected for both panama and cape size vessels. For the following sample calculation the following particulars were taken as an average for a panama vessel:
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Length (L) Beam (B) Draught (T) Mass (m) Block Coefficient
(CB)
230m 33m 14m 74000t 0.68
Initially the wave loading moment must be calculated, as this is used directly in Equation (13) for the still water bending moment.
Since 90m ≤ L ≤ 300m, Equation (2) can be used.
From Equation (5) a value of 1 is given to f1 assuming seagoing conditions. Now Equation (9) can be used to calculate the wave loading moment, firstly in a hogging direction:
For hogging (using Equation (8)):
Now Equation (9):
For sagging (using Equation (7):
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Now Equation (9):
After applying the distribution factor C2 the wave loading across the vessel can be seen in Figure 3-9.
Figure 3-9: Wave loading bending moment distribution through length of vessel
The still water moment initially requires the section modulus (Equation (10)). From Table 3-3 a kL value of 1 is used estimating a specified minimum yield stress of 235N/mm2.
For the continuous longitudinal structural members within 0.4 L amidships, the stress can be found as follows (from Equation (11)):
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For members outside of 0.4 L amidships, Equation (12) can be used:
Now the permissible still water bending moment can be found using Equation (13).
For hogging outside of 0.4 L amidships:
For hogging inside of 0.4 L amidships:
For sagging outside of 0.4 L amidships:
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For sagging inside of 0.4 L amidships:
The result can be seen in Figure 3-10.
Figure 3-10: Still water bending moment distribution through length of vessel
