The Effect of Subgrade Coefficient on Static Work of a Pontoon Made as a Monolithic Closed Tank
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applied
sciences
Article
The Effect of Subgrade Coefficient on Static Work of a Pontoon
Made as a Monolithic Closed Tank
Anna Szymczak-Graczyk
Department of Construction and Geoengineering, Faculty of Environmental and Mechanical Engineering,
Poznan University of Life Sciences, Piatkowska Street 94, 60-649 Poznan, Poland;
anna.szymczak-graczyk@up.poznan.pl; Tel.: +48-602-516-345
Abstract: This article presents the effect of taking into account the subgrade coefficient on static
work of a pontoon with an internal partition, made in one stage and treated computationally as a
monolithic closed rectangular tank. An exemplary pontoon is a single, ready-made shipping element
that can be used as a float for a building. By assembling several floats together, the structure can
form a floating platform. Due to the increasingly violent weather phenomena and the necessity
to ensure safe habitation for people in countries at risk of inundation or flooding, amphibious
construction could provide new solutions. This article presents calculations for a real pontoon made
in one stage for the purpose of conducting research. Since it is a closed structure without any joint
or contact, it can be concluded that it is impossible for water to get inside. However, in order to
exclude the possibility of the pontoon filling with water, its interior was filled with Styrofoam. For
static calculations, the variational approach to the finite difference method was used, assuming
the condition for the minimum energy of elastic deflection during bending, taking into account
the cooperation of the tank walls with the Styrofoam filling treated as a Winkler elastic substrate
and assuming that Poisson’s ratio ν = 0. Based on the results, charts were made illustrating the
Citation: Szymczak-Graczyk, A. The
change in bending moments at the characteristic points of the analysed tank depending on acting
Effect of Subgrade Coefficient on loads. The calculations included hydrostatic loads on the upper plate and ice floe pressure as well as
Static Work of a Pontoon Made as a buoyancy, stability and metacentric height of the pontoon. The aim of the study is to show a finished
Monolithic Closed Tank. Appl. Sci. product—a single-piece pontoon that can be a prefabricated element designed for use as a float for
2021, 11, 4259. https://doi.org/ “houses on water”.
10.3390/app11094259
Keywords: floating platforms; pontoon; finite difference method; rectangular tanks; hydrostatic load;
Academic Editor: bending moments; Winkler elastic substrate; substrate stiffness; houses on water
Francesco Tornabene
Received: 9 March 2021
Accepted: 4 May 2021
1. Introduction
Published: 8 May 2021
Monolithic rectangular tanks can be used as single, ready-made elements serving
Publisher’s Note: MDPI stays neutral
as floats for buildings or as components for constructing floating platforms intended,
with regard to jurisdictional claims in
among other purposes, for mooring vessels. Recently, stationary floating structures have
published maps and institutional affil- seen an increased global interest [1]. It can be estimated that there are 40,000 to 50,000
iations. of these objects in the world [2]. Such interest should not come as a surprise, as design
solutions supporting amphibious construction can be applied in a wide manner. This is
due to numerous reasons, ranging from the interest in the land—water interface as a very
attractive site for the tourism industry to the increased expansion of living space in densely
Copyright: © 2021 by the author.
populated countries. Another developmental contributor to amphibious construction
Licensee MDPI, Basel, Switzerland.
might be progressive climate change, which causes heavy rainfall, inundation and flooding.
This article is an open access article
It is anticipated that, by the end of this century, water levels will have risen by a meter [3].
distributed under the terms and Amphibious construction is one of several methods oriented towards urbanisation of
conditions of the Creative Commons floodplains [4,5]. Floats can be made of various materials depending on their design
Attribution (CC BY) license (https:// purpose. The market offers floats made of plastic, metal or reinforced concrete. Reinforced
creativecommons.org/licenses/by/ concrete pontoons consist of a box filled with polystyrene and a cover plate installed later.
4.0/). The solution always poses a risk of inaccuracies in assembly, subsequent leaks and shorter
Appl. Sci. 2021, 11, 4259. https://doi.org/10.3390/app11094259 https://www.mdpi.com/journal/applsci
Appl. Sci. 2021, 11, 4259 2 of 12
use [6]. A more advantageous solution, although more computationally and executively
challenging, is to design and produce a tank as a single, ready-made element, possible to
Appl. Sci. 2021, 14, x FOR PEER REVIEW
be built without any additional construction work [7,8]. Typisation and prefabrication2are of 13
two of the most important directions in construction today. By optimizing prefabricated
structures, the aim is to achieve the best material consumption while ensuring the load-
bearing
concretecapacity
pontoons of consist
the structure
of a box and—consequently—economic
filled with polystyrene and a cover benefits.
plate The pontoon
installed later.
mentioned in this article was a single-piece tank made in one stage,
The solution always poses a risk of inaccuracies in assembly, subsequent leaks and shorter i.e., a closed box
without any contacts or joints. This excluded the possibility of
use [6]. A more advantageous solution, although more computationally and executively leakage or flooding of
the structure. is
challenging, Totominimize
design and theproduce
risk of the pontoon
a tank filling ready-made
as a single, with water, itelement,
was tightly filledto
possible
with Styrofoam.
be built without The any literature
additionalon the subjectwork
construction provides
[7,8].few publications
Typisation on closed tasks
and prefabrication are
made in one stage [7,8] and taking into account the effect of the
two of the most important directions in construction today. By optimizing prefabricatedsubstrate, which—for
the pontoonthe
structures, in question—was
aim is to achieve Styrofoam [9]. Since
the best material the technical
consumption literature
while ensuring has the
notload-
yet
provided solutions for this type of tank taking into account the
bearing capacity of the structure and—consequently—economic benefits. The pontoon effect of its substrate, it
was decided that a dedicated numerical analysis would be performed
mentioned in this article was a single-piece tank made in one stage, i.e., a closed box with- for the purposes
ofout
this
anystudy.
contactsIn building
or joints. structures,
This excluded plates
the often rest on
possibility of a substrate
leakage made ofofmaterials
or flooding the struc-
other than the buildings. The most commonly applied substrate scheme
ture. To minimize the risk of the pontoon filling with water, it was tightly filled is a Winkler elastic
with
substrate
Styrofoam. [10].The
This static scheme
literature on theissubject
a system of infinitely
provides closely arranged
few publications springs
on closed of equal
tasks made
stiffness. The Winkler
in one stage [7,8] andmodeltakingspecifies a substrate
into account withofthe
the effect theuse of one parameter,
substrate, which—for assuming
the pon-
that the working load is equal to the product of plate deflection and substrate stiffness.
toon in question—was Styrofoam [9]. Since the technical literature has not yet provided
Thus, the deflection of the substrate occurs only on the surface on which the load acts.
solutions for this type of tank taking into account the effect of its substrate, it was decided
Figure 1 schematically presents the Winkler elastic substrate, while Equation (1) provides
that a dedicated numerical analysis would be performed for the purposes of this study. In
the aforementioned correlation [11].
building structures, plates often rest on a substrate made of materials other than the build-
ings. The most commonly applied substrate p = Kz ·scheme
w is a Winkler elastic substrate [10]. (1)
This static scheme is a system of infinitely closely arranged springs of equal stiffness. The
Winkler model specifies a substrate with the use of one parameter, assuming that the
where
working(kN/m
p—load load is2 ),equal to the product of plate deflection and substrate stiffness. Thus, the
deflection
w—deflection of (m),
the substrate
and occurs only on the surface on which the load acts. Figure 1
schematically presents the Winkler elastic substrate, while Equation (1) provides the
Kz —substrate stiffness (kN/m3 ).
aforementioned correlation [11].
