Finding the Weight Difference of a Rectangular Structure with a
Parabolic Arc for Mathematical Models Made of Steel and other
Composite Materials
TYMOR ABED ALSTTAR SEDIQER1, EMAD TOMA KARASH2*, JAMAL NAYIEF SULTAN3,
MAJID KHALEEL NAJEM3
1Department of Civil Technologies, Kirkuk Technical Institute,
Northern Technical University,
Mosul 41000,
IRAQ
]
2Department of Mechanical Technology, Mosul Technical Institute,
Northern Technical University,
Mosul 41000,
IRAQ
3Department of Power Mechical Engineering, Techical Engineering College,
Northern Technical University,
IRAQ
Abstract: - The most popular materials for lightweight constructions, including building and aircraft structures,
industrial, military, and aerospace technology, are armored composites. Composites made of carbon fiber are
typically employed in lightweight applications. The ANSYS program was used to produce four mathematical
models. Steel is used in the construction of the first and second versions whereas composite materials are used
in the third and fourth variants. To find all the deformations, stresses, and strains that appear on the four
models, as well as to calculate the weights of those four structures and compare them, these four models were
tested with the ANSYS 15.0 program to obtain equal deformation resistance for all models under the influence
of different loads. The results show that the composite models had lower strains, stresses, and deformations
than the steel models. Among other results, it was discovered that the weight of the third model made of
composite materials decreased by (32.72%) compared to the steel-based first model, and after doing the
necessary calculations and assessing the results, the fourth model made of composite materials' weight was
reduced by (19.21%) when compared to the second model made of steel.
Key-Words: - Steel, Parabolic arc, Carbon fiber, Composite materials, Beams, Finite element method.
Received: March 7, 2023. Revised: July 27, 2023. Accepted: September 8, 2023. Published: September 28, 2023.
1 Introduction
High-strength steel is increasingly being used in
construction, and steel is a common building
element. When utilized for vertical members in
high-rise structures, high-strength steel can
efficiently minimize cross-sectional size, maximize
space utilization, and save material costs. The most
often utilized form of steel in engineering is that
which is limited by ties. The ties-confined concrete
stress-strain relationship is taken into consideration
while analyzing the mechanical characteristics of
high-strength steel vertical members.
In several engineering disciplines, such as
shipbuilding, aviation, and civil engineering, thin
plates are structural components that are frequently
utilized. Ship plates with ribs and stiffeners, offshore
panels, and aerospace panels are frequently used and
always subject to partial edge traction on their plane.
This kind of loading may cause buckling, which has
a detrimental effect on how well the involved
structural elements function. In-plane compression
and shear loads can result in regional or global plate
instability. The corrugations that emerge from plate
instability could cause lasting damage to the entire
structure and lead it to lose efficiency given
analytical solutions in their research of thin plate
buckling under compression and shear, [1], [2].
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The three most important requirements for the
structural frame of buildings and other projects are
strength, stiffness, and stability. However, one of the
key challenges in creating logical and economical
constructions is the development of effective
structural elements with the right bearing capacity
and ideal size. This challenge is solved using a
variety of generic algorithm options, topological
optimization, the theory of adaptive optimization,
and other techniques, [3], [4], [5], [6]. In many areas
of technology, including mechanical engineering,
aircraft engineering, instrument engineering, and
construction, topological and parametric
optimization techniques are applied to create the
best possible construction projects, [7], [8], [9], [10].
