Clasification Society Rulefinder 2020 - Version 9.33 - Fix
Common Structural Rules - Common Structural Rules for Bulk Carriers and Oil Tankers, January 2019 - Part 1 General Hull Requirements - Chapter 5 Hull Girder Strength - Appendix 1 Direct Calculation of Shear Flow - 2 Example of Calculations for a Single Side Hull Cross Section

2 Example of Calculations for a Single Side Hull Cross Section

2.1 Cross section data

2.1.1 The cross section is shown in Figure 4. The coordinates of the node points marked by filled black circles in Figure 4 are given in Table 1, where the plate thickness and the line segments (marked by circles in Figure 4) of the cross section are given in Table 2.

The sample calculations are performed taking advantage of symmetry of the cross section.

Table 1 : Node coordinates of cross section

Node number Y coordinate (m) Z coordinate (m)
0 0.00 0.00
1 5.80 0.00
2 11.70 0.00
3 14.42 0.00
4 16.13 1.72
5 16.13 6.11
6 11.70 1.68
7 5.80 1.68
8 0.00 1.68
9 16.13 14.15
10 16.13 19.60
11 7.50 20.25
12 7.50 19.63

Table 2 : Calculation of cross sectional properties

Line no. Node i Node k Thickness (mm) Length (m) an50 (m2) sy-n50 (m3) iy0-n50 (m4)
1 0 1 17.0 5.80 0.099 0.000 0.00
2 1 2 17.0 5.90 0.100 0.000 0.00
3 2 3 17.0 2.72 0.046 0.000 0.00
4 3 4 17.0 2.43 0.041 0.035 0.04
5 4 5 18.0 4.39 0.079 0.309 1.34
6 5 6 19.0 6.26 0.119 0.464 2.00
7 6 7 21.0 5.90 0.124 0.208 0.35
8 7 8 21.0 5.80 0.122 0.205 0.34
9 5 9 18.0 8.04 0.145 1.466 15.63
10 9 10 21.0 5.45 0.114 1.931 32.87
11 10 11 24.0 8.65 0.208 4.139 82.47
12 11 12 24.0 0.62 0.015 0.297 5.92
13 12 9 15.0 10.22 0.153 2.590 44.13
14 2 6 15.0 1.68 0.025 0.021 0.02
15 1 7 15.0 1.68 0.025 0.021 0.02
Total 1.416 11.686 185.138

2.1.2 The Z coordinate of horizontal neutral axis and the inertia moment about the neutral axis are calculated as follow:

Figure 4 : Numbering of nodes and lines

2.2 Calculations of the determinate shear flow

2.2.1 The virtual slits are added to cut the walls of the closed cells as shown in Figure 5. And then, the line integrations specified in [1.2.2] are performed to obtain determinate shear flow, qD. The calculation results are shown in Table 3. The locations of the virtual slits and the paths of line integrations shown in Figure 5 are one such example. These definitions can be arbitrarily determined so as to calculate them easily.

Figure 5 : Ranges and directions of paths for line integrations

Table 3 : Calculation of determinate shear flow

Path no. Line no. Node i Node k qDi×10-6 (N/mm) qDk×10-6 (N/mm) Note
1 1 0 1 0.0 4.6 Start from the virtual slit
15 1 7 4.6 5.6 -
2 2 1 2 0.0 4.7 Start from the virtual slit
14 2 6 4.7 5.7 -
3 3 2 3 0.0 2.2 Start from the virtual slit
4 3 4 2.2 3.9 -
5 4 5 3.9 5.8 -
4 10 9 10 0.0 -5.6 Start from the virtual slit
11 10 11 -5.6 -19.2 -
12 11 12 -19.2 -20.2 -
13 12 9 -20.2 -27.7 -
9 9 5 -27.7 -29.2 -
5 6 5 6 -23.4 -20.5 Start with the sum of qDK at the ends of path 3 & 4
6 7 6 7 -14.8 -10.2 Start with the sum of qDK at the ends of path 2 & 5
7 8 7 8 -4.5 0.0 Start with the sum of qDK at the ends of path 1 & 6

2.3 Calculations of the indeterminate shear flow

2.3.1 To obtain the system of equations for indeterminate shear flows, the contour integrations around 3 closed cells as defined in Figure 6 are performed. The closed cell at the centre of double bottom is considered as an open shape since the symmetrical condition of the cross section is considered. The calculation results of contour integrations around the closed cells are shown in Table 4 to Table 6.

Table 4 : Contour integration of /tn50 and φ around cell 1

Line no. Node i Node k qDi×10-6 (N/mm) /tn50 φ×10-3 (N/mm) Note
2 1 2 0.0 347.1 0.81 -
14 2 6 4.7 112.0 0.58 Common wall with cell 2
7 6 7 -14.8 281.0 -3.50 -
15 7 1 -5.6 112.0 -0.58 -
Total 852.0 -2.68 -

Table 5 : Contour integration of /tn50 and φ around cell 2

Line no. Node i Node k qDi×10-6 (N/mm) /tn50 φ×10-3 (N/mm) Note
3 2 3 0.0 160.0 0.17 -
4 3 4 2.2 142.7 0.43 -
5 4 5 3.9 243.9 1.22 -
6 5 6 -23.4 329.7 -7.32 -
14 6 2 -5.7 112.0 -0.58 Common wall with cell 1
Total 988.3 -6.07 -

Table 6 : Contour integration of /tn50 and φ around cell 3

Line no. Node i Node k qDi×10-6 (N/mm) /tn50 φ×10-3 (N/mm) Note
10 9 10 0.0 259.5 -0.65 -
11 10 11 -5.6 360.6 -4.45 -
12 11 12 -19.2 25.8 -0.51 -
13 12 9 -20.2 681.5 -16.59 -
Total 1327.5 -22.19 -
2.3.2 The following system of equations can be developed by using the results of the contour integration around each closed cell:
  • Cell 1: 852.0 qI1 – 112.0 qI2 = 2.68 × 10-3
  • Cell 2: -112.0 qI1 + 988.3 qI2 = 6.07 × 10-3
  • Cell 3: 1327.5 qI3 = 2.219×10-2

The solution of this system gives indeterminate shear flows of the closed cell 1 to 3:

qI1 = 4.01 × 10-6, qI2 = 6.60 × 10-6, qI3 = 1.67 × 10-5

Figure 6 : Numbering of closed cells

2.4 Summation

2.4.1 The shear flow qv, at all locations of the cross section can be obtained by the summation of determinate shear flow, qD and indeterminate shear flow, qI as shown in Figure 7.

Figure 7 : Calculation results of shear flow qv, in 10-6 N/mm for vertical shear force with 1 N

Parent topic: Appendix 1 Direct Calculation of Shear Flow
Copyright 2020 Clasification Society, International Maritime Organization, International Labour Organization or Maritime and Coastguard Agency. All rights reserved. Clasification Society, its affiliates and subsidiaries and their respective officers, employees or agents are, individually and collectively, referred to in this clause as 'Clasification Society'. Clasification Society Register assumes no responsibility and shall not be liable to any person for any loss, damage or expense caused by reliance on the information or advice in this document or howsoever provided, unless that person has signed a contract with the relevant Clasification Society entity for the provision of this information or advice and in that case any responsibility or liability is exclusively on the terms and conditions set out in that contract.