Figure1.1.Winkler
Figure Winklersubstrate
substratemodel
model(drawn
(drawnby
bythe
theauthor).
author).
Another model of the two-parameter Winkler-type substrate is proposed in [12],
p=K ∙w (1)
where instead of springs, a system of densely spaced bars with a specific bending stiffness
wherewas introduced. In the Filonenka–Borodić model, it is assumed that a Winkler-type
value
p—load (kN/m
substrate 2),
is covered on its upper surface by an unlimited membrane, tensioned with evenly
w—deflection
distributed (m),forces
tensile and [11]. The model of Vlasov and Leontiev is one of the more accurate
Kz—substrate
models stiffness
of substrate (kN/m3It
mapping. ). consists of determining an elastic layer, the deflection of
Another model of the two-parameter
which is determined approximately using detailed Winkler-type substrate
material data is proposed
describing in [11].
this layer [12],
where instead substrate
Two-parameter of springs,models
a system of as
such densely
Vlasov,spaced bars or
Pasternak with a specific bendingsystems
Gorbunov–Posadov stiffness
ofvalue was introduced.
equations are much lessIn the Filonenka–Borodić
frequently used, due tomodel, it is assumed
the difficulty that a Winkler-type
in making calculations
substrate
with is covered
their use [13–16].onThe its differential
upper surface by anofunlimited
equation membrane,
a plate resting tensioned with
on a one-parameter
Winkler substrate is tensile
evenly distributed described by [11].
forces Equation (2). of Vlasov and Leontiev is one of the more
The model
accurate models of substrate mapping. It consists of determining an elastic layer, the de-
Dαt 2material data describing this
D∇2 ∇2 wapproximately
flection of which is determined + Kz w = q − using ∇ ∆T
(1 + νdetailed
) (2)
layer [11]. Two-parameter substrate models such as Vlasov, h Pasternak or Gorbunov–Po-
sadov systems of equations are much less frequently used, due to the difficulty in making
where
calculations with their use [13–16]. The differential equation of a plate resting on a one-
parameter Winkler substrate is described by Equation (2).
Dα
D∇ ∇ w + K w = q − (1 + ν) ∇ ∆T (2)
hAppl. Sci. 2021, 11, 4259 3 of 12
Eh3
D= 12(1−v2 )
—plate flexural rigidity,
2 ∂2
∇2 = ∂x∂
2 + ∂y2 —Laplacian,
∂4 ∂4 ∂4
∇2 ∇2 = ∂x 4 + 2 ∂x2 ∂y2 + ∂y4 —bi-Laplacian,
ν—Poisson’s ratio,
h—plate thickness,
w—plate deflection,
αt —coefficient of thermal expansion,
q—load perpendicular to the central surface of the plate,
∆T—difference in temperature between lower plate Td and upper plate Tg determined by
correlation: ∆T = Td − Tg , αt is the coefficient of thermal expansion of the plate material,
Kz —subgrade stiffness reaction.
The Winkler-type substrate model may raise doubts in terms of the assumption that
the substrate dislocates only at points where the load acts. In fact, if the substrate on
which the structure is placed is continuous, displacements also occur outside its load zone.
Although other, more accurate models do not have this drawback, they usually lead to the
same differential Equation (2), and the difference lies in the method through which their
coefficients are determined.
The purpose of this article is to analyse the statics of closed rectangular tanks with
an internal partition, the distribution of internal forces and their effect on the value of
bending moments, assuming the cooperation with the Styrofoam filling treated as an elastic
substrate. Additionally, the buoyancy and stability of the pontoon were checked against
standard guidelines from different countries. This work is of an application nature, and the
idea of this article is to develop a ready-made building element that can be built in without
any additional construction work.
2. Materials and Methods
2.1. Calculation Method
This work uses the finite difference method to solve the determined systems of dif-
ferential equations. Although this method is less frequently used, apart from the finite
element method, it can be successfully applied in static calculations. It is a universal way
of solving differential equations under certain boundary conditions, consisting in replacing
the derivatives present in the equation and boundary conditions by appropriate difference
ratios. Since the function describing the plate deflection is not known, unknown values are
assumed to be surface ordinates in a finite number of points (nodes), which are located
at the intersection points of the created division mesh for the object being calculated [11].
Tanks are complex structures, characterized by spatial work; therefore, too much simplifica-
tion in calculations or failure to take into account actual dimensions or material constants
may lead to erroneous results. Currently, with the possibility of using widely understood
computer-aided design, the calculation process is becoming less and less complicated.
However, most computational programs are based on the finite element method. This work
used the finite difference method as an alternative and as an effective means to solve the
determined systems of differential equations. The subject matter has been taken up in nu-
merous outstanding and fundamental scientific works [17–30]. The method has been used
in calculations regarding plate structures [9,31,32], tanks [7,8,33] or surface girders. This
work also used the condition for the minimum energy of elastic deformation accumulated
while undergoing bending in the plate resting on the elastic substrate. The calculations
were carried out traditionally, discretizing the object and creating systems of equations.Appl. Sci. 2021, 11, 4259 4 of 12
Then, using proprietary calculation solutions, the results were obtained, i.e., deflections at
each point of the division mesh. The energy functional is described by Equation (3).
Dx
( 2 2 2 " 2 #
∂2 w ∂2 w ∂2 w ∂2 w ∂2 w ∂2 w
V = +2 + + 2v −
2 ∂x2 ∂x∂y ∂y2 ∂x2 ∂y2 ∂x∂y
A
αt ∆T ∂2 w ∂2 w ∂2 ∆T
+2(1 + v) + 2 + dA (3)
h ∂x2 ∂y h
1x x
+ Kw2 dA − qwdA .
2
A A
where A is the plate area and other components of the formula are described in (2). Further
analysis was based on the adopted designations (4):
2 2 2
∂2 w ∂2 w ∂2 w ∂2 w ∂2 w
w2 xx = w2 xy = w2 yy = wxx = wyy = 2 (4)
∂x2 ∂x∂y ∂y2 ∂x 2 ∂y
On the assumption that the Poisson’s ratio was v = 0 and excluding the parameter of
temperature, the energy functional changed its form into Equation (3) [11].
