static finite element analysis of composite wing spar
TRANSCRIPT
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ISTANBUL TECHNICAL UNIVERSITY « FACULTY OF AERONAUTICS AND ASTRONAUTICS
GRADUATION PROJECT
JANUARY, 2021
STATIC FINITE ELEMENT ANALYSIS OF COMPOSITE WING SPAR UNDER AERODYNAMIC LOADS
Thesis Advisor: Prof. Dr. M.Orhan KAYA
SEYFİ TAŞKIN
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FEBRUARY 2021
ISTANBUL TECHNICAL UNIVERSITY « FACULTY OF AERONAUTICS AND ASTRONAUTICS
STATIC FINITE ELEMENT ANALYSIS OF COMPOSITE WING SPAR UNDER AERODYNAMIC LOADS
GRADUATION PROJECT
SEYFİ TAŞKIN (110170503)
Department of Aeronautical Engineering
Thesis Advisor: Prof. Dr. M.Orhan KAYA
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Thesis Advisor : Prof. Dr. M.Orhan KAYA .............................. İstanbul Technical University
Jury Members : Prof. Dr. İbrahim ÖZKOL ............................. İstanbul Technical University
Dr.Öğr Üyesi Özge ÖZDEMİR .............................. İstanbul Technical University
Seyfi TAŞKIN,student of ITU Faculty of Aeronautics and Astronautics student ID 110170503, successfully defended the graduation entitled “STATIC FINITE ELEMENT ANALYSIS OF COMPOSITE WING SPAR UNDER AERODYNAMIC LOADS”, which he prepared after fulfilling the requirements specified in the associated legislations, before the jury whose signatures are below.
Date of Submission : 25 JANUARY 2021 Date of Defense : 08 FEBRUARY 2021
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To my family,
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FOREWORD I would like to thank my consultant professor M.Orhan KAYA, who leads me in my graduation study and supported me during whole semester. Besides, I would like to thank my supportive family and my friends. JANUARY 2021
Seyfi TAŞKIN
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TABLE OF CONTENTS
Page
FOREWORD .............................................................................................................. v TABLE OF CONTENTS ......................................................................................... vii ABBREVIATIONS ................................................................................................... ix LIST OF TABLES ..................................................................................................... x LIST OF FIGURES .................................................................................................. xi SUMMARY ............................................................................................................ xiii 1. INTRODUCTION .................................................................................................. 1
1.1 Litarature Review ............................................................................................... 2 1.2 What is a composite? ......................................................................................... 3 1.3 What are the advanced composites? .................................................................. 3
2. MECHANICS OF COMPOSITE MATERIAL .................................................. 5 2.1 Hooke’s Law for a 2-D Angle Lamina .............................................................. 5 2.2 Strain and Stress in Laminates ........................................................................... 7
3. EULER-BERNOULLI THEORY ...................................................................... 12 3.1 Euler-Bernoulli Assumptions ........................................................................... 12 3.2 Displacement Field ........................................................................................... 13
3.2.1 Strain and Stress Fields ............................................................................. 13 3.2.2 Bending moment-curvature relationship ................................................... 14 3.2.3 Principle of Virtual Work .......................................................................... 14 3.2.4 Theoretical calculations of displacements ................................................ 15
4. 1-D EULER-BERNOULLI TWO-NODE COMPOSITE BEAM ELEMENT .................................................................................................................................... 16
4.1 Kinematics of A Plane Laminated Beam ......................................................... 17 4.1.1 Displacement fields ................................................................................... 17 4.1.2 Strain and Stress Field ............................................................................... 18 4.1.3 Displacement Functions of the Composite Beam Element ....................... 18
5. FINITE ELEMENT FORMULATION ............................................................. 19 5.1 Strain and Strain matrix ................................................................................... 21
6. 2-D EULER-BERNOULLI COMPOSITE BEAM ELEMENT ...................... 23 6.1 Bending-Torsion Coupling ............................................................................... 23 6.2 Finite element formulation for Bending-Torsion coupling .............................. 23
7. DETERMINING COMPOSITE BEAM PROPERTIES ................................. 28 7.1 Determining Loads acting on wing spar .......................................................... 28
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7.2 Determining Composite Beam Properties ....................................................... 29 8. RESULTS AND DISCUSSION .......................................................................... 30
8.1 Discussion ........................................................................................................ 39 9. REFERENCES ..................................................................................................... 40
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ABBREVIATIONS Hmcf : High modulus carbon fiber Std : Standart Ud : Unidirectional Cf : Carbon Fiber
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LIST OF TABLES
Page
Table 1: Material Properties of Analyzed composite materials. ............................... 29 Table 2: Comparison of analysis results and theoretical results for uncoupling
analysis. ....................................................................................................... 33 Table 3: Comparison of analysis results and theoretical results for coupling analysis.
..................................................................................................................... 38
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LIST OF FIGURES Page
Figure 1: Laminated Composite plate. ........................................................................ 4 Figure 2: Laminated Composite Beam ........................................................................ 4 Figure 3 : Local and Global coordinate system for an angle lamina (K.Kaw, 2006) . 5 Figure 4: Stress and Strain variation (K.Kaw, 2006) .................................................. 8 Figure 5: Presentation of plane section remain plane assumption (Erochko, 2020) . 12 Figure 6: Presentation of Bending moment and Stress sign (Oñate, 2013) .............. 14 Figure 7: Laminated Composite Beam (Xiaoshan Lin, 2020,) ................................. 16 Figure 8: Transverse displacement and buckling angle of std ud cf composite beam.
