515

1 INTRODUCTION

The ship's construction is formed from elements

consisting of plates and stiffened plates connected.

These elements support local and global forces to

withstand the loads on the ship. If the load works on

different types of construction, then the response of the

two types is also different. The two types of ship

construction are single and double hull construction.

Both types of hulls have advantages and

disadvantages. A single hull allows for greater cargo

capacity, but it is also more susceptible to structural

damage from grounding or collisions. While the

double hull has additional strength due to the inner

hull, which functions to prevent liquid from directly

entering the cargo space, the load carried is reduced

due to the additional construction element, namely the

inner hull. Damage that may occur to ship hulls is

generally asymmetrical. This damage caused a

reduction in the ship's strength. Thus, evaluating the

impact of damage on both single and double-hull ship

structures during a grounding or collision is crucial.

Numerical studies related to structural analysis

focused on structures in general and their application

to ships have also been carried out. Rakowski and

Guminiak [1] studied the free vibrations of

geometrically nonlinear elastic Timoshenko beams

with fixed supports and used numerical method

implemented in finite elements. Li and Chen [2]

concentrated on creating empirical formulas for

designing hull structures and estimating safety,

employing nonlinear finite element analysis to

evaluate plates under biaxial compression. The

methodology can be applied to bio-composites using

calculation methods detailed in current regulations,

following an approach similar to that used in the study

by Velasco-Parra et al [3] and their research focused on

assessing the feasibility of using jute fiber and

bioepoxy resin for constructing a boat hull. A finite

element study of a special channel beam with thin

Ship Hull Construction Analysis to the Ultimate

Strength Considering Damages

M.Z. Muis Alie

, A. Ardianti

, J. Juswan

, T. Rachman

, A. Alamsyah

, N. Indah

& N.S. Aulia

Hasanuddin University, Makassar, Indonesia

Institut Teknologi Kalaimantan, Balikpapan, Indonesia

ABSTRACT: Damage to ship construction causes its structure to lose its ultimate strength. The damage could

result from a collision or grounding. The ship's structural integrity following a collision or grounding must be

evaluated. This is done to satisfy the ship's structural design requirements. The objective of the present study is

to analyze the effect of damage implemented on single and double hull construction to the ultimate strength of

ship structure. The ultimate strength of a ship's hull girder following damage from a collision or grounding was

ascertained a numerical approach is used in this study. The cross-section sample of the ship is taken in this study

namely single and double hull of bulk carrier. The modeling of damage to the ship's bottom and side shell sections

from collisions and groundings are taken into account to know the influence of damage when it is applied to

single and double hull.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 19

Number 2

June 2025

DOI: 10.12716/1001.19.02.21

516

walls and reinforced webs was carried out by Grenda

and Paczos [4]. Zhong et al [5] used computational

methods to examine the overall strength characteristics

of a hull girder with a sandwich plate upper deck and

a laser-welded web core. Additionally, they examined

how the girder behaved under different load

combinations, torsion, and vertical and horizontal

bending. Quispe et al [6] tested a reduced-scale hull

box girder for four-point bending using both

experimental and computational modeling techniques.

The dynamic maximum load-bearing capability of

ultra-large container ships under actual loading

conditions was examined by Jagite et al [7] and their

study concentrated on assessing the hull girder under

lateral and localized loads brought on by various cargo

loading circumstances, in addition to a composite

bending moment obtained from hydro-elastic research

over an extended period of time. The maximum

strength of a ship hull girder model with apertures was

investigated by Zhao et al [8]. The effects of coupled

bending moments and lateral pressure were the main

focus of Ma et al [9] investigation into the load-bearing

properties and failure behaviors of hull girders at

various scaling factors. Cui et al [10] extended their

study by taking into account the impacts of elastic

shakedown in order to evaluate the maximum strength

of hull constructions. Deng et al [11] Using both

experimental techniques and finite element analysis

examined the ultimate strength and buckling failure

behavior of single-hull and double-hull girders with

broad deck apertures under cyclic ultimate bending

moments. Babazadeh and Khedmati [12] expanded on

earlier studies by examining the effects of fractures on

a ship's hull girder's ultimate longitudinal strength. To

ascertain the ultimate strength of an ISSC2000 bulk

carrier, they conducted a progressive collapse analysis,

taking into account crack damage at several points on

the hull girder, such as the deck, sides, bottom, and

double bottom.

