Master of Science (MSc) in Naval Architecture and OCEAN Engineering

ANALYSIS OF SHIP STRUCTURES

PROJECT WORK

Yordan Garbatov

2024/2025

yordan.garbatov@tecnico.ulisboa.pt 1
 GLOBAL SHIP HULL STRUCTURAL ANALYSIS BASED ON THE FINITE
ELEMENT METHOD

Announcement of the Assignment: Week No. 1 (February 17, 2025)

Delivery Deadline of the Project: Week No. 5 (March 23, 2025)

The finite element method will analyse a tanker ship hull structure. The analysis will
cover a 3-D amidships one-cargo-space module, and the commercial software ANSYS
will be used.

1 STRUCTURAL MODEL

Figure 1 Global ship structural model- one module, example.

The degrees of freedom of the nodal points at one of the ends of the hull module are
restrained in all directions. All other nodal degrees of freedom are set to be free.

yordan.garbatov@tecnico.ulisboa.pt 2
 Figure 2 Finite element model, example.

2 LOAD APPLICATION

The ship hull will be analysed in three load conditions: bending moments, shear forces
and torsion moment.

The material selected for this ship is steel with a Young Modulus of 200 GPa, a Yield
point of 235 MPa, and 𝑓1 = 1 for the entire section. The loads will be applied in the
selected nodes for each condition. Each node has an associated area, calculated as
equivalently distributed between nodes.

For a regular section:

𝐴 = 𝑡1 ∆𝑧1 (1)

where 𝑡1 is the thickness of the plate, assuming the same on both sides of the node 1,
n1, and ∆𝑧1 is the distance between node 2 and node 1 and the middle between node 1
and node 3.

Figure 3 Nodal area calculation, regular section

For all the other cases, the thickness of the plates is not always the same on both sides
of a node (sometimes it is), and therefore, for a “T”-joint section, it is calculated as:

yordan.garbatov@tecnico.ulisboa.pt 3
 ∆𝑧1 ∆𝑧2 ∆𝑧3 −𝑡2
𝐴 = 𝑡1 + 𝑡2 + 𝑡3 (2)
2 2 2

Figure 4 Nodal area calculation, “T”-joint section

For the “L” section:

∆𝑧2 +𝑡1 ∆𝑧1 −𝑡2

𝐴 = 𝑡2 2
+ 𝑡1 2
(3)

Figure 5 Nodal area calculation, “L”-section

Finally, the area of a cross-intersection is calculated as:

∆𝑧4 ∆𝑧2 ∆𝑧 −𝑡 ∆𝑧3 −𝑡3
𝐴 = 𝑡1 2
+ 𝑡3 2
+𝑡2 12 3 + 𝑡4 2
(4)

Figure 6 Nodal area calculation, cross-section

yordan.garbatov@tecnico.ulisboa.pt 4
2.1 Vertical and Horizontal Sectional Shear

To simulate vertical or horizontal sectional shear forces, a set of nodal forces are applied
at the nodal locations of the unrestrained end section of the hull module in the hull
depth or length direction, respectively.

Figure 7 Vertical (left), horizontal (centre) and nodal locations.

The applied force in each node, based on the shear force, is calculated as:
𝑄
𝐹𝑖 = 𝐴 𝐴𝑖 (5)
𝑠ℎ𝑒𝑎𝑟

2.2 Vertical and horizontal bending

The bending moment is generated by applying nodal axial forces at all unrestrained
module-end-section nodes.

Figure 8 Vertical (left) and horizontal (right) bending moment generation.