Dx 2 1x x
V= w xx + 2w2 xy + w2 yy + Kw2 dA − qwdA (5)
2 2
A A A
2.2. Verification of Pontoon Buoyancy and Stability
The dimensioning of building structures is performed by the limit state method.
Therefore, pontoons, apart from meeting the first limit state—load capacity—must also
comply with the second limit, state serviceability. For this type of structure, it is the
fulfilment of buoyancy and stability requirements. Pontoon buoyancy is determined by
calculating its immersion depth, i.e., freeboard (the vertical distance from the top edge of
the pontoon (deck) to the surface of water). Its stability is determined by the location of
the metacentric point and tilt angle. The metacentre is the theoretical intersection point of
the buoyant vector of the tilted floating vessel and its plane of symmetry. Its distance from
the centre of gravity of the vessel (e.g., pontoon), the metacentric height, is a measure of
its stability [6]. If the metacentre is above the centre of gravity, i.e., the metacentric height
is positive, then the balance of the vessel is stable. If the metacentric distance is negative,
then the equilibrium is unstable. According to recommendations adopted in Poland, in
compliance with [34] when designing platforms loaded with crowds, it should be assumed
that the load is 3.0 kPa, whereas the tilt angle should be verified at 1.0 kPa applied to half
the width of the platform. The results of the stability and metacentric height calculations
presented later in the article were obtained on the basis of the procedures provided for
in Australian Standard AS 3962-2001 [35]. The schematic drawings of the pontoon (cross
sections) used to verify its buoyancy and stability are shown in Figure 2a,b, respectively [8].compliance with [34] when designing platforms loaded with crowds, it should be as-
sumed that the load is 3.0 kPa, whereas the tilt angle should be verified at 1.0 kPa applied
to half the width of the platform. The results of the stability and metacentric height calcu-
lations presented later in the article were obtained on the basis of the procedures provided
Appl. Sci. 2021, 11, 4259 for in Australian Standard AS 3962-2001 [35]. The schematic drawings of the pontoon 5 of 12
(cross sections) used to verify its buoyancy and stability are shown in Figure 2a,b, respec-
tively [8].
Appl. Sci. 2021, 14, x FOR PEER REVIEW 5 of 13
(a) (b)
Figure
Figure 2. The schematic
2. The schematic drawing
drawingof ofthe
thepontoon
pontoonunder
underconstant
constantload
load(a)
(a)and
andthetheschematic
schematicdraw-
draw-
ing
ing of the pontoon
of the pontoon under
under constant
constant and
andvariable
variableload
load(b),
(b),where
whereG—gravity
G—gravitycentre
centreofofthe
thepontoon,
pontoon,
B—buoyancy
B—buoyancy centre of of the
the pontoon
pontoon at atrest,
rest,K—keel,
K—keel,hhdd—immersion
—immersiondue duetotothe
thedead
deadweight
weight load,
load,
hg—location of the gravity centre, C—centre
C—centre of of the
the visible
visible area
areaof
ofthe
thewater
watersurface,
surface,M—metacentre,
M—metacen-
tre,
W W1—total
1 —total weight
weight of of constant
constant andvariable
and variableloads,
loads, B’—buoyancy
B`—buoyancy centre of of the
theloaded
loadedpontoon,
pontoon,
F—floating forces
F—floating forces in
in water,
water, hhmc
mc —metacentric
—metacentric height
height above
abovethe gravity
the centre,
gravity h
centre, —metacentric
mbh
mb —metacentric
height above
height above the
the buoyancy
buoyancy centre, and hh1d—height
centre, and of the buoyancy centre.
1d —height of the buoyancy centre.
3. Results
3.1. Static Analysis
3.1. Static Analysis of
of the
the Pontoon
Pontoon
For the purpose
For the purpose of of this
thiswork,
work,appropriate
appropriatestatic
staticcalculations
calculationswere
were made
made in in order
order to
to verify the effect that an internal elastic substrate has on the value and distribution
verify the effect that an internal elastic substrate has on the value and distribution of bend-
of
ingbending
moments moments
in closedinrectangular
closed rectangular tanks. Calculations
tanks. Calculations used theused
finitethe finite difference
difference method
method in variational approach and were made for a closed
in variational approach and were made for a closed rectangular tank with rectangular tank with
axial axial
dimen-
dimensions l , l and l
x ylz being
sions lx, ly and being at a 4:1:0.5 ratio, respectively. The value of substrate
z at a 4:1:0.5 ratio, respectively. The value of substrate stiffness stiffness
(Styrofoam) 3 The tank featured an internal partition in
(Styrofoam) was assumed as
was assumed asKKzz == 5000
5000 kN/m
kN/m3.. The tank featured an internal partition in
the centre of its length. The discretization grid took into account the symmetry plane of
the centre of its length. The discretization grid took into account the symmetry plane of
the structure, and assuming that the mesh size was s = ly /12, the system of equations with
the structure, and assuming that the mesh size was s = ly/12, the system of equations with
823 unknowns was obtained. The following loads were included: hydrostatic load acting
823 unknowns was obtained. The following loads were included: hydrostatic load acting
on the tank walls, uniform load acting on the bottom (Figure 3a), uniform load acting on
on the tank walls, uniform load acting on the bottom (Figure 3a), uniform load acting on
the upper plate of the tank (Figure 3b) and ice floe pressure acting on the circumference of
the upper plate of the tank (Figure 3b) and ice floe pressure acting on the circumference
the tank at half its height (Figure 3c).
of the tank at half its height (Figure 3c).
(a) (b) (c)
Figure 3.
Figure 3. Schematic
Schematicdrawing
drawingofofthe
thehydrostatic
hydrostaticload
loadacting
actingonon
the tank
the tankwalls
wallsand
andthethe
bottom
bottom(a),(a),
uniform load acting on the upper plate (b) and ice floe pressure (c), for which static calculations
uniform load acting on the upper plate (b) and ice floe pressure (c), for which static calculations
were made.
were made.
Figure 44 shows
Figure showsaaschematic
schematicdrawing
drawingofof the
the pontoon
pontoon with
with its its internal
internal partition
partition withwith
the
the numbering of points for which the article gives coefficients proportional to bending
numbering of points for which the article gives coefficients proportional to bending moments.
moments.Appl.
Appl.Sci. 2021,11,
Sci.2021, 14,4259
x FOR PEER REVIEW 66of
of12
13
Figure 4. Schematic
Figure 4. Schematic drawing
drawingof
ofthe
thetank
tankwith
withits
itsinternal
internalpartition
partitionwith
withthe
thenumbering
numberingofof
points and
points
cross sections.
and cross sections.