................................................................................................................... 30 Figure 9: Angle due to bending (rad), beam material std ud cf(uncoupling). ........... 30 Figure 10: Transverse displacement (m) and buckling angle(rad), beam material
Hmcf fabric(uncoupling). .......................................................................... 31 Figure 11: Angle due to bending (rad), beam material Hmcf fabric(uncoupling). ... 31 Figure 12: Transverse displacement (m) and buckling angle(rad), beam material
graphite-epoxy(uncoupling). ..................................................................... 32 Figure 13: Angle due to bending (rad), beam material graphite-epoxy(uncoupling).
................................................................................................................... 32 Figure 14: Transverse displacement (m) and buckling angle(rad), beam material
hmcf fabric (coupling). .............................................................................. 35 Figure 15: Angle due to bending (rad), beam material hmcf fabric(coupling). ........ 35 Figure 16 Transverse displacement and buckling angle of std ud cf composite
beam(coupling). ......................................................................................... 36 Figure 17: Angle due to bending (rad), beam material std ud cf (coupling). ............ 36 Figure 18: Transverse displacement (m) and buckling angle(rad), beam material
graphite-epoxy (coupling). ........................................................................ 37 Figure 19: Angle due to bending (rad), beam material graphite-epoxy (coupling). . 37
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KOMPOZİT KANAT KİRİŞİNİN AERODİNAMİK YÜKLER ALTINDA STATİK ANALİZİ
ÖZET
Bu çalışmada 2 boyutlu Euler-Bernoulli kompozit kiriş elemanı, keşif insansız hava
aracının kanat kirişi olarak tasarlanmıştır. Kanat kirişinin sonlu elemanlar yöntemi
kullanılarak, MATLAB kodu programlanarak eğilme ve burulma analizi yapılmıştır.
Çalışmadaki asıl amaç kompozit kanat kirişinin aerodinamik yükler altında eğilme
deplasmanı w, eğilme açısı ! ve burulma açısını " hesaplamaktır. Kiriş elemanı
ankastre kiriş olarak düşünülmüştür. Çünkü kanat kirişi kök kısmından gövdeye
ankastre olarak monte edilmektedir. Bu yaklaşım kanat kirişi analizi için en uygun
yaklaşımdır. MATLAB kodu Ferreira’nın kitabının 1-D Euler-Bernoulli Beam
başlığının altinda bulunan koddan türetilmiştir. Değişiklikler yapılarak kirişi kompozit
kiriş olarak hesaplayacak fonksiyon eklenmiştir. İki ucundan destekli olan kiriş
ankastre kiriş haline getirildi. Eğilme deplasmanı ve eğilme açısına, burulma açısı
eklenerek 2 boyutlu hale getirildi. Ortalama bir keşif insansız hava aracının seyir
uçuşunda yarattığı taşıma kuvveti hesaplandı. Yayılı yük ve yayılı burulma momenti
hesaplanarak kanat kirişinin üzerine eklendi. Yayılı yük, taşıma kuvvetinin, seyir
halindeki hava aracının ağırlığını karşılayan kuvvet olarak hesaplanmasıyla
bulunmuştur. Burulma momenti yunuslama momentinin hesaplanması ile kanat
üzerinde yayılı moment olarak kabul edilmiştir. Bu yükler MATLAB programında
kullanılmıştır.
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STATIC FINITE ELEMENT ANALYSIS OF COMPOSITE WING SPAR UNDER AERODYNAMIC LOADS
SUMMARY
In this study 2-D laminated Euler-Bernoulli composite beam element is considered as
the wing spar of an Unmanned Aerial vehicle. The wing beam component is analyzed
using finite element method by programming a MATLAB code. Main goal in this
project is to calculate transverse displacement, angle due to bending and twisting angle
of the cantilever beam considered as wing beam under the aerodynamic forces.
Transverse displacement w, angle due to bending (slope) !, torsion " are calculated
for the composite wing spar of an UAV under the aerodynamic forces in the regime
of level flight. Beam element is considered cantilever because the root of the beam is
fixed to the fuselage of the aircraft. This approach is convenient when the wing of the
aircraft taken into account as a composite beam.MATLAB code is created from
Ferreira’s book which is under the title of 1-D Euler-Bernoulli Beam. Changes are
done and beam is changed to composite beam and torsion is added. After performing
these changes, the beam element became 2-D.
Lift of an average reconnaissance unmanned aerial vehicle is considered as the
distributed load acting on the laminated composite beam element. Pitching moment
(buckling moment) is considered on the wing as distributed along the x-axis.
Considering a specific weight and wing span for an average unmanned aerial vehicle,
lift and pitching moment are calculated. These loads are used in the analysis computed
in MATLAB.
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1. INTRODUCTION
For so many structural and non-structural applications, composite materials are
increasingly becoming the number one material of choice. They are ideal for structural
applications where high strength-to-weight and stiffness-to-weight ratios are required.