To estimate the forces subjected to nodal location, first, the moment should be
calculated and in the case of the vertical bending moment:
𝑀𝑣 (𝑦𝑖 −𝑦𝑛𝑎 )
𝐹𝑖 = 𝐼𝑛𝑎
𝐴𝑖 (6)

and in the case of horizontal bending moment:

yordan.garbatov@tecnico.ulisboa.pt 5
 𝑀ℎ (𝑥𝑖 )
𝐹𝑖 = 𝐴𝑖 (7)
𝐼𝑐𝑙

2.3 Torsion loading

The torsion loading is generated by a torsion moment applying a set of nodal forces at
only four corner nodes such that the following equilibrium condition is fulfilled:

𝑀1 = 𝐹1 𝑌𝑛𝑎 (8)

𝑀2 = 𝐹2 𝐵/2 (9)

𝑀3 = 𝐹3 (𝐷 − 𝑌𝑛𝑎 ) (10)

𝑀4 = 𝐹4 𝐵/2 (11)

and the magnitude of the torsion moment applied to the hull module is:

𝑀𝑡 = 𝑀1 + 𝑀2 + 𝑀3 + 𝑀4 (11)

Figure 9 Torsion moment generation.

3 LOADINGS

3.1 Hull-Girder Section Modulus- Minimum Requirements

The longitudinal strength of a ship is specified by the minimum ship hull-girder section
modulus 𝑊. The hull-girder section modulus amidships is not to be less than that
estimated from the following equation:
1
𝑊 = 𝑓 0.01𝐶𝑤 𝐿𝐵𝑃 2 𝐵(𝐶𝐵 + 0.7), 𝑐𝑚2 . 𝑚 (12)
1

where 𝐿𝐵𝑃 is the vessel’s length, B is the vessel’s breadth, 𝐶𝑏 is the block coefficient at
the summer load waterline, which is not to be less than 0.60.

yordan.garbatov@tecnico.ulisboa.pt 6
 300−𝐿𝐵𝑃 1.5
10.75 − ( 100
) , 90 ≤ 𝐿𝐵𝑃 ≤ 300, 𝑚
𝐶𝑤 = 10.75, 300 ≤ 𝐿𝐵𝑃 ≤ 350, 𝑚 (13)
𝐿 −350 1.5
10.75 − ( 𝐵𝑃150 ) , 350 ≤ 𝐿𝐵𝑃 ≤ 427, 𝑚
{

3.2 Sill water bending moment

The still water-bending moments of amidships 𝑀𝑠 (sagging and hogging) are typically
not to be taken less than:

𝑀𝑠𝑜 = −0.065𝐶𝑤 𝐿𝐵𝑃 2 𝐵(𝐶𝑏 + 0.7), 𝑘𝑁𝑚, in sagging (14)

𝑀𝑠𝑜 = 𝐶𝑤 𝐿𝐵𝑃 2 𝐵(0.125 − 0.015𝐶𝑏 ), 𝑘𝑁𝑚, in hogging (15)

when required in connection with stress analysis, the still water bending moments at
arbitrary positions along the length of the ship are typically not to be taken less than:

𝑀𝑠 = 𝑘𝑠𝑚 𝑀𝑠𝑜 (16)

𝑘𝑠𝑚 = 1.0 within 0.4 𝐿𝐵𝑃 amidships (17)

𝑘𝑠𝑚 = 0.15 at 0.1 𝐿𝐵𝑃 from AP or FP (18)

𝑘𝑠𝑚 = 0 at >FP and AP (9)

3.3 Wave-induced bending moment

The rule vertical wave-induced bending moments amidships are given by:

𝑀𝑤𝑜 = −0.11𝛼𝐶𝑤 𝐿𝐵𝑃 2 𝐵(𝐶𝑏 + 0.7), in sagging, kNm (10)

𝑀𝑤𝑜 = 0.19𝛼𝐶𝑤 𝐿𝐵𝑃 2 𝐵(𝐶𝑏 + 0.7), in hogging, kNm (11)

𝛼=1 for seagoing conditions (12)

𝐶𝑏 is not taken less than 0.6 (13)

when required in connection with stress analysis, the wave-induced bending moments
at the arbitrary position along the length of the ship are typically not to be taken less
than:

𝑀𝑤 = 𝑘𝑤𝑚 𝑀𝑤𝑜 , kNm (14)

𝑘𝑤𝑚 = 1.0 between 0.4 𝐿𝐵𝑃 and 0.65 𝐿𝐵𝑃 from AP (15)

𝑘𝑤𝑚 = 0 at FP and AP (16)