In
In order
orderto tocreate
createaanew newstructural
structural element
element that cancan
that be be
successfully
successfully used in construction
used in construc-
and because the idea of this article is to apply the obtained numerical
tion and because the idea of this article is to apply the obtained numerical calculation calculation results to
practice, the specific dimensions of the pontoon were assumed. The
results to practice, the specific dimensions of the pontoon were assumed. The dimensions dimensions enable its
production, transport and use as a single element or as a connected
enable its production, transport and use as a single element or as a connected floating floating platform, and
they were adopted
platform, and theyas length
were lx = 10asm,length
adopted widthlxly==102.5m,m,width
heightly l=z 2.5
= 1.25
m, m, thickness
height of all
lz = 1.25 m,
walls h = 0.08 m and substrate shear modulus (Styrofoam) K = 5000 kN/m 3 .
z
thickness of all walls h = 0.08 m and substrate shear modulus (Styrofoam) Kz = 5000 kN/m3.
The
The results
results of of the
the calculation
calculation of of coefficients
coefficients proportional
proportional to to the
the values
values ofof bending
bending
moments
moments at select points for the pontoon with and without the cooperation withits
at select points for the pontoon with and without the cooperation with itselastic
elastic
substrate
substrate are
are presented
presented in in Table
Table1.1. InIn order
ordertoto analyse
analysethe theeffect
effectof oftaking
takinginto
intoaccount
accountthe the
substrate
substrate on on the
the change
change in in the
the values
values of of deflections
deflections andand bending
bending moments,
moments, charts
charts were
were
drawn up showing the distribution of the values of bending moments in longitudinal and
drawn up showing the distribution of the values of bending moments in longitudinal and
transverse sections (Figures 5–8). The designations of points are in line with Figure 4, and
transverse sections (Figures 5–8). The designations of points are in line with Figure 4, and
as the most important, the hydrostatic load on the walls, the uniform load on the bottom of
as the most important, the hydrostatic load on the walls, the uniform load on the bottom
the pontoon and the uniform load of the top plate were selected.
of the pontoon and the uniform load of the top plate were selected.
By analysing the data obtained from the numerical calculations listed in Table 1, it
By analysing the data obtained from the numerical calculations listed in Table 1, it
can be concluded that, at each point of the partition grid as well as for each type of load,
can be concluded that, at each point of the partition grid as well as for each type of load,
the bending moments are reduced, taking into account the elastic substrate. Hence, the
the bending moments are reduced, taking into account the elastic substrate. Hence, the
tank components were assumed to rest on polystyrene, which while filling the inside of
tank components were assumed to rest on polystyrene, which while filling the inside of
the tank, reduces the values of bending moments, which in turn are used to calculate
the tank, reduces the values of bending moments, which in turn are used to calculate the
the required reinforcement area for the reinforced concrete structure of the tank. The
required reinforcement area for the reinforced concrete structure of the tank. The conclu-
conclusions resulting from Table 1 can be used to support a decision to reduce the amount
sions
of resulting
reinforced from
steel used Table
in the1 can be usedprocess
production to support
and athusdecision to reduce
for material andthe
costamount
savings.of
reinforced steel used in the production process and thus for material and cost savings.
3.2. Calculation Results of Buoyancy and Stability
Following the recommendations of Australian standard AS 3962-2001 [35], the buoy-
ancy for the analysed tank at self-load and at service load evenly distributed on the top
plate, amounting to 3.0 kN/m2 was calculated. Additionally, the metacentric height and tilt
angle were calculated at half load of the top plate equal to 1.0 kN/m2 . Since the second limit
state, i.e., serviceability, needed to be verified, it was necessary to assume for calculations
the load coefficient γF = 1, in accordance with [36,37].
As earlier, for detailed calculations the following data was used: length lx = 10 m,
width ly = 2.5 m, height lz = 1.25 m, thickness of tank walls and the upper plate h = 0.08 m,
reinforced concrete class C35/45 and Styrofoam filling with a volumetric weight 0.45 kN/m3 .Appl. Sci. 2021, 11, 4259 7 of 12
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Table 1. Comparison of coefficients proportional to the values of bending moments for the closed tank with an internal
omparison
son of coefficients
coefficients
partition,
of coefficients withtodimensions.
proportional
proportional the to theof
values
lbending
x = 10
values of m, ly = 2.5
bending
moments
m and
moments
forclosed
lz =the
for
the closed
1.25 m calculated
closed tankanwith
with
an and without
internal partition,taking intolxaccount
with dimensions. lx =lits
10 elastic
m, mly =and
2.5lm and
1.25lm z = 1
f coefficients
son of proportional to the
proportional values
to the of of
values bending
bending moments
moments for
forthe
the closed tank tank
tank with with
an internal
an internal
internal partition,
partition,
partition, with dimensions.
withdimensions.
with dimensions.
lx =lx10
= 10 =ly 10
m, m, m,
=ly2.5
= 2.5 =m2.5
my and and mz = m
lz = 1.25
lz = 1.25
with
able and
1. Table 1.
without
Comparison substrate.
Comparison
taking
of The
into of size
account
coefficients of
its mesh
coefficients
elastic
proportional division
proportional
substrate.
to the s =
to
The
values l
the/12.
values
size
of of
bending of
mesh bending
division
moments
d without taking into account its elastic substrate. The size of mesh division s = ly/12.
y moments
s
for= l
the
y /12. for
closed the closed
tank with tank
an with an
internal internal
partition, partition,
with with
dimensions. dimensions.
lx = 10 m, llxy == 10
2.5m,m la
thout taking
d without intointo
taking account its elastic
account substrate.
its elastic The
substrate. Thesize ofofmesh
size divisionss==llyy/12.
meshdivision /12.
alculatedcalculated
with and with andtaking
without without taking
into intoits
account account
elastic its elastic substrate.
substrate. The size of The sizedivision
mesh = ly/12. s = ly/12.
of meshsdivision
lue
yzed Value Analyzed
4 Analyzed Value
Value acc.
acc.