Studies have shown that the use of fiber-reinforced
pile-reinforced polymer systems in the building,
construction, and industrial fields to externally
strengthen concrete sections and pipes is a practical
alternative. The effectiveness of these systems has
been confirmed by numerous experimental
experiments, [11], [12], [13]. Due to their lower
dead weight than reinforced cement composites,
composite structures are substantially less
susceptible to seismic conditions. Currently, high-
rise structure design and construction research
employs cutting-edge techniques, a variety of design
software, and sophisticated experimental techniques
to get reliable results. This is accomplished by
taking into account both the task benefits of using
steel reinforcement and the advantages of using
various types of concrete. These elements have an
impact on the composite columns' high capacity and
axial compression, despite their various shapes, [14],
[15], [16]. Advanced composite materials with
excellent mechanical properties and low specific
weight make up the majority of today's thin-walled
contractions. Materials with thin stiffening walls are
the major focus. They are created and constructed as
closed or open profiles with intricate slatted shapes
and different shapes, [17], [18], [19], [20], [21],
[22]. The stirrups and steel coil in SRC columns
may be able to handle the internal concrete
deformation and strength. The impact of stirrups
having concrete borders has been extensively
researched, [23], [24], [25]. Epoxy resin composites
have been a mainstay in the engineering and
industrial fields for a good while. Parts with superior
mechanical, thermal, and electrical ties have been
created using epoxy-based component
manufacturing, [26]. To enhance the characteristics
of epoxy resins, it is now customary to add a second
phase (such as inorganic fillers). Recent studies have
demonstrated that carbon fiber-reinforced polymer-
based composites greatly enhance the mechanical,
thermal, and barrier properties of pure polymer
matrices, [27], [28]. Matrix strengthening is
undoubtedly one of the procedures used to enhance
the compressive and flexural properties of
composites made from fiber fabrics, [29]. It has
been demonstrated that a polymer matrix's
mechanical characteristics can be greatly improved
by the addition of a tiny quantity of stiff
nanoparticles, [30], [31].
The goal of this article is to develop composite
material structures for buildings, halls, and
warehouses that are resistant to deformations,
stresses, and strains at levels comparable to those
found in steel structures under the same loads.
Along with other requirements like resistance to
corrosion, vibrations, and fatigue stresses to which it
is exposed, halls, warehouses, etc., are also included.
2 Materials and Model Analysis
The four three-dimensional models of the four
various types of beams were made using the
ANSYS-15 program. The first and second models
are built of traditional materials (steel), whereas the
third and fourth models are made of composite
materials. The upper surfaces of the two models (M1
and M3) are precisely subjected to a 50 kN force
along the y-axis. A distributed load was applied
from above along all top structures and at a value of
(2308 N/m) for the two models (M2 & M4). Figure
1 shows the shapes and dimensions of the fourth
model.
Describe the mechanical characteristics of steel,
epoxy resin, and carbon fiber composition in Table
1. The results of the all mechanical properties of the
composite materials as calculated using the
Mathcad-15 program are shown in Table 2. Table 3
shows the models, codes, specific disciplines,
element types, and load types used.
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Fig. 1: It indicates the model form, cross-sectional area, and dimensions used in the tests
Table 1. Describe the mechanical characteristics of steel, epoxy resin, and carbon fiber, [32], [33]
Model
Materials
Density,,
(Kg/m3)
Modulus of
Elasticity,
E, (GPa)
Passion’s
Ratio
Modulus of
Rigidity,
G, (GPa)
Price
Kilograms,
$
M-1
Steel
7870
205
0.3
80
M-2
M- 3
T300 Carbon Fiber and
7901 Epoxy Resin
T300 carbon fiber,
40 %
1765
125
0.24
5.43
14
M- 4
7901 Epoxy Resin,
60 %
1299
11.3
0.31
3.98
0.05
Table 2. The findings of the composite materials' mechanical properties as determined
Models
Materials
Code used in Mathcad-15 program [0]
Code used in Mathcad-15 program [90]
E ii, (MPa)
G ij, (MPa)
E ii, MPa
G ij, (MPa)
Model - 3
T300 Carbon Fiber
and 7901 Epoxy
0.281
0.301
Model - 4
a.
b.
c.
d.
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Table 3. Models, codes, Individual disciplines, element types, as well as the several load types
No.
Model
Code
Individual -
Disciplines
Type of Element
Loads
1
Model -- 1
---------
Structural
Beam 188
50 KN
2
Model -- 2
----------
Structural
Beam 188
2308 N/m
3
Model -- 3
[0/90]70
Structural
SHELL 281
50 KN
4
Model -- 4
[0/90]84
Structural
SHELL 281
2308 N/m
3 Results and Discussion
A curved column of the same length was designed
for four different mathematical models, the first and
second models of steel with a hollow section, and
the third and fourth models of composite materials
with a solid section. Loads were applied to the four
models using the ANSYS program. Where a load of
(50 KN) was applied as shown in Figure 1-a on the
first model. The results of deformations, strains, and
stresses that appeared on the model were obtained,
as the value of deformation was (140.104 mm).