For instance, composites for wing skins and other control surfaces have been used by
the aircraft industry to save fuel consumption and weight. Because aircrafts and
spacecrafts are weight sensitive structures in which composite materials are cost-
effective. (Pratik R. Patil1, 2019)
Composites are becoming an important component of the materials of today because
they offer features such as reduced weight, resistance to corrosion, increased fatigue
strength, faster and easier assembly, etc. They are used as products that range from
aircraft frames to golf clubs, electronic packaging and medical packaging, equipment,
and space vehicles to home design. (K.Kaw, 2006)
One of the basic structural or system element is the beam. Composite beams are
lightweight structures that can be used in many different applications, including in the
civil, aerospace, submarine, medical and automotive industries. Examples of beam
usage in structural engineering include homes, steel-framed structures, and bridges.
Beams serve in these implementations as structural components or sections that
sustaining the whole structure. Moreover, as a beam, the whole system can be modeled
at a preliminary stage. In addition, the entire wing of a plane can be also modeled as a
beam. The finite element method is widely accepted as an effective method to solve
engineering problems. In the modeling of engineering systems behaviors, including
composite structures such as laminated beams and plates, it has been one of the
prominent methods that extensively and successfully applied. For effective finite
element analysis, the use of simple and efficient elements is the most important step.
Various forms of finite elements have been developed to analyze and simulate the
structural behavior of composite beams such as one-dimensional (1-D), two-
dimensional (2-D), and three-dimensional (3D) elements (Xiaoshan Lin, 2020,).
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1.1 Litarature Review
Like all structures, in composite materials, local variations of the stress distribution
have to be precisely calculated to prevent local failure phenomena such as
delamination, matrix breakage, and local excessive plastic deformations due to the
distinct variations of mechanical properties from a layer to the adjacent one. Therefore,
it is a key issue to select an acceptable mathematical model to accurately predict the
local behaviors and efficiency of the described systems in different cases. The classical
beam theory (CBT) developed by Bernoulli-Euler (Bernoulli, 1964) is the most widely
used theory for the bending study of beams. This theory neglects the effect of shear
deformation and rotary inertia, so this theory is generally valid for thin beams and less
accurate for thicker beams. By enabling the effect of rotary inertia of the beam cross-
sections, Rayleigh (Rayleigh, 1877) improved the classical theory. This theory
neglects the effects of shear deformation and rotary inertia, but this theory is generally
precise for thin beams and less precise for thicker beams. By integrating the effect of
rotary inertia of the beam cross-sections, Rayleigh improved the classical theory.
Boley (1963) studied the precision of the CBT for the variable section beams. On the
basis of the two-dimensional principle of plane stress, the stresses and deflections of
the beam are observed. A modern beam theory was developed by Timoshenko
(Timoshenko, 1921), which was regarded as a modification of the classical beam
theory. This theory consists of first-order shear effects as well as the kinetic energy
effect of rotational inertia. This theory is also also known as the theory of first-order
shear deformation or Timoshenko beam theory (TBT).
Irschik (1991) developed a correlation between Levinson's (Levinson, 1981) refined
beam theories and classical Bernoulli-Euler theory, using the principle of virtual work
to expand the acceptance of refined higher-order theories. Under detailed review, the
rectangular cross-section beams with clamped and hinged boundary conditions are
taken into consideration.
For the linear static study of beams made of isotropic materials based on Carrera's
Unified Formulation, Carrera (E. Carrera, 2010) and co-authors developed many
refined theories (CUF). Using Timoshenko beam theory, Lin and Zhang (X. Lin, 2011)
developed a new displacement-based beam element with two degrees of freedom per
node for finite element studies of isotropic and composite beams. To explain the
layered properties of the composite beams, a layered approach is used. For a cantilever
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beam with different cross-sections, Miranda et al. (SD. Miranda, 2013) created a
simplified beam theory considering the influence of shear deformation.
In previous experiments, 2-D and 3-D elements were also used for finite-element
analysis of composite beams. Ferreira et al. (A.J.M Ferreira, 2001,), for instance,
modeled FRP-reinforced concrete composite beams based on the first-order shear
deformation theory, using degenerated 2-D shell components. For the study of
composite laminated plates and beams, Yu (Yu, 1994) suggested a six-nodded higher-
order triangular layered shell element with six degrees of freedom at each node.
While 3-D elements are typically able to make numerical predictions that are more
precise and reliable than 2-D elements, in terms of both formulation and modeling,
they are far more complicated. Also, due to the vast number of nodes and degrees of
freedom, 3-D components will cost much more computing space and energy. On the
other hand, for the study of beam-like structures, the 2-D aspect is more economic and
effective in computational terms (Xiaoshan Lin, 2020,).
1.2 What is a composite?
A composite is a structural material that consists of two or more mixed components
that are not soluble in one another are combined at a macroscopic stage. One
component is reinforcing phase like fiber and the other is called matrix which hold the
composite material together. Fibers, fragments, or flakes can be in the form of the
reinforcing phase material. In general, the matrix phase materials are continuous.
Examples of composite structures include steel-reinforced concrete and graphite-
fiber-reinforced epoxy, etc. (K.Kaw, 2006).
1.3 What are the advanced composites?
Composite materials that are traditionally used in the aerospace industry are called
advanced composites. These composites in a matrix material such as epoxy and
aluminum have high strength reinforcements while having a thin diameter.