. Figure 4 Figure 4 4
Figure
K =K5000
K= 5000
= 5000K
kN/m
kN/m = 35000
kN/m 3 3 kN/m3 K
K == 5000
K = 5000
5000 kN/m
kN/m K kN/m=3 350003 kN/m3 K =K5000 KkN/m
= 5000 = 5000kN/m
3K = 50003 kN
kN/m3
0.69868
0.69868 0.69868
qsqs22 qs 2 0.43023
0.43023 qs 2 2 K qs
0.43023
qs = 5000
2 K
K = 5000 kN/m
=
kN/m 5000
−8.39831q 3
−8.39831q
3 kN/m
1 s−8.39831q
2
1
3
s 2 1 s 2 −6.05757−6.05757
K = 5000 q 1 s 2−6.05757
kN/m q 1 s 23 K =
q 15000
s K =
kN/m
2 −0.27892 50003
q
−0.27892
2 skN/m −0.27892
q 2
3
s q s kN/m q2s q2s K =q
−0.34647
K = 5000
2 q
−0.34647
2 s
3 −0.34647
0.69868 qs2 0.43023 qs 2 −8.39831q 1s2 −6.05757 q 1s2 −0.27892 q 2s −0.34647
MA3qs
MA3−8.97746 22 0.69868 0.69868
qs 2
−6.79305 qs qs 22
0.43023 qs0.43023
2
1.20015 q1qs 22
q1s2 −8.39831q
s1.20015 q1s22 1.04529 −8.39831q
1s2 1s2 2
q1s2 −6.05757
q1s2 21.04529 −6.05757
q1s2 −0.27892 q2s q−0.27892
q1s−0.27892
2 1s2 −0.27892 −0.27892
2s−0.34647 sq2s q2sq2s −q
q2−0.34647
qs−8.97746
−8.977462 qsM qs2 −6.79305 −6.79305 2−6.79305
2 qs qs2 1.20015
qs2 1.20015 −1.04529 qq2 2ss q2s q− q2 s q2s−0.34647
2
−8.97746 A3 0.69868 qs
qs 0.43023 q 1s2−8.39831q1 s 1.04529 q
6.057571s q1 s −
−0.27892
0.27892 −0.34647
0.34647
M
MA4−0.03089
−0.03089 qs
−0.03089
A4 22 −8.97746
qs 2 −8.97746
qs 2
−0.05852 qs 2
2 2
−0.05852 −6.79305
qs 2 −6.79305
qs 2
−5.88475 q qs
s 2 2
1−5.88475 1.20015
q s q 1.20015
1s
2 −4.57988 2 q 1sq 21s 2
−4.57988 1.04529
q s q1.04529
1s
2 −0.25756 2 q 2s q−0.25756
1s2 −0.27892
q s −0.27892
q
−0.30472 2s q2s q−0.30472
2s
q2s −q
qs2 qs2MA4 −0.05852 qs −5.88475 q 1s −4.57988 q1s2 q12s −0.25756 q 2s −0.30472
2 1 2 1 2
−0.03089 −0.05852
−8.97746 qs
qs22 2 −5.88475
−6.79305 qs22 q1s221.20015 q1 s2 −4.57988 1.04529
2 q1s2qq11ss2
−0.25756
− 0.27892 qq2ss
2 q2s2 q1s2
−0.30472
−0.34647 q2 s q2s
M
MB1−6.43673 qs −0.03089 −0.03089
−5.23986
qs 2 qs −0.05852 −0.05852
0.42435
qs q qs
s 2
−5.88475 −5.88475
0.47834
q s −4.57988 −4.57988
−0.25756
q s −0.25756
−0.30472
−0.25756 q s q s q s
q2s −q
qs−6.43673 qs −5.23986 qs 0.42435 q1s q1s2 20.47834 q1s q2ss−0.25756 q2s −0.30472 −0.30472
B1 2 2 1 2 1 2 1 2 2 2
−6.43673
−6.43673 2 qs
2 −5.23986
−5.23986 qs 2 qs
2
0.424350.42435q1s2− q1s 2
0.47834 0.47834 q1s 2 −0.25756
−0.25756 q 2s −0.30472
MB20.30288M qs 2 MB1
−6.43673 −−6.43673
0.03089
−0.12571
qs qsqs 2 2 − 0.05852
−5.23986 qs 2
1.95192
−5.23986
qs q 1sqs 2 5.88475
2 q1 s22 1.39644
0.42435 q −2 4.57988
0.42435
s q 1sq1s q21 s22 0.47834 − 0.25756
−1.03379
q 2 q2qs2q
0.47834
s s 2 −0.30472
−0.96737
−0.25756 qs2qs2s q2sq2s
−0.25756
q q2s −q
2 2
qs2 0.30288 qs qs−0.12571 qs q1qs12s1.95192 q1s q11ss221.39644 q1s q2s−1.03379 q2s −0.96737 q2s−0.96737
B2 2 2 1 2 1 1 2
0.30288
0.30288 qs 2 −0.12571
−0.12571 2 qs
2
1.951921.95192 q1s 2
1.39644 1.39644 q1s −1.03379
−1.03379 q 2s −0.96737
MC−0.32098M qs 2
M 0.30288 − −0.18282
0.30288
qs
6.43673
2 qsqs qs
2 22 −
−0.12571
5.23986−0.61660
qs
qs 22
−0.12571 qs 22
0.42435 q 2
1.95192
s −0.43870
q 1.95192
s q
0.47834
2 q q s 2s2
1.39644 0.16188
− 1.39644
0.25756
q s 2 q 2sq qs s 2 − 0.14114
0.30472
−1.03379 q q sq
−1.03379s2s q s −q
qs−0.32098
2 B2 qs −0.18282 qs −0.61660 qq1s1−0.61660 q1 1s −0.43870 q1s1s2−0.43870 q1s q2s 0.16188 q2s q2s 0.14114
C 2 2 2 1 1 2 1 2 1 2 2 2
−0.32098
−0.32098 2 qs −0.18282
−0.18282 qsqs 2 qs
2 −0.61660 2 q 1s
2 −0.43870 2 q 1s
1 2
0.161880.16188 q2s 0.141140.14114 q2s
−0.06428
My −0.06428M qs
28 2
−0.32098 0.21213
−0.32098
qs 0.21213 qs 22
−0.18282 −1.28998
−0.18282
qs−1.28998 s
qs 22 −0.91521
−0.61660 q q s 2 0.95926
−0.43870 q s q s 2 0.90710 q
0.16188 s q s
q1s2 −0.61660 q1−0.91521
s1.39644−0.91521
q1qs1s2 −0.43870 q1s0.95926 q0.16188 q2sq0.90710 q0
28 2 2 2 2 2 2 2 2 2 2
−0.06428 qs−0.06428
2 qs C qs
y M
2 2 0.30288
0.21213 qs2qs0.21213
qs qs2 −
qs0.12571
2 qs
−1.28998 qq11ss−1.28998
21.95192 qq1 s1s
2
−0.91521
1 q1s2 −0.95926
1.03379 q2 s 0.95926
1
q2s 2−
s 0.96737 2s 0.90710
2
q2s
Mz280.29028 qs
28 2
Mz0.29028 2 28 −0.06428
0.05238
−0.06428
qs 2 qs
22
0.21213 7.81254
qs0.21213
2 qs
22
−1.28998 5.27326
−1.28998
q 1s2
qq1s2
1s2
q 1s2 22 −0.91521−0.19180
−0.91521
q 1s2
q2qs 2sq1s2 0.95926
0.90710
−0.08074
q q2s q2sq2s
0.95926
2s 0q
0.29028
0.29028 qs2 413 qs M qs 2
0.05238
−0.32098qs
0.05238 qs2 2 0.05238
2 qs 2 qs 2
−0.18282 qs 2
7.812547.81254qq1s1s2− 7.81254
q s 2
0.61660 q1 s
1 q 1 s 22
5.27326 5.27326
−0.43870 5.27326
qq 1 s q 1s 2
−0.19180
0.16188 q2qs2s −0.19180
q 2 s q 2 s
0.14114 −0.08074
q2 s −0.08074
q 2s
My4130.15780