In the second model, Figure 2-b, a distributed
load was applied to the model from its beginning to
its end. Many simulations were carried out by
reducing the load, to obtain a load that results in the
same deformation obtained in the first model, whose
value was (140.104 mm). After making several
attempts to change the applied load, a load that
achieves this was obtained, and the value of this
load was (2308 N/m).
After designing the third model consisting of
composite materials, the same load was applied in
the first model, whose value was (140.104 mm). The
number of layers was changed and the results were
extracted to the stage of obtaining the same
deformation in the first model, after making quite a
few attempts. The same deformation was obtained in
the first model, which was (140.104 mm). This was
achieved when the number of layers reached (140
layers), and each layer had a thickness of (1 mm).
In the fourth model, which is also composed of
composite materials, the same load was applied to
the second model, whose value was (140.104 mm).
The number of layers was changed until a
deformation equal to the deformation in the other
models was obtained. After conducting a large
number of attempts using the ANSYS 15.0 program,
the optimal solution was obtained, in which the
number of layers consisting of the arched column
reached (168 layers).
Figure 2, Figure 3, Figure 4, Figure 5, Figure 6,
Figure 7, Figure 8, Figure 9, Figure 10, Figure 11,
Figure 12, Figure 13, Figure 14, Figure 15, Figure
16 and Figure 17 shows all the results obtained
through the four model tests under the influence of
the loads applied to them. These figures show the
deformations, stresses, and strains that appeared on
the four models after loads were applied to them, by
using the ANSYS program.
(a)
(b)
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(c)
(d)
Fig. 2: The displacements () distribution across all
models
(a)
(b)
(c)
(d)
Fig. 3: The deflections (δ) results across all models
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(a)
(b)
(c)
(d)
Fig. 4: The displacements (Ux) results across all
models
(a)
(b)
(c)
(d)
Fig. 5: The displacements (Uy) results across all
models
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(a)
(b)
(c)
(d)
Fig. 6: The normal stresses (σx) results across all models
(a)
(b)
(c)
(d)
Fig. 7: The shear stresses (τxy) results across all models
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(a)
(b)
(c)
(d)
Fig. 8: The shear stresses (τxz) results across all models
(a)
(b)
(c)
(d)
Fig. 9: The intensity stresses (σint.) results across all models
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(a)
(b)
(c)
(d)
Fig. 10: The intensity stresses (σvon.) results across
all models
(a)
(b)
(c)
(d)
Fig. 11: The normal strains (εx) results across all
models
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(a)
(b)
(c)
(d)
Fig. 12: The shear strains (εxy) results across all models
(a)
(b)
(c)
(d)
Fig. 13: The shear strains (εxz) results across all models
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(a)
(b)
(c)
(d)
Fig. 14: The first strain (εfirst) results across all models
(a)
(b)
(c)
(d)
Fig. 15: The third strain (εthird) results across all models
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(a)
(b)
(c)
(d)
Fig. 16: The intensity strains (εint.) results across all
models
(a)
(b)
(c)
(d)
Fig.17: The von strains (εvon.) results across all
models
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Figure 18 displays the load distribution on the
first and second models as well as the shape of the
load distribution and the places with the highest
shear forces. The maximum bending moment is
shown in Figure 19, along with the distribution of
moments on the first and second models and where
it occurs.
(a)
(b)
Fig. 18: The shear forces (S-F) results, for models (1, 2)
(a)
(b)
Fig. 19: The bending moments ( results, for
models (1, 2)
Based on the deformations, stresses, and strains that
appeared on the models after loading, the ANSYS
program's results for the four models are
summarized in Table 4.
Table 4. A summary of the findings from stress, strain, and deformations on the four models is displayed
No.
