Modeling of Dynamic Axial Crushing of Thin Walled Structures by LS-DYNA and Genetic Programming

3rd
.International Conference on Researches in Science & Engineering
31 Aug. 2017, Kasem Bundit University, Bangkok, Thailand
Modeling of Dynamic Axial Crushing of Thin Walled Structures
by LS-DYNA and Genetic Programming
M.Hasanlu*1
, M.Rostami2
,A.Abazarian3
,M.Soltanshah4
, M.Zamanian5
1,2,3
Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, P.O. Box 3756, Rasht,
Iran
4
Faculty of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran
5
Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran.
*
Corresponding Author: mhasanlu@webmail.guilan.ac.ir
Abstract
Energy absorption capability of thin-walled structures with various cross sections has been
considered by researchers up to now. These structures as energy absorbers are used widely in
different industries such as automotive and aerospace and protect passengers and goods
against impact. In this paper, mechanical behavior of subjected thin-walled aluminum tubes
with and without polyurethane filler foam under axial impact has been investigated. The tubes
are very thin so that (D/t) ≈ 550 governs for cylindrical specimen. Structure behavior was
analyzed through finite element analysis by LS-DYNA. In addition, relations governing
energy absorption of thin-walled structures were extracted using genetic programming. These
relationships can be used to predict the behavior of structures under investigation. Trying to
obtain the relationships it has been attempted to extract functions with the simplest possible
form. Circular, hexagonal, and square cross sections with the same length, thickness, and
circumference of sections were studied. The results show that circular structure has higher test
energy absorption while experiences the lowest change in length compared to hexagonal and
square structures. Besides, the effects of stress concentration in hexagonal and square sections
can be observed on the corners of walls. Also under the dynamic loading circular structure
was crushed more axisymmetric, while hexagonal and square structures tended to the
buckling.
Keywords: Thin-Walled Structure, Polyurethane Foam, Energy Absorption, Impact,
Mechanical Behavior, Genetic Programming, LS-DYNA

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1. Introduction
Polyurethane is a thermosetting polymer created from combination of Methyl Di Isocynate
with Polyol and some other chemical additives. By choosing these additives and changing
chemical and physical conditions of reaction process, various properties can be created for
various applications. Flexible polyurethane foam is used in automobile industry to
manufacture interior trim parts for cars such as seats, steering wheel, knobs, etc. In addition,
polyurethane is used as structural foam to manufacture parts such as bumpers.Energy
absorption is absorber of a kind of energy and converts it to other energy or energies. Energy
absorptions from solar energy absorptions to mechanical shock absorptions such as
earthquake dampers in buildings and impact absorptions in automobiles have been always of
human interest. The first absorptions used in automobiles such as tire and the spring mounted
under the chassis mostly embrace comfort and convenience for travelers. With the growing
number of accidents and deaths caused by sudden depreciation, kinetic energy, and stopping
the vehicle through a collision, engineers thought of designing bumpers with high-energy
absorption in automotive. Hanssen et al [1] designed an automotive bumper with aluminum
filler foam and put it under dynamic loading experimentally. Moreover, LS-DYNA was used
to simulate the impact process and the prediction of bumper behavior while collision was
obtained with acceptable accuracy by software simulation compared to laboratory samples. In
[2], Aluminum Honeycomb structure was analyzed under axial impact with high speed.
Experiments showed that Aluminum Honeycomb absorbency improves as speed increases.In
many engineering systems, energy absorption during impact is one of the necessities. A
variety of mechanisms is used to absorb and dissipate impact energy among which friction,
breaking, bending, or twisting plastic and crush can be mentioned. Thin-walled structures
filled with foam are lightweight and have higher energy absorption capacity. In addition,
mechanical properties and energy absorption capacity depend on the properties and geometry
of materials used. Thin-walled structures are the most common impact protection systems
converting kinetic energy to irreversible energy of plastic deformation. Laboratory results in
[3] showed that deformation during axial loading could take place in three ways:
Axisymmetric Concertina, Non-axisymmetric Diamond, and hybrid. Deformation in
structures depends on ratio of diameter to thickness and diameter to length of the structure.
Seitzberger et al [4] tested on thin-walled structures with circular and square cross sections.
The structures were of mild steel filled with aluminum foam. They showed that the interaction