M qs2 y 0.29028
y0.15780 qs0.06149
0.29028
2 qsqs2 0.05238 1.29175
qs0.05238
2 qs
2
2
7.81254 0.70420
q7.81254
s 2 qq1s1sq
22 1
1s2 2 5.27326
−0.19180
−0.05392
q 2 q2s q1s2
5.27326
s −0.19180
−0.08074
−0.02615 q2s q2sq2s
−0.19180
q s −
0.15780
0.15780 qs2 413 qs 28
2 qs 2
0.06149
−0.06428
0.06149 qsqs2qs0.06149
2 qs 2 qs 2
2 1.29175
0.212131.29175 qq1s1s2− 1.29175
q1s 2 q 1 s 2
2 1
0.70420−0.70420 0.70420
q1s2 q 1s 2 1
−0.05392
0.95926 −0.05392
q2qs2s q2s q 2 s
0.90710
2
q2−0.02615
sq2s q2s q2s−0.02615 q
7.40284 qs2Mz
Mx7.40284 5.09798
0.15780 qs
2
2
qs
0.63264
0.06149 qs
2 1.28998 q1 s
2 0.19518
1.29175 qq1s1sq
0.91521 2 2 q1 s
s 2
−0.05392
−0.19180 2 q2s q1s2
0.70420
−0.02615
−0.08074
−0.05392 q s
Mx4137.40284 0.15780
qs qs 2
5.09798 0.06149
qs qs 2 1.29175 q s 2 0.70420 q s −0.05392 q s −
qq1s1s20.63264 q1s 0.19518 q1s −0.19180 q2s −0.08074 −0.08074 q
2 2 2 1 1 2 1 2 2
7.40284 qs2 429 qs 2
5.097985.09798
qs qs 2
2 0.63264 q1s 2 0.19518 q1s 2 −0.19180 q 2s −0.08074 q2s
1.18255 qs2My 413 0.65208
0.29028 qs2qs22 0.052380.63264
0.25493
qs 27.81254 q1 s2 0.19518
0.10836 qq1s1s2 2q12s2
5.27326 −0.19180
−0.05392
− 0.19180 q2qsq2 2ss 2 −0.02615q2qs2s q2s
−0.08074
My1.18255
My4291.18255 qs 2 7.40284
qs 2 7.40284
qs 2
0.65208qs0.65208
qs 2 5.09798
qs 2 qs5.09798
2
0.25493 qs 2
0.25493
q 1s2 q0.63264
1s2 q0.63264
1s2
0.10836 q0.10836
1s
q 1s2 q0.19518
1s2 q0.19518
1s2
−0.05392 q−0.05392
1s
q 2s
−0.19180
q 2s −0.19180
q2−0.02615
s q−0.02615
2s
q 2s
−q
1.18255 qs 429
−0.190952 qsM 413
2 0.65208
−0.10572qs 2
2qs2 2 0.25493
−0.22838
2 q1qs1s1.29175
2 2 0.10836
q1 s2 2 −0.12710 q11ss q 2s2
22 −0.05392
0.01971
− 0.05392 q2sqq2ss 2 −0.02615
0.01140
−0.02615 qs2s q2s
Mx429 M−0.19095 x 1.18255 0.15780
1.18255
qs
qs2 −0.10572 2 qsqs
−0.10572
0.06149
0.65208 qs
qs0.65208
qs2 −0.22838 2 qs 2
q1s2 0.25493
−0.22838 q1s −2.03845 1s2
0.70420
q0.25493 q2−0.12710
1s1 0.10836 q0.10836
q1s2 1.18854 1s2 2q1s
0.01971 −0.05392
q2s 1.07084 q2qs0.01140
−0.05392
2 q2qs 2s −q
q2s q2s 0.01140
x
−0.19095
−1.51448
−0.19095 qs 2 qs qs2 2 −0.10572
−0.61475qs 2 2qs
qs
2 −0.22838
−3.16197 q q1 1s22
s −0.12710−0.12710qq1s1 s 2 q 1s
2
0.019710.01971
q 2sq 2 s q2s 0.01140
M 437 M 429 −0.19095 2 2
qs 2 2
−0.10572 qs 2 2
−0.22838 q s 22 − −0.12710
0.19180 q sq s 2 − 0.080740.01971
q s q s
Mx437 −0.19095 qs
7.40284
2 qs −0.10572
5.09798 qsqs 0.63264 −0.22838
q s q s0.19518
2 q s −0.12710 q s 2 0.01971 q s q0
qs−1.51448
qs2 2y qs qs−0.61475 qs −3.16197 q1qs−3.16197 q1 1s −8.58140 qq1s1s2−2.03845 q1s 0.28786 q2sq2s 1.18854 q2s 0.18698 q2s q2s 1.07084
x 2 2 2 1 1 2 1 2 1 2 2 2
−1.51448
−0.36444
−1.51448 2 qs −0.61475
−0.08510
−0.61475 2 2qs
qs
2 −3.16197
−10.87368 12s2q1s
2
−2.03845−2.03845 2 q 1s
1 2
1.188541.18854 q2s 1.070841.07084 q2s
437 M
Mz −10.24996 437
M22x −1.51448
429 −1.51448
qs 2 2qs 2
−0.61475 −0.61475
qsqs
2 2 qs 2
2 −3.161972 −3.16197
q1−8.58140
s0.10836
2 q s 2
q2 1qs1s2 −2.03845
2 −2.03845
q1s0.28786
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q1.18854
2− 1.18854
q2sq0.18698 q s q1
qs−0.36444 qs2 −0.08510 qs −0.08510 qs2 −10.87368 qq−10.87368
q1qs1s2 −0.41450 qq1s1s2−8.58140 q1s2 0.28786 q2sqq22ss 0.28786 s 0.02615 q2s 0.18698
z 1 1 2
−0.364442 qs qs 1.18255
−0.08510
−8.16864 2 2qs
2 0.65208 −10.87368 s0.25493
22 q1s q2s 2 s2s q2s
−0.36444 qsqs −0.87588 11s −8.58140 0.28786 0.18698
0.18698 q
Mx797 M
−10.24996 x797 2 2 437−0.36444
−10.24996
qsqs qs 2 −0.36444
qs 2
−8.16864 qs 2
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q s 2 −10.87368
q 1s2 q 1s2 2 2
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q s 2 −8.58140
q 1s2 q 1s2
0.28786 q0.28786
s q0.28786
2s q 2s
0.18698 q0
2 qs qs−0.87588 q 1s −0.41450 1s1s2qq 1s 0.287862ss2s q2s q20.18698
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2 2
0.13284
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2 −0.10572 1.62748
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2 1.16812
q1 s −0.41450
1 − qq
0.12710 2
1s 1 −0.95842
0.01971
0.28786 qq2q 2 −0.90726
0.01140
0.18698
Mx0.13284
Mx8130.13284 qsMz 437 −10.24996
qs 2 −10.24996
qs 2
−0.22996 qs 2
−0.22996 −8.16864
qs 2 −8.16864
qs 2 qs 2
1.62748 −0.87588
q s 2 −0.87588
q 1 s 2 q 1s2 2
1.16812 −0.41450
q s 2 −0.41450
q 1 s 2 q 1s2
−0.95842 0.28786
q s q0.28786
2s q 2s
−0.90726 0q
qs22 qs qs21.62748 q1s2−q3.16197 1s −1.16812q1s2 qq211ss2 −0.95842
q2qs2s q2s q2−0.90726 q2s q2s
2 2 2 1 1 2
0.13284 qs2 821 −1.51448
−0.22996 qs −0.614751.62748 q1 s2 1.16812 2.03845 1.18854