3.4 Total bending moment

The following equation determines the total bending moment:

yordan.garbatov@tecnico.ulisboa.pt 7
 𝑀𝑡 = 𝑀𝑠 + 𝑀𝑤 (17)

3.5 Shear forces in still water

The design values of still water shear forces along the length of the ship are typically not
to be taken less than:

𝑄𝑠 = 𝑘𝑠𝑞 𝑄𝑠𝑜 , kN (18)

𝑀𝑤𝑜
𝑄𝑠𝑜 = 5 𝐿
, kN (19)

𝑘𝑠𝑞 = 0 at AP and FP (20)

𝑘𝑠𝑞 = 1 between 0.15 𝐿𝐵𝑃 and 0.3 𝐿𝐵𝑃 from AP (21)

𝑘𝑠𝑞 = 0.8 between 0.4 𝐿𝐵𝑃 and 0.6 𝐿𝐵𝑃 from AP (22)

𝑘𝑠𝑞 = 1 between 0.7 𝐿𝐵𝑃 and 0.85 𝐿𝐵𝑃 from AP (23)

3.6 Wave-induced shear forces

The vertical wave-induced shear forces along the length of the ship are given by:

𝑄𝑤 = 0.3𝛽𝑘𝑤𝑞 𝐶𝑤 𝐿𝐵𝑃 𝐵(𝐶𝑏 + 0.7), kN (24)

𝛽 = for seagoing conditions (25)

𝑘𝑤𝑞 =0.7 between 0.4 𝐿𝐵𝑃 and 0.6 𝐿𝐵𝑃 from AP (26)

𝑘𝑤𝑞 = 0 at FP and AP, (27)

𝐶𝑏
𝑘𝑤𝑞 = 1.59 𝐶 between 0.2 𝐿𝐵𝑃 and 0.3 𝐿𝐵𝑃 from AP, (28)
𝑏 +0.7

𝑘𝑤𝑞 = 0.7 between 0.4 𝐿𝐵𝑃 and 0.6 𝐿𝐵𝑃 from AP, (29)

𝑘𝑤𝑞 =1 between 0.7 𝐿𝐵𝑃 and 0.85 𝐿𝐵𝑃 from AP, (30)

3.7 Horizontal wave-induced bending moment

The horizontal wave-induced bending moments along the length of the ship are given
by:
9
𝑥
𝑀𝑤ℎ = 0.22𝐿4 (𝑇 + 0.3𝐵)𝐶𝑏 (1 − 𝑐𝑜𝑠 (360 𝐿 )), kNm (31)

where 𝑥 is the distance in [m] from AP to the section considered.

yordan.garbatov@tecnico.ulisboa.pt 8
3.8 Horizontal shear forces

The wave-induced horizontal shear force along the length of the ship due to the
horizontal wave is given by:
𝑥
𝑄ℎ = 0.6𝑄𝑚𝑎𝑥 𝑠𝑖𝑛 (360 𝐿 ), kN (32)

𝑄𝑚𝑎𝑥 = 𝑄𝑠,𝑚𝑎𝑥 + 𝑄𝑤,𝑚𝑎𝑥 , kN (33)

3.9 Torsion moment

The wave-induced torsion moments along the length of the ship due to the horizontal
wave and inertia forces and the rotational wave and inertia moment loads are given:
5 4
𝑀𝑤𝑇 = 𝐾𝑇1 𝐿4 (𝑇 + 0.3𝐵)𝐶𝑏 𝑧𝑛𝑎 ± 𝐾𝑇2 𝐿3 𝐵2 𝐶𝑤𝑝 , kNm (34)

𝑥
𝐾𝑇1 = 1.40𝑠𝑖𝑛 (360 𝐿 ) (35)

𝑥
𝐾𝑇2 = 0.13 [1 − 𝑐𝑜𝑠 (360 𝐿 )] (36)

𝐴𝑤𝑝
𝐶𝑤𝑝 = 𝐿𝐵
(37)

where 𝐴𝑤𝑝 (m2) is the water plane area of the vessel at draught 𝑇, and 𝑧𝑛𝑎 is the position
of the neutral axis in (m), 𝐶𝑤𝑝 is assumed to be 0.86.