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between the foam and structure wall reduces folding distances. Therefore, the energy
absorbed with structure increases. In [5], multi-objective genetic algorithm was used to obtain
optimal form of a filled square aluminum structure. Zarei and Kroger [6, 7] used multi-
objective optimization design technique to maximize energy absorption and minimize the
weight of thin-walled aluminum structures. Energy absorption in quasi-static impact of
cylindrical and conical thin-walled structures with and without polyurethane foam has been
studied in [8 & 9]. The experimental and numerical results showed the effectiveness of filler
foam in increased energy absorption in these structures. Gupta and Velmurugan [10] studied
composite conical thin-walled structures. The results of their experiments showed that energy
absorption increases significantly in conical structures filled with foam compared to hollow
structures. In [11], structures with hexagonal, octagonal, and star cross sections were
investigated under quasi-static load. The results showed that increasing the number of interior
corners in star cross sections leads to improved energy absorption structures. Tamashita et al
[15] investigated axial impact loading on hollow thin-walled structures with circular section
and polygon sections of 4, 5, and 12 of extruded alloy aluminum 6063. The results showed
that increasing the number of sides leads to increase in resistance to crush and increase in
energy absorption. Reddy and Wall [12] investigated thin-wall cylinders with polyurethane
foam under dynamic and quasi-static loading conditions. Reid et al [13] investigated dynamic
and static axial loading of thin-walled structures with rectangle and square sections with and
without polyurethane foam. The results of their experiments showed that energy abruption
increases as foam density increases. In addition, deformations and folding were more
symmetrical and regular in samples filled with foam. Gameiro and Cirne [14] investigated the
behavior of thin-walled aluminum cylinders with cork filler under dynamic loading. Nazari et
al also studied using genetic programming in buckling of oil and gas pipelines buried in the
seabed [19]. Pipelines buried are under buckling due to changes in temperature and high
pressure. In aforementioned article, models predicted from the buckling are presented by
genetic programming. Moreover, models obtained from genetic programming had acceptable
outcome and accuracy compared with finite element models. Castelli et al [20] have also
taken advantage of genetic programming in predicting energy efficiency of residential
buildings. They aimed to express a model to predict heating and cooling loads of buildings.
Finally, the method proposed by Castelli had higher accuracy compared with other modern
techniques. In another study, Zaji and Bonakdari used genetic programming and artificial

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neural networks in order to estimate longitudinal speed at the intersection of open channels
[21]. In another study on genetic programming, in order to create path milling machine was
used to manufacture parts with prismatic form [22] in which a simple model of machining
process has been presented aiming to create path for milling machine by genetic
programming, M.Rostami et al investigated about dynamic axial crushing of thin walled
structures based on neural network modeling [23]. In this study, modeling and analysis of
structures were done using finite element via LS-DYNA and the relations governing their
energy absorption were obtained using genetic programming. Energy absorption of structures
can be predicted by these relations. Genetic programming extracts models with the lowest
error, while in this model, simplicity of functions obtained is also of significance. Functions
with high complexity have lower fitness compared with other functions. In fact, accuracy and
simplicity of the function are important for modeling. Thin-walled aluminum structures with
circular, hexagonal, and square sections have been studied with and without polyurethane
foam. The structures are very thin so that for cylindrical sample the relation is (D/t) 550
where D is diameter and t is thickness of cylindrical sample wall. This study aimed to
investigate the influence of different sections in mechanical behavior after structures impact.
Permanent deformation, stress distribution in the wall, and energy-absorbing capability of
structures are shown in this research. In the following section, finite element modeling is
described. The third section shows permanent deformation of structures after impact. The
fourth and fifth sections investigate the results of simulations by software and discuss them.
2. Modeling
Modeling and structure properties in numerical simulation are the same as properties of
materials used in the experiment of Ressy and Wall [12]. In this experiment, behavior of a
thin-walled cylinder with thickness of 0.12 mm, original length of 148 mm, and diameter of
65 mm, made of aluminum alloy with density of 2900 Kg/m3
, yield stress of 550 MPa,
Young's modulus of 78.5 GPa, and Poisson coefficient of 0.33 (a behavior like aluminum
2090-86T) and polyurethane foam with density of 80 Kg/m3
, Young's modulus of 25 MPa,
and Poisson coefficient of 0, which is placed in thin-walled cylinder, are examined. Stress-
strain curves of aluminum structure and filler foam are shown in Figures 1 and 2,
respectively. A 52-kilogram weight has impact on the filled cylindrical sample from 1.75 m
over filled cylinder and with a speed of 5.2 m/s. According to ref [12], actual speed of weight