Graphite/epoxy, Kevlar/epoxy, and composites of boron/ aluminum are examples
(K.Kaw, 2006). In this study laminated composite is the specified subject among
composite materials. Laminated composites are widely used in the advanced industries
such as aerospace industry, automobile industry etc. The reason of the highly usage of
laminated composite materials is that these materials are stiff, light, able to resist
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corrosion and easy to assemble. These properties of the laminated composite materials
are better than the other materials.
A laminated composite material is the material that is produced by stacking two or
more layers of lamina. Lamina is the one layer of composite material consist of two
components called reinforcing phase and matrix. Reinforcing phase which is fiber in
laminated composite can be in different orientation. The difference between each
lamina like the angle of fibers, the dimension of the fibers are the cases which specify
the properties of the material.
Figure 1: Laminated Composite plate.
In Figure 1 a laminated composite material is shown. In the Cartesian coordinate
system x-axis coincides with the beam axis and the mid-plane of the laminate is
located on the x-axis. is the fiber angle between the x-axis and the fiber direction.
z-axis is the lamina stacking direction and the thickness of the laminate is measured
by z-axis. The mid-plane of laminate is located on the origin of z-axis. Below z-axis
is represented as negative direction, just the contrary above of the origin of mid-plane
is represented positive direction. Top laminate is the number one laminate and the
bottom laminate is the last laminate. For instance; [0/30/45] denotes that the top
laminate has the 0° ply angle, mid-plane has 30° ply angle and the bottom lamina has
45° ply angle.
Figure 2: Laminated Composite Beam
α
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2. MECHANICS OF COMPOSITE MATERIAL
2.1 Hooke’s Law for a 2-D Angle Lamina
In general, laminates are not always consists of unidirectional lamina which means
their fiber directions are parallel to the x axis of the lamina. Because in transverse
direction of the unidirectional lamina, material properties can be lower than
longitudinal direction. Therefore, most of the laminates have laminae placed at a
different angle than unidirectional laminae. (K.Kaw, 2006)This section gives the
stress-strain relation for laminate consist of angle lamina.
There are two coordinate system, one of them is called local coordinate system and
shown in Figure 3 as 1-2 coordinate system. Local coordinate axis 1 shows the fiber
direction (parallel to fiber) and 2 shows normal (perpendicular) to the fiber. The other
coordinate system is the global coordinate system and shown in figure as x-y
coordinate system. x-axis is the longitudinal direction and y is the transverse direction
of the lamina. ! is the angle of the fiber between x axis and 1 axis of the local
coordinate system.
Figure 3 : Local and Global coordinate system for an angle lamina (K.Kaw, 2006)
To determine stress-strain relation between local coordinate and the global coordinate
system, stress relation is given as,
#$!$"%!"
& = [)]#$ #$$$%%$%& (2.1)
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where [T] is transformation matrix and defined as,
[)]#$ = #+% ,% −2,+,% +% 2,+,+ −,+ +% − ,%
& (2.2)
where c = cos(!) and s = sin(!).
Stress-strain equation between local strains and global stress can be written as,
#$!$"%!"
& = [)]#$[7] #ε$ε%9$%
& (2.3)
Global strains and local strains are related through the transformation matrix as,
#:$:%9$%
& = [;]#$[)][;] #:!:"9!"
& (2.4)
where R is the Reuter matrix defined as (K.Kaw, 2006),
[;] = #1 0 00 1 00 0 2
& (2.5)
Stress-strain relation with respect to global axis,
#$!$"%!"
& = [)]#$[7][;][)][;]#$ #:!:"9!"
& (2.6)
The multiplication of 5 matrices in front of the strain matrix is defined as new matrix
and called transformed reduced stiffness matrix as,
#$!$"%!"
& = >7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@ #:!:"9!"
& (2.7)
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2.2 Strain and Stress in Laminates
Stress and Strain equation of a laminate gives the global stresses in each lamina.
Before calculation the strains should be known at any point along the thickness of the
laminate. (K.Kaw, 2006) Stress- strain equation is,
A$!$"%!"
B = >7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@ A:!:"9!"
B (2.8)
where [7?] is transformed reduced stiffness matrix.
The strain displacements can be written as,
A:!:"9!"
B = C:!'
:"'
9!"'D + F C
G!G"G!"
D (2.9)
where {:!':"'9!"' }( is the midplane strains and {G!G"G!"}( is the midplane
curvature which given as (K.Kaw, 2006),
C:!'
:"'
9!"'D =
⎩⎪⎪⎨
⎪⎪⎧
NO'NPNQ'NR
NO'NR
+NQ'NP ⎭
⎪⎪⎬
⎪⎪⎫
(2.10)
midplane curvatures,
CG!G"G!"
D =
⎩⎪⎪⎨
⎪⎪⎧ −
N%V'NP%
−N%V'NR%
−2N%V'NRNP⎭
⎪⎪⎬
⎪⎪⎫
(2.11)
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So stress-strain equation can be written as (K.Kaw, 2006),
A$!$"%!"
B = >7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@ C:!'
:"'
9!"'D + >
7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@ CG!G"G!"
D (2.12)
The stresses may be different in every ply because transformed reduced stiffness
matrix changes from ply to ply. Transformed reduced stiffness matrix depends on the
material properties and the fiber orientation of the ply (K.Kaw, 2006).