−0.95842 1.07084
−0.90726s
Mx821 Mx 0.13284 qs 0.13284
2 qs2 −0.22996 −0.22996
qs2 qs2 1.62748 q1.62748 1s2 q1s 1.16812 q1.16812
1s2 q1s2 −0.95842 −0.95842
q2s q2s −
−10.87368
Mx 797 −0.36444 qs2 −0.08510 qs2 −8.58140 q1 s2 0.28786 q2 s 0.18698 q2 s
q1 s2
Appl. Sci. 2021, 14, x FOR PEER REVIEW 8 of 13
Mx 813 −10.24996 qs2 −8.16864 qs2 −0.87588 q1 s2 −0.41450 q1 s2 0.28786 q2 s 0.18698 q2 s
Mx 821 0.13284 qs2 −0.22996 qs2 1.62748 q1 s2 1.16812 q1 s2 −0.95842 q2 s −0.90726 q2 s
Figure 5.
Figure Charts of
5. Charts of bending
bending moments
moments due due toto the
the hydrostatic
hydrostatic load
load of
of the
the walls
walls and
and the
the uniform
uniform load
of theofbottom
load without
the bottom taking
without into account
taking the effect
into account of an elastic
the effect substrate
of an elastic and for
substrate thefor
and Poisson’s ratio
the Pois-
ν = 0 ratio
and taking intotaking
account theaccount
effect ofthe
aneffect
elasticofsubstrate 3 and for the
son’s ν = 0 and into an elasticKsubstrate
= 5000 kN/m
K = 5000 kN/m3 and Poisson’s
for
the Poisson’s
ratio ratio νsection
ν = 0 in cross = 0 in cross section4).1–1 (Figure 4).
1–1 (FigureFigure 5. Charts of bending moments due to the hydrostatic load of the walls and the uniform
Appl. Sci. 2021, 11, 4259 load of the bottom without taking into account the effect of an elastic substrate and for the Pois-
8 of 12
son’s ratio ν = 0 and taking into account the effect of an elastic substrate K = 5000 kN/m3 and for
the Poisson’s ratio ν = 0 in cross section 1–1 (Figure 4).
Appl. Sci. 2021, 14, x FOR PEER REVIEW 9 of 13
Figure 6.
Figure Charts of
6. Charts of bending
bending moments
moments duedue toto the
the hydrostatic
hydrostatic load
load of
of the
the walls
walls and
and the
the uniform
uniform load
of theofbottom
load without
the bottom takingtaking
without into account the effect
into account of an elastic
the effect substrate
of an elastic and for
substrate thefor
and Poisson’s ratio
the Pois-
= 0Poisson’s
νthe and taking into account the effect of an elastic substrate K = 5000 kN/m 3 and for the Poisson’s
son’s ratio ν = ratio
0 and taking into account the effect of an elastic substrate K = 5000 kN/m 3 and
ν = 0 in cross section 2–2 (Figure 4). The internal partition lies in the symmetry for
ratio
axis. ν = 0 in cross section 2–2 (Figure 4). The internal partition lies in the symmetry axis.
Figure7.
Figure 7. Charts
Charts of
of bending
bending moments due to the uniform load of of the
the top
top plate
plate without
without taking
takinginto
into
accountthe
account theeffect
effectof
ofan
anelastic
elasticsubstrate
substrateand
andfor
forthe
thePoisson’s
Poisson’sratio
ratioνν==00and
andtaking
takinginto
intoaccount
accountthe
the effect
effect of anofelastic
an elastic substrate
substrate K =K5000
= 5000
kN/mkN/m 3 and for the Poisson’s ratio ν = 0 in cross section 1–
3 and for the Poisson’s ratio ν = 0 in cross section 1–1
1 (Figure 4).
(Figure 4).Figure 7. Charts of bending moments due to the uniform load of the top plate without taking into
Appl. Sci. 2021, 11, 4259 9 of 12
account the effect of an elastic substrate and for the Poisson’s ratio ν = 0 and taking into account
the effect of an elastic substrate K = 5000 kN/m3 and for the Poisson’s ratio ν = 0 in cross section 1–
1 (Figure 4).
Figure
Figure8.8.Charts
Chartsof ofbending
bendingmoments
momentsdue duetotothe
theuniform
uniformload
loadofofthe
thetop
topplate
platewithout
withouttaking into
taking into
account
accountthe
theeffect
effect of
of an
an elastic
elastic substrate
substrate and
and for
for the
the Poisson’s ratio ν
Poisson’s ratio ν == 00 and
and taking
taking into
into account
account the
effect of an elastic substrate K = 5000 kN/m3 and for the Poisson’s ratio ν = 0 in cross section 2–2
(Figure 4). The internal partition lies in the symmetry axis.
In order to reduce the tilt angle in the designed pontoon, the weight of the pontoon was
increased. The calculations took into account an increase in the weight of the pontoon due
to the water absorption of concrete, which is about 4–5%. Table 2 also shows the pontoon
buoyancy and stability calculations, taking into account an increase in the pontoon’s weight
caused by 5% concrete absorbability. The results of calculations are summarized in Table 2.
Table 2. Results of buoyancy and stability calculations made for the analysed tank.