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is 88% of speed obtained from equation 1. In numerical simulation, due to achieving identical
changes in length of filled and hollow sections, simulation duration from the moment of
weight release to loading end is considered to be 0.03 s for filled structures and 0.24 s for
hollow structures.
(1)
where m is mass of the weight, g is acceleration of gravity, h is distance of the weight from
the cylinder, and v is velocity of the weight.
Figure 1 Stress – Strain Curve for PU[12].
In simulation of filled and hollow thin-walled structures with hexagonal and square sections,
structure length and cross-sectional area and other mechanical properties are identical to
length, area, and mechanical properties of circular section so that energy absorption and
permanent deformation can be investigated in identical conditions.
2.1.Axial loading of thin-walled circular structure
Deformation of hollow circular structures in axial loading has been studied by Pugsley and
Macaulay [18]. In [16], Pugsley suggested an equation for expressing compression loading,
Pc, which can be written as follows.
(2)
Figure 2 Engineering Stress – Strain Aluminum Structure[12].

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In the above equation, is fully plastic bending moment on width of the circular
structure. The structure is of t thickness and yield stress.
Figure 3 Idealized deformation model of buckling of circular tube [12].
Applied load on aluminum structure is shown by the following equation.
( ) [ ]
(3)
where D is diameter of circular structure and h is optimal value of fold (figure 3). In the
optimal value, minimum amount of mean load is created in which h fold is completed. In
addition, applied load up to can be obtained by integration from equation 3 and from =0
to = . The load is obtained from equation 4.
( ) [ ]
(4)
When , equation 4 changes into equation 5.
[ ( ) ]
(5)
The equations were obtained for hollow circular structure. It should be noted that relations
governing filled structures are slightly different from the above equation so that the length of
fold in filled structure compared to hollow structure becomes 2h [12]. After impact and crush
of the structure, the foam is compressed by folds in structure wall. The angle of the folds is

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< . In [12], Reddy and Wall proposed a relation to express the mean load applied on filled
circular thin-walled structures.
( *
(6)
Where and are density of cell walls of rigid polyurethane foam and density of
polyurethane foam, respectively. Therefore, applied load on the structure shown by equation
is expressed as a function of density of foam. Applied load on the polyurethane foam is as
follows.
(7)
Where is plateau stress [12]. Therefore, crushing force of circular filled structure is shown
as equation 8.
( ) (8)
Also, energy absorbed by a structure with original length of under maximum deformation
of is as follows.
(9)
Above equations have been obtained for quasi-static loading but it should be noted that
deformation modes are the same for samples under dynamic and quasi-dynamic loading [12].
Therefore, the effects of inertia can be neglected and effects of strain rate can be considered.
Strain rate depends on thin-walled structure described by Cowper-Symonds [18] and through
equation 10.
(
̇
)
(10)
In this equation, is yield stress at strain rate ̇ and p=4 and c=6500 s-1
are constant
numbers for aluminum alloys [12]. Strain rate is different at folds of structure wall. Moreover,
this fold changed in one fold. Therefore, for ith
fold with mean strain rate of ̇ti the mean
dynamic load in circular structure is as follows.