Shi and Gao [13] used a steel model with

superstructures to conduct a collapse experiment. The

experiment's outcomes confirmed important factors

including starting flaws and welding residual stress

that were part of the modeling of nonlinear finite

elements. To determine the crashworthiness of double-

hull constructions, a conceptual design method for

evaluating grounding and collisions was put forth by

Liu et al [14]. Using the section modulus as a strength

indicator, Zhang et al [15], introduced a unique

technique for assessing strength decrease using

stiffness loss analysis. According to their findings, a

hull girder's decreased rigidity after damage does not

always indicate a decline in strength. They confirmed

the idea that strength and stiffness loss are not always

equal by analyzing 13 standard cross-sections (UNSS).

Using both experimental and computational

techniques, [16] Wang and Wang (2020) examine the

torsional failure response of a single-compartment hull

girder that is intended to serve as a scaled model of the

mid-ship section of a container vessel. The simplified

progressive collapse approach was refined by Li et al

[17] to better forecast how ship hull girders would react

to cyclic loading. The cyclic progressive failure method

is the name given to this improved strategy. In one

study, Wang and Wang [18] examined changes in the

thickness and length of the plate geometry at various

scales using several genuine girders from the hull of a

10,000 TEU cargo ship. By analyzing the scaling

properties of ultimate strength and collapse behavior

in hull girders, this work sought to create a more

precise scaling criterion for comparing the ultimate

strength of scale models with full-sized ships. In

contrast to situations when the bending force grows

gradually, research shows that cyclic loading can lower

a ship hull girder's ultimate strength. This serves as the

foundation for previous research conducted by Liu and

Guedes Soares [19]. Examining the collapse process of

a hull girder with large deck apertures under torsional

stress and determining the critical element causing the

change from warping failure to shear failure were the

objectives of the study.

According A double-hull tanker under biaxial

bending was examined by Kuznecovs et al [20] in four

different situations: an intact hull, a hull with collision

damage, a newly built hull, and a hull impacted by

corrosion. The purpose of the study was to evaluate the

accuracy and processing requirements of two distinct

approaches. Vu Van et al [21] study aimed to evaluate

how different corrosion levels and beginning defects

affected the maximum bending moment in two

different bulk carrier sizes and kinds. Zhang et al [22]

used both finite element modeling and experimental

testing to expose scaled doublehull side structures to

quasi-static impacts at the mid-span using conical and

knife-edge indenters. Investigating fracture behavior

and related energy dissipation mechanisms was the

study's goal. Using a numerical approach, this study

investigates the ultimate strength of ship hull girders

in single and double hull configurations. The analysis

considers the damage caused by longitudinal bending

during both the hogging and sagging phases. The

impact of damage on the ultimate strength of ship

constructions with one or two hulls is another novel

topic covered in the study.

2 SHIP PARTICULAR

This research examines how single and double-hull

girder designs affect ultimate strength through

analytical techniques. The study examines the cross-

section of bulk carriers with single and double hull

constructions, analyzing them under conditions of

hogging and sagging. The grounding damage is

located in the lower part and is assumed to be evenly

distributed in both single and double hull

configurations. The collision damage is found on the

external surface of both single and double hull designs.

The collision damage is believed to be situated on the

shear strake at the corner of the deck. The ship

measures 32.2 meters in width and 19.062 meters in

depth. Along the longitudinal direction, the frame

spacing is fixed at 5.1 meters. The ship's single and

double hulls are the same size. Both single and double

hull designs have the same material characteristics,

such as density, yield strength, Poisson's ratio, and

Young's modulus (see Table 1). The analysis excludes

considerations of initial deflection, corrosion, or cracks.

It is assumed that the cross-section remains planar

during assessing progressive collapse. The final

strength assessment for both single and double hull

bulk carriers is performed under conditions of hogging

and sagging.

517

Table 1. Material properties

Item

Value

Unit

Text

Density

7.89 × 10-9

N/mm

Young Modulus

206000

N/mm

Poisson Ratio

0.3

Yield Strength

315/355

N/mm

3 DAMAGED MODELLING ON SHIP CROSS

SECTION

Figure 1. Section view of a single-hull bulk carrier

Figure 2. Section view of a double-hull bulk carrier

In this study, the cross sections of single and double

hull constructions for intact vessels are shown in

Figures 1 and 2, respectively, along with their

dimensions. Figures 3 and 4 illustrate grounding and

collision damages for both single and double hull

constructions of bulk carriers. It is crucial to emphasize

that, as seen in Figure 3, the symmetrical grounding

damage is located in the cross-section's lower outer

region in both single and double bulk carriers. In the

event of collision damage, it is assumed to be

positioned at the corner of the deck, as illustrated in

Figure 4. The applied rotational force is given at one

side of the cross section, while the other side is set up

to be constrained.

Figure 3. Grounding damages on single hull and double hull

bulk carriers

Figure 4. Collision damages on single hull and double hull

bulk carriers

4 MODEL, LOAD AND BOUNDARY CONDITION

The finite element models for the single-hull and

double-hull bulk carrier designs are shown in Figure 5.