4 AMIDSHIP SECTION MODULE

Determine the midship section modulus. Based on the elastic beam theory, define the
normal flexural stresses along the ship’s depth, assuming the equivalent thickness for
the longitudinal structural components of the midsection.
𝐵
𝐵1 = (49)
2

𝐵
𝐵2 = 2
− 𝑚𝐶1𝑏 (50)

where 𝑚 is assumed as 2 for 𝐿𝐵𝑃 ≤ 203 𝑚, otherwise it is 3.

𝐶1𝑏 = 𝑛 𝑠 (51)

where 𝑛 is assumed as 4.

2.08𝐿 + 438, 𝑚𝑚, 𝐿𝐵𝑃 ≤ 270, 𝑚

𝑠={ (52)
1000, 𝑚𝑚, 270 < 𝐿𝐵𝑃 ≤ 427, 𝑚
𝐿𝐵𝑃 −40 𝑇
𝐻= + 40𝐵 + 3500 (53)
0.57 𝐿𝐵𝑃

𝐻1 = 0.2𝐷 (54)

yordan.garbatov@tecnico.ulisboa.pt 9
 𝐻2 = 0.5𝐷 (55)

𝐻3 = 1.5, 𝑚 (56)

The thicknesses of longitudinal structural components are defined as a function of the

equivalent plate thickness of the deck and bottom:

𝛿𝑑 = 0.035𝐿𝐵𝑃 + 5, 𝑚𝑚 (57)

𝛿𝑏 = 0.04𝐿𝐵𝑃 + 6, 𝑚𝑚 (58)

Figure 10 Equivalent thickness

1t
10
1200
tdeck
1200

t4 t5
H1

t2

W
H2

t3 t5
t2

t3 t5
t2 t7

t8 t5 t5 t5 t6 H

0.004L+0.9
C1 C1 C1

B/2 tbottom t1

Figure 6 Midship section.

𝑡𝑑𝑒𝑐𝑘 = 0.035𝐿 + 5, (𝑚𝑚) (38)

𝑡𝑏𝑜𝑡𝑡𝑜𝑚 = 0.04𝐿 + 6, (𝑚𝑚) (39)

𝑡1 = 𝑡𝑏𝑜𝑡𝑡𝑜𝑚 + 3, (𝑚𝑚) (40)

𝑡2 = 0.5(𝑡3 + 𝑡5 ), (𝑚𝑚) (41)

𝑡3 = 0.8𝑡𝑏𝑜𝑡𝑡𝑜𝑚 , (𝑚𝑚) (42)

𝑡4 = 0.9𝑡𝑑𝑒𝑐𝑘 , (𝑚𝑚) (43)

yordan.garbatov@tecnico.ulisboa.pt 10
 𝑡5 = 0.7𝑡𝑏𝑜𝑡𝑡𝑜𝑚 , (𝑚𝑚) (44)

𝑡6 = 1.2𝑡𝑑𝑒𝑐𝑐𝑘 , (𝑚𝑚) (45)

𝑡7 = 0.9𝑡𝑑𝑒𝑐𝑐𝑘 , (𝑚𝑚) (46)

𝑡8 = 1.2𝑡𝑏𝑜𝑡𝑡𝑜𝑚 , (𝑚𝑚) (47)

The final thicknesses of the ship hull have to be defined to satisfy the minimum midship
section modulus requirement, 𝑊.