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(
̇
)
(11)
Finally, dynamic load of ith
fold in filled structure is shown by equation 12.
(12)
Where is yield stress of polyurethane foam in ith
fold.
2.2.Material and properties
For thin-walled aluminum structure, MAT_PIECEWISE_LINEAR_PLASTICITY is used.
Also, MAT_CRUSHABLE_FOAM is suitable for polyurethane foam [16]. For impacting
weight, MAT_RIGID with properties similar to steel properties is considered. In Table 1,
properties of materials used are presented.
Table 1 Mechanical Properties
Definition and detection of contact surfaces are particularly important in problem solving. For
aluminum contact surface with impacting mass and for aluminum contact surface with
underlying plate, CONTACT_AUTOMATIC_SURFACE_TO_SURFACE is used. Contact
surface of combined aluminum structure folds is CONTACT_AUTOMATIC_SINGLE_
SURFACE. CONTACT_ERODING is also suitable for foam contact surfaces. Foam contact
surface with thin-walled structure with foam contact surface with underlying plate and
impacting math are considered as CONTACT-ERODING-SURFACE-TO-SURFACE. Also,
for foam contact surface on it, CONTACT-ERODING-SINGLE-SURFACE is suitable. Each
filled structure with maximum mean number of 75500 elements was completely solved. In
order to select optimum number of elements, the number of filled square structure elements to
energy absorbed is shown in Figure 4. Choosing the optimum element is also effective in
increasing problem solving accuracy in reducing problem solving duration.
Poisson's Ratio
Young Modulus
(MPa)
Density
(ton/mm3
)
Material
0
25
Polyurethane
0.33
Aluminum
0.3
Steel (Master)

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Figure 4 Variation of Absorbed Energy to Number of Elements (Filled Square Tube)
3. Structure deformation
Deformation and stress distribution in filled circular structure 0.018 seconds after loading is
shown in the first image of Figure 5. In the second image of Figure 5, filled hexagonal
structure 0.018 seconds after loading can be observed. This section has little tendency to
buckle during loading and maximum stress can be observed on the corners of structure wall.
This cannot be seen in circular section due to not having corner. Square section has the least
energy absorption among other filled sections. This section also buckles during loading,
which can be seen in the third image of Figure 5. In filled square section, maximum stress can
be seen on corners the same as hexagonal section
Figure 5 Filled Circular Tube 0.018 sec after Impact
3 4 5 6 7 8 9 10 11
x 10
4
520
525
530
535
540
545
550
555
Energy Absorption Variations
Number of Elements
Energy
Absorbed
-
J
LS-DYNA

3rd
Figure 6 Filled Hexagonal Tube 0.018 sec after Impact
Figure 7 Filled Square Tube 0.018 sec after Impact
In permanent deformation of filled structures with increasing number of sides for the section,
tendency to buckle decreases. In Figure 6, sectioned view after loading of three filled sections
can be observed. Deformation and stress distribution of hollow structures are shown in Figure
7 0.012 seconds after loading.

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Figure 8 Cross-Section of Filled Circular Tube after Impact
Figure 9 Cross-Section of Filled Hexagonal Tube after Impact
Figure 10: Cross-Section of Filled Square Tube after Impact

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Figure 11: Hollow Circular Tube 0.012 sec after Impact
Figure 12: Hollow Hexagonal Tube 0.012 sec after Impact
Figure 13: Hollow Square Tube 0.012 sec after Impact
In figure 8, deformation of hollow structures is shown at the end of loading.