Figure 4: Stress and Strain variation (K.Kaw, 2006)
Considering a laminate made of n plies and each plies has a thickness of t, the thickness
of laminate is h, the midplane location is h/2. Thickness of the laminate locate along
the z-coordinate. The thickness of the n-plied laminate is written as,
ℎ =XY)
*
)+$
(2.13)
Calculation of the resultant force per unit length along the x-y plane can be
represented by integrating global stresses in each lamina,
Z! = [ $!\F
,/%
#,/%
(2.14)
Z" = [ $"\F
,/%
#,/%
(2.15)
Z!" = [ %!"\F
,/%
#,/%
(2.16)
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where h/2 is the half thickness of the laminate.
Resulting moments can be calculated in similar way, by integrating the global
moments per unit length in the x-y plane along the thickness.
]! = [ $!F\F
,/%
#,/%
(2.17)
]" = [ $"F\F
,/%
#,/%
(2.18)
]!" = [ %!"F\F
,/%
#,/%
(2.19)
where;
Z!,Z" = normal force per unit length,
Z!" =shear force per unit length,
]!,]" =bending moment per unit length,
]!"=twisting moment per unit length.
The midplane strains and plane curvatures do not depend on the z-coordinates and the
transformed reduced stiffness matrix is constant for every single ply. (K.Kaw, 2006)
Resultant forces and moments can be written in terms of midplane stress and
curvatures as,
CZ!Z"Z!"
D =X [ >7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@
)
>:!'
:"'
9!"'@ \F
,!
,!"#
*
)+$
+X [ >7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@
)
>G!G"G!"
@ F\F
,!
,!"#
*
)+$
(2.20)
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and resultant moments,
C]!]"]!"
D =X [ >7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@
)
>:!'
:"'
9!"'@ F\F
,!
,!"#
*
)+$
+X [ >7?$$ 7?$% 7?$&7?$% 7?%% 7?%&7?$& 7?%& 7?&&
@
)
>G!G"G!"
@ F%\F
,!
,!"#
*
)+$
(2.21)
Rearranging equations resultant forces and moments can be written as (K.Kaw,
2006),
>Z!Z"Z!"
@ = #^$$ ^$% ^$&^$% ^%% ^%&^$& ^$& ^&&
& >:!'
:"'
9!"'@ + #
_$$ _$% _$&_$% _%% _%&_$& _$& _&&
& >G!G"G!"
@
(2.22)
>]!]"]!"
@ = #_$$ _$% _$&_$% _%% _%&_$& _$& _&&
& >:!'
:"'
9!"'@ + #
$̀$ $̀% $̀&
$̀% `%% `%&$̀& $̀& `&&
& >G!G"G!"
@ (2.23)
where
^). =X[(7?).)])(ℎ) − ℎ)#$)
*
)+$
, b = 1,2,6; e = 1,2,6, (2.24)
_). =X[(7?).)])(ℎ)% − ℎ)#$
%)
*
)+$
, b = 1,2,6; e = 1,2,6, (2.25)
)̀. =X[(7?).)])(ℎ)/ − ℎ)#$
/)
*
)+$
, b = 1,2,6; e = 1,2,6, (2.26)
[A] is called extensional stiffness matrix, [B] is called coupling stiffness matrix, [D]
is called bending stiffness matrix. Combining resultant force and moment matrices
gives 6x6 matrix as,
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⎩⎪⎪⎨
⎪⎪⎧Z!Z"Z!"]!]"]!"⎭
⎪⎪⎬
⎪⎪⎫
=
⎣⎢⎢⎢⎢⎡^$$ ^$% ^$& _$$ _$% _$&^$% ^%% ^%& _$% _%% _%&^$& ^$& ^&& _$& _%& _&&_$$ _$% _$& $̀$ $̀% $̀&_$% _%% _%& $̀% `%% `%&_$& _%& _&& $̀& `%& `&&⎦
⎥⎥⎥⎥⎤
⎩⎪⎪⎨
⎪⎪⎧:!'
:"'
9!"'
G!G"G!"⎭
⎪⎪⎬
⎪⎪⎫
(2.27)
[A] which is extensional stiffness matrix gives relation between the resultant in-plane
forces and in-plane strains, [B] which is coupling stiffness matrix gives relation
between force, moments pairs and strains, midplane curvature pairs, [D] which is
bending stiffness matrix gives relation between bending moment and the plate
curvatures.
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3. EULER-BERNOULLI THEORY
3.1 Euler-Bernoulli Assumptions
Considering a beam has length of L and the cross-section A, properties of the beam
are, isotropic and homogenous. The xz plane is the plane that loads are acting.
Therefore, there are three main assumptions which create the Euler-Bernoulli theory.
1.Plane section remain plain after deformation
2. The displacement along the y-axis (v) equal to zero.
3. Deformed beam angles or slopes are small.
First assumption implies that after the beam deforms, any portion of a beam (i.e., cut
through the beam at a certain point along with its distance) that was a flat plane before
the beam deforms would remain a flat plane after deformation. Therefore, it is clear
that the beam should be exposed only bending moments, it means there is no shear on
transverse planes.