In the whole concept, a shell element is used for both

kinds of bulk carriers. The loading and boundary

conditions for single-hull and double-hull bulk carriers

are shown in Figure 6. While the other side of the cross-

section has an applied load in the form of a rotating

force, the other side has a rigid body connection. In this

instance, the Multiple Point Constrained (MPC)

method is applied with the rotation force attached.

Figure 5. Finite Element Model of Bulk Carrier

Figure 6. Load and Boundary Conditions

5 DAMAGES MODEL

The 3D Finite Element Model used to build single and

double hulls in bulk carriers while taking collision and

grounding damage into account is seen in Figures 7

and 8. The shell elements are implemented into the

whole cross-section of bulk carriers. Assuming that the

elements are removed from the 3D models of bulk

carriers. The elements at the outer bottom plate,

including stiffeners, are eliminated, and the length of

the damage is 2 meters. The assumption of grounding

damage is measured from the center line to the left and

the right, as shown in Figure 7, marked by the circle

line. The elements at the deck side corner, represented

as collision damages, have also been removed,

including their stiffeners, as expressed in Figure 8. The

material is assumed to be homogenous and isotropic.

As the fundamental case, the analysis does not

consider the initial deflection, strain-hardening effects,

and cracks.

Figure 8. Collision Damages of Bulk Carriers

518

6 SIMPLIFIED ANALYTICAL METHOD

The procedure for the analytical method is derived

irrespective of the influence of neutral axes and

damages.

1. The cross-section is divided into elements, which

consist of unstiffened and stiffened plates.

2. The average stress-average strain relationships are

derived for individual elements, including the

effects of buckling and yielding, as shown in Eq. 1.

( )



(1)

The axial stress corresponds to the axial strain,

determined by the average stress-average strain

relationship calculated in advance for the individual

elements. Generally, the average stress-average strain

relationship in accordance with buckling and yielding

is a nonlinear function of strain. It is assumed that the

cross-section remains plane, therefore, the axial strain

at the i-th structural element due to the horizontal and

vertical curvatures are stated as follows Eq.2:

( )

i i i i H i V

y z y z

   

= + +

(2)

P, MV and MH are the axial force, vertical, and

horizontal bending moments, respectively, and they

are obtained by integrating axial stresses over the intact

cross-section as stated in Eqs. 3, 4, and 5.





(3)

H i i i

M y A





(4)

V i i i

M z A





(5)

where N is the number of intact elements and Ai is a

cross-section of the individual elements. However, to

obtain the stress and moment at the deck and bottom,

the longitudinal bending of the hull girder needs to

fulfill the condition of the zero-axial force stated in Eq.

3. Eqs. 1 and 2 are substituted into Eqs. 3~5, thereby

forming a set of nonlinear simultaneous equations

concerning the axial strain



0 and curvatures



V and



This is used to determine the relationship between

crosssectional forces and deformations. The equation

states the location of the neutral axis in the y-z plane on

a straight line Eq. 6:

i H i V

  

+ + =

(6)

3. Derive the tangential axial stiffness of individual

elements Di, in Eq. 1, from the recent average stress-

average strain curve Eq. 7.



 = 

(7)

where



4. Calculate the position of the neutral axis yG and zG,

Eqs. 8 and 9

i i i

y D A















(8)

i i i

z D A















(9)

yG and zG the coordinates at the neutral axis are

measured from the origin at the bottom keel. Here,

yi and zi are the coordinates of individual elements.

5. Evaluate the flexural stiffness of the cross-section

regarding the neutral axis, Eq. 10

HH HV

VH VV



   





   

   





   



   

(10)

where, the stiffness is calculated as follows:

( )

( )( )

AA i i

HH i i G i

VV i i G i

HV VH i i G i G i

D D A

D D y y A

D D z z A

D D D y y z z A

=−

= = − −



where



MH,



MV,



V are the incremental of the

horizontal and vertical bending moment, including

horizontal and vertical curvatures, respectively.

6. Calculate the individual elements' strain, curvature,

and stress increments using the slope of the average

stress-average strain curve.

7. Generate a curve of bending moments versus

rotations and convert it to curvature by dividing the

rotation by length.

8. Plot a curve of the bending moments against the

curvatures.