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Figure 14 : Hollow Circular Tube after Impact
Figure 15 : Hollow Hexagonal Tube after Impact
Figure 16: Hollow Square Tube after Impact
4. Genetic programming
Genetic programming (GP) is an optimization method derived from Darwin's principles of
evolution. Unlike traditional optimization methods that manipulated parameters of an initial
response estimation, genetic programming retains a potential model population that have tree
structure with initial functions. The functions link input nodes to constants. The probability of
a model remaining to transfer to the next generation depends on its performance that is
assessed by fitness function and the most successful answers have more chance to
reproducing. In synthetic development, biological reproduction is done by operators such as

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bound and rebound. With selection based on fitness, answers are combined by the genetic
operators and create the next generation from the generated answers. Since better answers
have higher probability for reproduction, we expect the answers to improve after some
generations.
The first step in genetic programming is to create an initial population. There are different
ways to do so. For example, trees with completely random size can be created that is called
grow method (Figure 18).
The result of this tree is a function: x+2y.
Another method is that all branches grow to the same depth, which is called Full Method
(Figure 19).
𝑢
+
𝑢
𝑐
𝑢
y
2
x
Figure 17 A sample tree in GP[24]
Figure 18 Grow method[24]

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Here, we combine the two methods, which is called Ramped half and half. In order to create
trees, there is a need to initial functions and terminals, which are presented in the following
tables.
Table 2: Terminal Examples
Type Terminal
Example
External Outputs
x , y
Non-Arguments Functions
Rand, Go Left
Constants Values
3, 0.45
After generating initial population with operations such as bound and rebound, new tress will
grow. The more their fitness is, the more chance they have to be selected.
Crossover:
In Crossover operation, two trees are selected and a branch of each is selected. The branches
are displaced in the two trees and thus two new trees are generated (Figure 20).
Figure 20: Crossover operation[24]
*
*
1 0
x y
Figure 19 Full method[24]

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Mutation:
In Mutation operator, a tree is selected based on fitness. Then, a branch of this tree is chosen
randomly and is substituted randomly with a sub-tree (Figure 21).
The remarkable thing is that the depth of trees should not exceed the maximum depth
considered for the algorithm. In order to create next generation, the population of previous
generation and generated population are arranged by bound and rebound operators based on
fitness. Then, the population of next generation is selected. According to complexity of the
model, number of initial population and maximum depth of trees are determined. The
algorithm goes on until it reaches the arbitrary precision in the desired function. In this article,
after obtaining points on energy and force curves (obtained from LS-DYNA), functions
governing these points are extracted by GP in order to predict the behavior of the structures
[24].
Figure 21: Mutation operation[24]
5. Simulation results
Among filled structures, thin-walled structure with circular section has the most energy
absorption with 2.1% error to the number mentioned in Ref [12] so that energy absorption in
[12] is 625 J and energy absorption in the sample examined in numerical simulation is 612 J.
Offspring
x y
3
Mutation Point
Parent
Mutation Point
Random Generated Subtree x
x y
y
2
*
x
y
2
*

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After circular section, thin-walled structures with hexagonal and square section have the most
energy absorption. Absorbed energy (EA) is obtained from the following equation.
.
A
E P d
  (13)
Where P and are power and displacement, respectively. In Figure 22, time to absorbed
energy curve is observed. Thin-walled structure with hexagonal section absorbs 580 J energy
in 122 mm length reduction, which is 32 J less than circular and 26 J more than square
section. Thin-walled structure with square section despite lower energy absorption
experiences greater reduced length compared with two other structures. 127 mm length
reduction that is 7 mm and 5 mm greater than circular and hexagonal sections, respectively. In
addition, absorbed energy by filled structures is drawn in Figure 24. Function extracted for
absorbed energy can be provided by filled structures for the three sections as follows. As can
be seen from these relationships, the relationships are obtained non-linearly by genetic
programming. Therefore, we have:
Figure 22: Filled Tubes Energy Absorption
0 0.005 0.01 0.015 0.02 0.025 0.03
0
100
200
300
400
500
600
700
LS DYNA - Genetic Programming
Time - sec
Absorbed
Energy
-
J
LS-DYNA Circular
GP Circular
LS-DYNA Hexagonal
GP Hexagonal
LS-DYNA Square
GP Square