This assumption is reasonably true for bending beams unless the beam has
considerable shear or torsional stresses relative to the axial stresses. This assumption
is shown in figure representing clearer.
Figure 5: Presentation of plane section remain plane assumption (Erochko, 2020)
Assuming that the slope angles of the beam are small gives imported simplicity to the
analysis. Small angle approximation is also valid which assume that ,bl(!) ≈ ! and
+n,(!) ≈ 1. Considering x as the location along the length of the beam V(P)
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represents the displacement of the beam, and the angle due to bending (slope) is !(P).
Slope can be represent as,
!(P) =\V\P
(3.1)
3.2 Displacement Field
Considering the Euler-Bernoulli assumption displacement field can be written as,
O(P, R, F) = −F!(P) (3.2)
Q(P, R, F) = 0 (3.3)
V(P, R, F) = V(P) (3.4)
Firs assumption support that the rotation is equal to the slope of the beam axis, so it
gives (Oñate, 2013),
! =\V\P
(3.5)
O = −F\V\P
(3.6)
3.2.1 Strain and Stress Fields
Starting from the strain field for a 3D solid,
:! =oOoP
= −F\%V\P%
(3.7)
:" = :0 = 9!" = 9!0 = 9"0 = 0
The beam is under pressure a pure axial strain, considering the Hook’s law axial stress
can be written as (Oñate, 2013),
$! = pε1 = −Fp\%V\P%
(3.8)
where E is the Young modulus.
14
3.2.2 Bending moment-curvature relationship
For homogeneous material, a specific cross section, the bending moment can be
written as,
] =qF$!2
\^ =rqF%2
\^sp\%V\P%
= pt"\%V\P%
= pt"G (3.9)
where t" = ∬ F%2 \^ is the moment of inertia with respect to y axis and G = 3$43!$
is
the bending curvature of the beam axis. In figure 4 the presentation of moment and
axial stress in a cross section, and the distribution of stresses can be seen obviously.
Figure 6: Presentation of Bending moment and Stress sign (Oñate, 2013)
3.2.3 Principle of Virtual Work
The principle of virtual work states that in equilibrium the virtual work of the forces
applied to a system is zero. It is written as,
vw$!:! \x5
= [ [wVy0 + w z\V\P{|] \PXwV)}0!
)
6
'
+Xw z\V\P{.].7
.
(3.10)
Where }0! is point loads and ].7 is concentrated moments, y0 is distributed load.
Virtual strain work which is showed in Eq can be simplified as follows,
∭ w$!:! \x5 = ∫ w6' Ä3
$43!$
Å Ç∬ F%2 \^Ép 3$43!$
= [ w6
'r\%V\P%
spt\%V\P%
\P = [ w6
'G]\P (3.11)
15
Therefore, between the bending moment and the virtual curvature, the virtual strain
work is expressed as the integral along the beam axis of the material.
[ w6
'G]\P = [ [wVy0 + w z
\V\P{|] \PXwV)}0!
)
+Xw z\V\P{.].7
.
6
'
(3.12)
3.2.4 Theoretical calculations of displacements
For cantilever Euler-Bernoulli beam under uniformly distributed load, bending
displacement of the free edge can be calculated by solving theoretical formula which
is given as (Oñate, 2013),
V(P) =ÑÖ8
8pt" (3.13)
where q is the distributed load and L is the length of the beam. Bending slope can be
calculated by solving following equation,
!(P) =ÑÖ/
6pt" (3.14)
Theoretical formula of buckling angle is given as,
"(P) =)Öáà
(3.15)
where T is the torsion, G is the shear of the beam and J is the polar moment of
inertia.
16
4. 1-D EULER-BERNOULLI TWO-NODE COMPOSITE BEAM ELEMENT
Assume a laminated beam element with two nodes and two degrees of freedom at each
node. The length of composite beam is L, thickness is t and the width of beam is b as
shown in Figure 7.
Degrees of freedom at each node is represented as the transverse displacement w and
the rotation due to bending !. Cross section of the beam is consisting of stacked layers
which have specific width. Each layer has the same properties and the beam is
homogenous.
Figure 7: Laminated Composite Beam (Xiaoshan Lin, 2020,)
Assume an equation for the deflected beam shape;
V(â) = ä' + ä$â + ä%â% + ä/â/ (4.1)
3rd order polynomial with four unknown coefficient and there are four degrees of
freedom.
Considering boundary conditions at each node;
V(â) = V$ at â = 0 949:= !$ at â = 0
V(â) = V% at â = Ö 949:= !% at â = Ö
Solving for ä' at â = 0 it is known from boundary condition that V(â) = V$, and
substituting gives;
V$ = ä' + ä$ × 0 + ä% × 0% + ä/ × 0/ (4.2)
Equation gives;
V$ = ä'
17
Solving for ä$ at â = 0 it is known from boundary condition that ;4;:= !$, and
substituting gives;
!$ = ä$ + 2 × ä% × 0 + 3 × ä/ × 0% (4.3)
Equation gives;
!$ = ä$
Solving for ä%and ä/ considering the boundary conditions also gives V%and
!%respectively.
V% = ä%
!% = ä/
Substituting ä', ä$, ä%, ä/ and the shape functions to equation 1.13 and rearrange it
gives;
V(â) = Z$V$ + Z%!$ + Z/V% + Z8!% (4.4)
w(â) is the transverse position of the beam at any point â along the beam.