9. Proceed to the next step.

7 RESULT AND DISCUSSION

Single-hull construction has just one watertight outer

layer extending across the entire structure. Because

there is only one layer, single-hull ships present a

higher risk to the marine environment in the event of

an accident. In contrast, double hull construction

incorporates an additional layer, with the space

between the two layers serving as ballast tanks. These

ballast tank spaces run the full length of the cargo area,

519

providing a significant safety advantage. Single-hull

designs lack these ballast spaces. However, double-

hull construction requires more steel, making the

building process longer. The ballast compartments in

double-hull ships are more susceptible to hull fractures

and small failures than singlehull designs. Operators of

double-hull ships often report cargo leakage into

ballast tanks due to stress, fatigue, or construction

flaws. This study examines the damage to the outer

bottom part of both singlehull and double-hull bulk

carriers.

By showing the moment-curvature relationship for

both hogging and sagging circumstances, which are

pertinent to single and double-hull designs of bulk

carriers, this paper illustrates the maximum strength

achieved by the numerical technique. According to

Babazadeh and Khedmati [12], Kuznecovs et al [20],

and Liu et al [14], the moment-curvature curve shows

the hull girder's bending moment capacity under both

tension and compression. An alternative method was

taken by Yao and Nikolov [24], who demonstrated the

load-carrying capacity based on the connection

between bending moment and curvature by

integrating Smit's technique into the software

"HULLST." For simple computations, the midship

section's distance of one frame represents the ship's

length Yao and Nikolov [24], Cui et al [10], Kuznecovs

et al [20]. In bulk carriers exposed to vertical bending

moments during hogging and sagging circumstances,

the ultimate strength of single and double hulls is

divided into two categories intact and damaged.

Figures 9 and 10 use FEM and HULLST, respectively,

to compare the ultimate strength of single and double-

hull bulk carriers in their undamaged states.

Figure 9. Comparison of moment-curvature of single hull

Bulk Carrier (Intact)

Figure 10. Comparison of moment-curvature of double hull

Bulk Carrier (Intact)

Figure 11. Comparison of moment-curvature of single hull

Bulk Carrier (Collision)

Figure 12. Comparison of moment-curvature of double hull

Bulk Carrier (Collision)

Figures 11 and 12 illustrate the comparison of

maximum strength between FEM and HULLST when

exposed to collision damage in single and double hull

bulk carriers. The moment-curvature curves produced

by FEM are represented with solid lines, whereas those

generated by HULLST are indicated with dashed lines.

It has been noted that the final strength determined

using the numerical method through FEM is slightly

higher than that obtained from HULLST. This is also

seen in the previous condition namely the intact

condition where this phenomenon is found. The single

and double hull construction gives influence to the

ultimate strength. By the additional of construction

element in term of an inner hull, it affects the inertia

and section modulus. This undoubtedly affects the

changes in the neutral axis's location.

Figure 13. Comparison of moment-curvature of single hull

Bulk Carrier (Grounding)

520

Figure 14. Comparison of moment-curvature of double hull

Bulk Carrier (Grounding)

The ultimate strength analysis using another

sample like box girder was also investigated by Ao and

Wang [23], Deng et al [11] and Quiespe et al [6] using

numerical analysis. In the present study, by using

numerical analysis and simplified method/analytical

method, the influence of damages to the single and

double hull constructions of bulk carriers are

investigated. Figures 13 and 14 compare the ultimate

strength of bulk carriers with single and double hulls

while accounting for the impact of grounding damage.

The comparison of the ultimate strength single and

double hull under hogging and sagging conditions

subjected to grounding damages are 4.3% and 3.8%,

respectively. While for collision damages, the

comparison of the ultimate strength between single

and double under hogging and sagging are 2.2% and

4.5%. It has been noted that adding an inner hull in

double hull construction affects the ultimate strength,

even though the damage from grounding happens at

the outer bottom plate. In this instance, double-hulled

ships exhibit greater bending moment strength than

single-hulled ships.

8 CONCLUSIONS

This study used numerical analysis to evaluate the

ultimate strength of bulk carriers with single and

double-hull configurations. The maximum strength of

both ships was assessed through an analytical

approach, and these results were subsequently

compared with numerical analysis. From the

perspective of ultimate strength, a bulk carrier

designed with a double hull and an inner hull is

stronger than one with a single hull construction. As

calculated using numerical methods, the maximum

strength of the two ships exceeds that obtained

through analytical methods. This is probably due to

removing certain elements and how stress is

distributed. The total strength under hogging and

sagging situations is largely determined by the design

of the single and double hulls. It is also found that the

ultimate strength obtained by numerical method is in

good agreement with analytical method. This study

contributes to the guidelines for designing and

constructing ship hulls.

ACKNOWLEDGEMENT

The Ministry of Research, Technology, and Higher Education

of the Republic of Indonesia (DRTPM), through Hasanuddin

University, provided funding under contract numbers

050/E5/PG.02.00.PL/2024 and 02035/UN4.22.2/PT.01.03/2024.

The authors are grateful for this assistance.

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