Pontoon Buoyancy
Bottom Pontoon Stability with Half of the Upper Plate
At Uniform Load
Thickness of At Self-Weight Load Loaded 1.0 kN/m2
3.0 kN/m2
the Analysed
Tanks Metacentric
Immersion Freeboard Immersion Freeboard Tilt Angle Freeboard
Height
(m) Depth hd (m) hf (m) Depth hd (m) hf (m) ϕ (◦ ) hf (m)
hmc (m)
0.08 0.72 0.53 1.02 0.23 0.33 7.00 0.33
0.08 * 0.75 * 0.50 * 1.05 * 0.20 * 0.32 * 6.92 * 0.30 *
* results for the pontoon with 5% concrete absorbability.
The presented calculations show that increasing the weight of the pontoon reduces
the tilt angle. Therefore, if it was required to reduce the tilt angle to a value below 6◦ ,
the thickness of the bottom plate should be increased while maintaining the remaining
parameters [8].Appl. Sci. 2021, 11, 4259 10 of 12
4. Discussion
The prefabricated pontoon analysed in the study as a ready-made element makes a
good alternative to currently available reinforced concrete platforms, consisting of a box
and a separate cover. The article presents the results of calculations made for a monolithic
rectangular tank intended to be used as a pontoon. There are still problems related to the
analysis, design and execution of concrete floating structures [38]. The literature on the
subject provides few works related to this subject. This article presents a pontoon that was
made on a 1:1 scale. The authors of [7] presented the series of closed tanks for use as a
pontoon with the following axial dimensions: lx , ly and lz at a ratio of 1:1:0.5; lx , ly and lz at a
ratio of 2:1:0.5; lx , ly and lz at a ratio of 3:1:0.5; and lx , ly and lz at a ratio of 4:1:0.5. However,
the effect of Poisson’s ratio and the subgrade coefficient was not taken into account. When
analysing the tanks of increasing length, it can be concluded that, for long tanks with size
proportions lx , ly and lz at γ:1:0.5 for γ > 3; the effect of disturbances resulting from the
gable walls’ load disappear in their central part; and the central part works similar to a
closed frame, i.e., a rod system. The paper [8] shows the effect of the increasing values
of bottom thickness on the values of deflections and moments. By thickening the bottom
at all points of the calculated tanks, the following regularity was noticed: deflections
decreased when the bottom plates became thicker. It was observed that the hydrostatic
load acting on the tank walls and the uniform load acting on the bottom combined with an
increase in bottom thickness led to an increase in bending moments acting in the middle
of the bottom plate span towards a shorter span, while reducing bending moments at
other points of the tank. In contrast, the uniform load acting on the upper plate led to
an increase in bending moments, but only in the bottom, both for fastening moments
and for the moment working in the centre of the plate towards a shorter span. Bending
moments at other points of the tank practically did not change. Reference [9] indicates
that taking into account the effect of the thermal insulation layer made of spray-applied
polyurethane foam in the plate calculations causes that with an increase in the modulus of
subgrade reaction, deflections and bending moments to decrease. Pontoons are structures
exposed to numerous loads, difficult to define; therefore, conducting research on real-scale
objects helps to understand the specifics of their structure [7,39]. The consideration of the
Poisson’s ratio or subgrade coefficient in calculations has practically no effect on buoyancy
and stability calculations. The tilt angles for the calculated pontoon were 7.0◦ and 6.92◦
(Table 2). According to the standard [36], the freeboard should be at least 0.05 m at a tilt
angle not exceeding 6◦ . The standard [34] states that this angle should not exceed 10◦ .
Simplified calculations are allowed by the standard [35] and then the tilt angle should not
exceed 15◦ . The calculation procedure for the purpose of this article was carried out in
accordance with [35]. Reference [40] relates to purposefulness of designing heavy floating
systems. A pontoon should be of such height as to keep the deck above the water level and
should be heavy enough to balance the upsetting moment in violent winds.
This article includes calculations for a unique rectangular tank with a partition in its
centre, made at a 1:1 scale, for which the effect of filling with Styrofoam as an elastic sub-
strate was taken into account. Figure 9 shows the pontoon during execution and launching.exceed 15°. The calculation procedure for the purpose of this article was carried out in
accordance with [35]. Reference [40] relates to purposefulness of designing heavy floating
systems. A pontoon should be of such height as to keep the deck above the water level
and should be heavy enough to balance the upsetting moment in violent winds.
This article includes calculations for a unique rectangular tank with a partition in its
Appl. Sci. 2021, 11, 4259 centre, made at a 1:1 scale, for which the effect of filling with Styrofoam as an elastic sub-11 of 12
strate was taken into account. Figure 9 shows the pontoon during execution and launch-
ing.
Figure 9. The pontoon during execution with visible reinforcement and a partition in the centre
Figure 9. The pontoon during execution with visible reinforcement and a partition in the centre (left) and during
(left) and during launching (right).
launching (right).
5. Conclusions
The calculation results presented in the article were obtained using the finite differ-
5. Conclusions
ence method in variational approach, assuming Poisson’s ratio ν = 0 and subgrade coeffi-
The calculation results presented in the article were obtained using the finite dif-
cient Kz = 5000 kN/m3. The following real dimensions of the pontoon were adopted: length
ference
lx = 10 method
m, width in
ly =variational
2.5 m, heightapproach, assuming
lz = 1.25 m and Poisson’s
thickness ratio
of all walls h=ν = m.
0.08 0 and subgrade
coefficient Kzon
= the
5000 3 . The following real dimensions of the pontoon were adopted:
kN/mcarried
Based analyses out, the following can be concluded:
length
• lx = 10into
taking m, width
accountlyin= calculations
2.5 m, height thelzcooperation
= 1.25 m and thickness
of the of all
tank walls walls
with h = 0.08 m.
its elastic
Based on the analyses carried out, the following can be concluded:
substrate filling reduces bending moments and, in some cases, changes the sign of
• bending
taking intomoments
account(Table 1). For example,
in calculations the bendingof
the cooperation moment Mx813
the tank (according
walls with itstoelastic
designations in Figure 4) occurring in the middle of the
substrate filling reduces bending moments and, in some cases, bottom length, abovethe
changes thesign of
bending moments (Table 1). For example, the bending moment Mx 813 (according
to designations in Figure 4) occurring in the middle of the bottom length, above
the support, which due to the hydrostatic load acting on the walls and the bottom,
without taking into account its elastic substrate, was Mx 813 = 10.24996 qs2 while, taking
into account its elastic substrate, was reduced to Mx 813 = 8.16864 qs2 , so it decreased
by 20%;
• taking into account in calculations an internal Styrofoam filling treated as an elastic sub-
strate reduces the values of bending moments and consequently the area that requires
necessary reinforcement and thus the solution becomes more financially beneficial;
• taking into account in calculations the cooperation of the pontoon’s structure with an
internal filling does not affect its buoyancy and stability;
• the pontoon presented in the article can be a ready-made, prefabricated reinforced
concrete shipping element, prepared to be built in at its final location, which can be
used as a float in amphibious construction.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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