4.1 Kinematics of A Plane Laminated Beam
In this chapter kinematics of the laminated beam is considered. Geometry of the beam
is introduced in Figure 7.
4.1.1 Displacement fields
Displacement fields are expressed as,
O(P, F) = −F!(P) (4.5)
V(â, F) = V(P) (4.6)
where V(P) is transverse displacement, !(P) is rotation due to bending, Qis the
lateral displacement along the y axis is zero. Because of normal orthogonality
condition which is cross sections normal to the beam axis remain plane and orthogonal
to the beam axis after deformation, the rotation is equal to the slope of the beam as
shown below,
!(P) = !"!# (4.7)
and the transeverse displacement is equal to,
V(P) = V'(P) (4.8)
18
4.1.2 Strain and Stress Field
Considering Euler-Bernoulli beam theory and laminated composite beam, stress-strain
equations can be written as shown below. Axial strain equation is,
:! =oOoP
=oO'oP
− Fo!oP
(4.9)
Because of O' is 0, equation (4.9) is taking the form of,
:! = −Fo!oP
(4.10)
The axial stress $! is related to ε1 by using Hook Law as,
$! = [Q]ε1 (4.11)
[7]is the reduced transformed stiffness matrix of the laminated composite beam.
4.1.3 Displacement Functions of the Composite Beam Element
Each layer is considered to be in a state of plane stress and the material properties are
constant along each layer's thickness. The material properties of the actual cross
section are then obtained by summing the contribution of each layer algebraically as,
)̀. =13X[7).]
*
)+$
(F)<$/ − F)
/) (4.12)
Equation (4.12) shows )̀. which is the bending stiffness matrix. z=<$ and z= are the
coordinates of the upper and the lower surface of the ith layer respectively. [7=>]is the
reduced transformed stiffness matrix of the ith layer.
Using the displacements and rotations at the two ends of the composite beam element,
the rotations and transverse displacements at any point along the x direction can be
described. For the composite beam element with length L, width b, and height h, the
formulas of deflection w and rotation ! are given as,
V = Z$V$ + Z%!$ + Z/V% + Z8!% (4.13)
! = Z?V$ + Z&!$ + Z@V% + ZA!% (4.14)
19
5. FINITE ELEMENT FORMULATION
In this section the finite element formulation of composite beam will be introduced.
The composite beam element with two nodes has two degrees of freedom at each
node.w represents the transverse displacement and θ represents rotation due to
bending. Assuming the transverse shear stress is zero according to Euler-Bernoulli
beam theory, transverse displacement w is given by;
V = ä' + ä$â + ä%â% + ä/â/
(5. 1)
Rotation due to bending ! = 9B91
is given by,
! = ä$ + 2ä%â + 3ä/â%
(5. 2)
Considering two nodes, the first one is in location x=0 and second is in x=L , the nodal
displacements are determined and shown below in matrix form as the displacement
values as follows,
ê
V$!"$V%!"%
ë = í
1 0 0 00 1 0 01 Ö Ö% Ö/
0 1 2Ö 3Ö%ì C
ä'ä$ä%ä/
D
(5. 3)
Expressing the displacement field in terms of the nodal displacements gives,
{Ñ} = [Z]{ÑC}
(5. 4)
where {Ñ} is the displacement field, {ÑC} is the nodal displacement, [N] is the shape
function matrix. The expressions are explained as follows;
{Ñ} = {V !}(
(5. 5)
[N] = {ZD ZE}(
(5. 6)
{ÑC} = {V$ !"$ V% !"%}
(5. 7)
20
Following equations are the Hermit shape functions that are derived;
Z$ = 1 −3âÖ%
%
+2âÖ/
/
(5. 8)
Z% = â −2âÖ
%
+âÖ%
/ (5. 9)
Z/ =3âÖ%
%
−2âÖ/
/
(5. 10)
Z8 = −âÖ
%+âÖ%
/ (5. 11)
After giving the shape function equation transverse displacement and the rotation can
be express again in a different way which is in matrix form;
ïV!ñ=ó
Z$ Z% Z/ Z8Z? Z& Z@ ZA
ò C
V$!$V$!$
D (5. 12)
Equation (4.7) gives the relationship between the rotation and the transverse
displacement; same relation can be used between the shape functions. Considering
relation between two function shape functions related to ! can be expressed as,
Z? = −6âÖ%+6âÖ/
%
(5. 13)
Z& = 1 −4âÖ+3âÖ%
%
(5. 14)
Z@ =6âÖ%−6âÖ/
%
(5. 15)
ZA = −2âÖ+2âÖ%
%
(5. 16)
21
5.1 Strain and Strain matrix
In this section strain-displacement relationships will be introduced. The strain energy
of the composite beam element is represented as,
:! = −zoθox
(5. 17)
Writing in matrix form the equation becomes,
:! = [Z]úÑ(C)ù (5. 18)
where [Z] is the shape function matrix, úÑ(C)ù is the element nodal displacement which
is
Ñ(C) = Ñ) and i=1, 2. Also expressing the notation Ñ) = {V) !) }T . Shape function
matrix is given as considering equation (5. 18)
[Z] = [Z$ Z% Z/ Z8] (5. 19)
The strain energy of the composite beam element is represented as,
û =12q[{:}(
!2
{$} \P \^ (5. 20)
Rearranging equation (5. 20) regarding composite stress-strain equation and bending
stiffness matrix,
û =12[ü{:}(`!