The final thicknesses of the ship hull should be defined in a way to satisfy the minimum
midship section modulus requirement calculated as:

𝑊 = 0.01𝐶1 𝐿2𝐵𝑃 𝐵(𝐶𝐵 + 0.7), 𝑐𝑚2 𝑚 (70)

where:

300−𝐿𝐵𝑃 1.5
10.75 − ( 100
) , 90 ≤ 𝐿𝐵𝑃 ≤ 300, 𝑚
𝐶1 = 10.75, 300 ≤ 𝐿𝐵𝑃 ≤ 350, 𝑚 (71)
𝐿 −350 1.5
10.75 − ( 𝐵𝑃 ) , 350 ≤ 𝐿𝐵𝑃 ≤ 427, 𝑚
{ 150

5 PROJECT INPUT
Ship 𝑳𝑩𝑷 , m, 𝑩, m, 𝑫, m 𝑻, m 𝑪𝒃
nº Length Breadth Depth Draught Block coefficient
1 149 19.2 10.1 7.3 0.78
2 157 20.7 10.9 7.9 0.78
3 164 22.0 11.5 8.4 0.78
4
5 177 24.2 12.7 9.2 0.78
6
7
8 191 26.9 14.1 10.3 0.77
9 196 27.7 14.5 10.6 0.77
10 200 28.5 14.9 10.9 0.77
11
12 207 29.9 15.7 11.4 0.77
13
14
15
16
17
18 225 33.4 17.5 12.7 0.77
19
20
21
22 235 35.4 18.6 13.5 0.78
23
24
25
26
27
28 248 38.1 19.9 14.5 0.79

yordan.garbatov@tecnico.ulisboa.pt 11
 29
30
31
32
33
34 259 40.4 21.1 15.4 0.81
35 261 40.7 21.3 15.5 0.81
36 263 41.1 21.5 15.7 0.81
37 264 41.4 21.7 15.8 0.81
38 266 41.8 21.9 15.9 0.82
39 268 42.1 22.1 16.0 0.82
40 269 42.4 22.2 16.2 0.82
41 261 40.7 21.3 15.5 0.81
42 182 25.2 13.2 9.6 0.78
43 187 26.1 13.7 9.9 0.77

44 222 32.9 17.2 12.5 0.77

6 STUDENTS
Ship nº Nº Name
1 76295 Simion Petru Stefan
2 78348 Tiago Vargas Vitorino
3 89996 Martim De Noronha Bretão Coelho dos Reis
4 93553 João Francisco Ribeiro Reis
5 99812 Miguel Duarte Dias de Figueiredo Gomez
6 99898 Bruno Miguel Guerreiro Ramos
7 100536 Martim José Baptista Gomes Alves
8 100537 Miguel Antunes Ramos
9 100542 Pedro Manuel Lopes Santareno
10 100543 Pedro Miguel de Araújo Ferreira Carapeto
11 102502 Ricardo da Rosa Rodrigues
12 102772 Matilde Borges da Fonseca Passanha
13 102814 Rodrigo António Costa Querido
14 102841 Miguel Ângelo Ferreira Reis
15 102916 Jessika Fabiana Gomes da Costa
16 102961 Luís Filipe Ferreira Maia Barbosa
17 103144 Edgar Henrique Teixeira Teles
18 103322 Pedro Jorge Cardoso Coutinho Lopes
19 103413 Dino Talpa
20 103535 Manuel Laing Correia de Matos Quintas
21 103875 David Santos Sánchez
22 107579 Pedro de Melo Felisberto
23 112010 José Gonçalves Adão Francisco
24 112139 Joshua Unger
25 112158 Filippo Violante
26 112159 Francesco Pittaluga
27 112180 Edoardo Braggio
28 112207 Petros Linaras
29 112308 Giuseppe Maria Piacenza
30 112324 Michele Zito
31 112396 Rafael Barrabas Striani
32 112704 Ola Westersjø Nesheim

yordan.garbatov@tecnico.ulisboa.pt 12
 33 112794 Magnus Behrens
34 113007 Alberto Utrera García
35 115206 Leonardo Castanheira de Almeida Cabaça
36 115389 Mathilde Ellen Marie Hancock
37 115404 Luc Chen
38 115405 Emile Jean Auguste Laroussinie
39 115488 Aarne Sakari Laitakari
40 115584 Daniel Petcu
41 115618 Corentin Valade
42 115621 Enguerrand Marie Patrick Le Poittevin De La Croix De Vaubois
43 115347 Evangelia Eleni Sapountzaki

yordan.garbatov@tecnico.ulisboa.pt 13