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Figure 23: Energy Offset of GP into LS-DYNA in Filled Tubes
Energy functions for filled circular section:
(14)
( ) ( | |)
Energy functions for filled hexagonal section:
(15)
( ) ( ( ))
Energy functions for filled square section:
(16)
and are describable.
(17)
[ ( )
]
(18)
( ( ))
In Figures 22 and 24, the results of LS-DYNA and results from genetic programming can be
observed. In these figures, it can be seen that the results from GP have acceptable accuracy
0 0.005 0.01 0.015 0.02
0
5
10
15
20
25
Time- sec
Absorbed
Energy
Error
-
J
Comparison LS DYNA - GP
Circular
Hexagonal
Square

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and are close to results obtained from LS-DYNA. In Figures 23 and 25, the error results of
GP to LS-DYNA are expressed for energy-time and power-time curves in each section.
Figure 24: Load–Time Curve of Filled Tubes
Figure 25: Load Offset of GP into LS-DYNA in Filled Tubes
Power function for filled circular section:
* √( ) ( ) | √ | +
(19)
0 0.005 0.01 0.015 0.02 0.025 0.03
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Time - sec
Load
-
N
LS-DYNA Circular
GP Circular
LS-DYNA Hexagonal
GP Hexagonal
LS-DYNA Square
GP Square
0 0.005 0.01 0.015 0.02
0
200
400
600
800
1000
1200
1400
Time- sec
Load
Error
-
N
Circular
Hexagonal
Square

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Power function for filled hexagonal section:
(20)
, , , and parameters are describable.
(21)
| |
(22)
(23)
(24)
( )
Power function for filled square section:
(25)
and are describable.
(26)
( ( ( )))
(27)
( ) ( )
Circular section among hollow structures has the most energy absorption the same as filled
structures. In addition, thin-walled structure with hexagonal section absorbs 84 J energy in
122 mm length reduction, which is 39 J less than circular section and 14 J higher than square
section. In Figure 26, time-energy curve is observed. In addition, energy absorbed by hollow
structures is drawn in Figure 28.

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Figure26: Hollow Tubes Energy Absorption
Figure 27: Energy Offset of GP into LS-DYNA in Hollow Tubes
Power function for hollow square section:
(28)
(
( )
)
, , and are describable.
(29)
(
( )
( )
)
(30)
(( ( )) )
0 0.005 0.01 0.015 0.02
0
20
40
60
80
100
120
140
Time - sec
Absorbed
Energy
-
J
LS-DYNA Circular
GP Circular
LS-DYNA Hexagonal
GP Hexagonal
LS-DYNA Square
GP Square
0 0.005 0.01 0.015 0.02
0
5
10
15
20
25
30
Time- sec
Absorbed
Energy
Error
-
J
Circular
Hexagonal
Square

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(31)
((| | ) ) ((
( ( ))
* )
(32)
((| | ) )
Power function for hollow hexagonal section:
(33)
( ( ))
( ( ) )
and parameters are describable.
(34)
( )
(35)
( ( ))
( )
| ( ) | ( )
(36)
In Figures 26 and 28, results obtained from LS-DYNA and results of genetic programming
can be observed. In the mentioned sections, it can be seen that the results from GP have
acceptable accuracy and are close to results obtained from LS-DYNA. In Figures 27 and 29,
the error results of GP to LS-DYNA are expressed for energy-time and power-time curves in
each section. In the following, functions obtained by GP can be observed. The functions for
each section are obtained in hollow and filled states (equations 14 to 51). In obtaining these
functions, in addition to function accuracy it has been attempted to consider its simplicity.
This can put the subject of the present research in multi-objective optimization category.
Figure 28: Load–Time Curve of Hollow Tubes
0 0.005 0.01 0.015 0.02
0
500
1000
1500
Time - sec
Load
-
N
LS-DYNA Circular
GP Circular
LS-DYNA Hexagonal
GP Hexagonal
LS-DYNA Square
GP Square