Model
(mm)
(mm)
(mm)
(MPa)
(MPa)
(MPa)
(MPa)
(MPa)
εfirst
εthird
εint.
εvon.
1.
M1
140.10
4
57.809
19.569
259.625
0.745
*10-9
0.738
*10-9
267.986
267.986
1.266
*10-3
9.45
*10-13
9.36
*10-13
0.255
*10-3
1.307
*10-3
1.307
*10-3
1.307
*10-3
2.
M2
140.10
4
0.53
1.336
49.866
8.682
8.602
52.803
52.084
0.243
*10-3
0.11
*10-3
0.109
*10-3
0.629
*10-3
0.255
*10-3
0.261
*10-3
0.267
*10-3
3.
M3
140.10
4
44.608
11.271
177.549
36.054
0.03
219.19
210.766
1.581
*10-3
1.436
*10-3
0.431
*10-18
1.583
*10-3
0.63
*10-3
2.303
*10-3
2.614
*10-3
4.
M4
140.10
4
0.948
1.663
19.503
14.812
1.878
60.332
56.361
0.270
*10-3
1.1
*10-3
0.937
*10-19
1.266
*10-3
1.968
*10-3
0.858
*10-3
0.794
*10-3
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Table 5. It shows the conclusions drawn from the data analysis and a comparison of them
Model
Materials
Density
(Kg/m3)
Price Kilograms,
$
Number of Layers
Length
(m)
Width
(mm)
Thickness
(m)
The volume of the
column
(m3)
Density
(Kg/m3)
Weight of the column
(Kg)
Total price
($)
Difference in weight
%
W1
W2
T1
T2
M-1
Steel
7870
3.5
1
23.1
0.1524
0.1347
0.2032
0.1855
0.1382
7870
1088
3808
-----
M-2
M-3
T300
Carbon
Fiber and
7901 Epoxy
Resin
T300 carbon
fiber, 40 %
1765
10
140
23.1
0.1524
0
0.140
0
0.4929
1485.4
732
2950
32.72
M-4
7901 Epoxy
Resin, 60 %
1299
0.05
168
23.1
0.1524
0
0.168
0
0.5914
1485.4
879
3542
19.21
Table 5 shows the findings as well as a
comparison of them using various mathematical
models and their corresponding weights. In
addition, the weights of the four models were
compared, and the difference between the weights
of the steel-made first and second models and the
composite-material third and fourth models weight
was determined.
4 Conclusions
The ANSYS program's testing of the four models
under varied loads revealed that the overall
deformation value in each model was equal at
(140.104 mm). The following are the most
important conclusions obtained from these testing
follows:
The first model had the largest distortion along
the Y axis, measuring (19.596 mm), and the
greatest deformation along the X axis,
measuring (57.809 mm).
In the first model, 259,625 MPa was the
greatest normal stress measured in the X-axis's
direction. The largest shear stress was found in
the XY plane (36.054 MPa) for the third
model, and in the XZ plane (8.602 MPa) for
the second model. The maximum intensity
stress value for the first model was (267.986
MPa), and it also had the highest von Mises
stress value.
The results of the strains show that the third
model had the highest value of the normal
strain toward the X axis, as well as the
maximum value of the shear strain in the XY
plane, with a value of (0.001436), while the
fourth model had the highest value of the shear
strain in the XZ plane, with a value of
(0.00109). According to the first elastic
principle strain, the third model had the
maximum value at (0.00153), while the highest
value of the third-second elastic principle strain
was in the third model, with a value of
(0.001968). The results of the tests also show
that the third model had the greatest levels of
intensity strain and Von Misses Strain, with
values of (0.002303) and (0.002614),
respectively.
One of the most significant findings from these
tests, and after examining the outcomes of the
various tests, is the reduction in the weight of
the structures in the third and fourth models
compared to the first and second models. The
third model's weight decreased by 32.72%
compared to the first model's weight, which
had the same load in the middle, and the fourth
model's weight decreased by 19.21% compared
to the second model's weight.
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Emad Toma Karash, Jamal Nayief Sultan,
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