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Figure 29: Load Offset of GP into LS-DYNA in Hollow Tubes
Power function for hollow circular section:
(37)
( )
, k, , and parameters are describable.
(38)
* ( ( ( ( ( ( (
( )( )
* )))))) +
(39)
( ( ) ) (
( )
*
(40)
( ( ) )
(41)
( ( ) ) ( ( ) )
Power function for hollow hexagonal section:
(42)
| |
, , o, and parameters are describable.
(43)
( ( ) )
(44)
(
| |
)
(45)
( )
(46)
| |
0 0.005 0.01 0.015 0.02
0
1
2
3
4
5
6
7
Time- sec
Load
Error
-
N
Circular
Hexagonal
Square

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(47)
| |
, , , and parameters are describable.
(48)
( ( √ ) )
(49)
( ( √ ) ) ( ( ( ) ))
(50)
( √ ) ( √ | | )
(51)
( ( √ ))
Comparing circular filled and hollow sections, it can be seen that filled sections absorb 612 J
energy and hollow sections absorb 123 J energy. Therefore, energy absorption increases by
4.9 times in filled state. In filled state, hexagonal sections absorb 580 J energy and in hollow
state, they absorb 84 J energy. In addition, filled square sections absorb 554 J energy and
hollow square sections absorb 70 J energy. Filled structures are capable of energy
amortization in a longer time compared to hollow structures in identical displacement. Also,
the curve of power absorbed changes in filled structures is smoother in filled structures than
in hollow structures. In addition to the results that compare structures in energy absorption,
the results from GP and obtained relations can significantly reduce problem-solving time.
Meanwhile, they can provide acceptable answers. By having results governing each impact
phenomenon, the behavior of that phenomenon can be predicted and solved in a fraction of a
second.
Conclusion
In this article, mechanical behavior of filled and hollow thin-walled structures was examined.
In the structures, the more the sides of the section become the more energy absorption and
amortized force increase in the structure. Finally, energy absorption reaches its maximum rate
in circular section. Also, the less the sides of the section become, the more the desire to
buckling during loading increases. In addition, the effects of stress concentration in square
and hexagonal sections are obvious during loading. However, this was not observed in
circular section. Filled circular structure has the most energy absorption, while it experiences
the least changes in length. Moreover, the most regular folds are associated with filled thin-
walled structure with circular section. Comparing energy absorption in filled thin-walled
structures with hollow structures, it can be found that energy absorption in filled state is at
least 5 times the hollow state. When the number of sides of the sections is less, this effect is

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more obvious so that energy absorption is increased by 8 times in square section. In addition
to higher energy absorption by filled structures, energy absorption are capable of energy
amortization in a longer time compared to hollow structures in identical displacement. Also,
after obtaining points on energy and force curves (obtained from LS-DYNA), functions
governing these points are extracted by GP in order to predict the behavior of the structures.
The results from GP and obtained relations can significantly reduce problem-solving time.
Meanwhile, they can provide acceptable answers.
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soil using genetic programming. Engineering Structures. Volume 101, 15 October 2015, Pages 306–317.
[20] M. Castelli, L. Trujillo, L. Vanneschi, A. Popovič, Prediction of energy performance of residential
buildings: A genetic programming approach. Energy and Buildings. Volume 102, 1 September 2015, Pages 67–
74.
[21] A. H. Zaji, H. Bonakdari. Application of artificial neural network and genetic programming models for
estimating the longitudinal velocity field in open channel junctions. Flow Measurement and Instrumentation.
Volume 41, March 2015, Pages 81–89.
[22] J. Barclay, V. Dhokia, A. Nassehi. Generating Milling Tool Paths for Prismatic Parts Using Genetic
Programming. 9th CIRP Conference on Intelligent Computation in Manufacturing Engineering - CIRP ICME
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