{:}\P (5. 21)
where D is the bending stiffness matrix (XuanWang, 2015).
⎩⎪⎪⎨
⎪⎪⎧Z!Z"Z!"]!]"]!"⎭
⎪⎪⎬
⎪⎪⎫
=
⎣⎢⎢⎢⎢⎡^$$ ^$% ^$& _$$ _$% _$&^$% ^%% ^%& _$% _%% _%&^$& ^$& ^&& _$& _%& _&&_$$ _$% _$& $̀$ $̀% $̀&_$% _%% _%& $̀% `%% `%&_$& _%& _&& $̀& `%& `&&⎦
⎥⎥⎥⎥⎤
⎩⎪⎪⎨
⎪⎪⎧:!'
:"'
9!"'
G!G"G!"⎭
⎪⎪⎬
⎪⎪⎫
It yields,
]! = $̀$G!
(5. 22)
Substituting equation gives,
û =12[úÑ(C)ù!
(Z′′([ $̀$]üZ′′úÑ(C)ù \P (5. 23)
22
Element stiffness matrix is,
[°C] = [Z′′(6
'
[ $̀$]üZ′′ \P (5. 24)
It gives,
[°C] = $̀$üÖ/
í
12 6Ö −12 6Ö6Ö 4Ö% −6Ö 2Ö%
−12 −6Ö 12 −6Ö6Ö 2Ö% −6Ö 4Ö%
ì (5. 25)
Considering virtual work approach,
°(C)úÑ(C)ù − úy(C)ù = 0
(5. 26)
úy(C)ù is the force vector.
The work done by the distributed force is defined as,
w¢C = [£wV \P
6
'
(5. 27)
where p is the distributed force substituting (A.J.M Ferreira, 2001,)
[£wV \P
6
'
= wúÑ(C)ù([£Z( \P
6
'
(5. 28)
The vector of nodal forces derived for distributed force is obtained as (Ferreira &
Media., 2009),
[yC] = [£Z( \P
6
'
(5. 29)
For a beam element which has q distributed force acting,
[yC] = £
⎣⎢⎢⎡Ö/2Ö%/12Ö/2
−Ö%/12⎦⎥⎥⎤=
⎣⎢⎢⎡£Ö/2£Ö%/12£Ö/2
−£Ö%/12⎦⎥⎥⎤
(5. 30)
23
6. 2-D EULER-BERNOULLI COMPOSITE BEAM ELEMENT
6.1 Bending-Torsion Coupling
For symmetric composite beam simplified linear analysis derivation of bending-
torsion coupling is represented in this section.
Determining bending stiffness, effective torsional stiffness and the bending-twisting
structural coupling for composite Euler-Bernoulli beam element,
⎩⎪⎪⎨
⎪⎪⎧Z!Z"Z!"]!]"]!"⎭
⎪⎪⎬
⎪⎪⎫
=
⎣⎢⎢⎢⎢⎡^$$ ^$% ^$& _$$ _$% _$&^$% ^%% ^%& _$% _%% _%&^$& ^$& ^&& _$& _%& _&&_$$ _$% _$& $̀$ $̀% $̀&_$% _%% _%& $̀% `%% `%&_$& _%& _&& $̀& `%& `&&⎦
⎥⎥⎥⎥⎤
⎩⎪⎪⎨
⎪⎪⎧:!'
:"'
9!"'
G!G"G!"⎭
⎪⎪⎬
⎪⎪⎫
So it becomes for laminated composite,
•]!]!"
¶=ó $̀$ $̀&
$̀& `&&ò •G!G!"
¶
(6.1)
Where G! is the bending curvature, G!" is the twisting derivation.
6.2 Finite element formulation for Bending-Torsion coupling
For 2-D Euler-Bernoulli composite beam element, there is an extra degree of freedom
than 1-D composite beam which is introduced before. Torsion is added and two
nodded composite beam element has 3 degrees of freedom. The displacement field has
determined as,
V = ä' + ä$â + ä%â% + ä/â/ (6.2)
! = ä$ + 2ä%â + 3ä/â% (6.3)
" = ä8 + ä?â (6.4)
Displacement fields are determined according to two nodes on the composite beam
element which are located in x=0, and x=L.
Matrix form is given below as first node is at x=0, and the second node is at x=L.
24
⎩⎪⎨
⎪⎧V$!$"$V%!%"%⎭⎪⎬
⎪⎫
=
⎣⎢⎢⎢⎢⎡1 0 0 0 0 00 1 0 0 0 00 0 0 0 1 01 Ö Ö% Ö/ 0 00 1 2Ö 3Ö% 0 00 0 0 0 1 Ö⎦
⎥⎥⎥⎥⎤
⎩⎪⎨
⎪⎧ä'ä$ä%ä/ä8ä?⎭⎪⎬
⎪⎫
(6.5)
The relation between the displacement field and the nodal displacement is,
{Ñ} = [Z]{ÑC} (6.6)
Expressing the equation for the composite beam element, {Ñ} is displacements, {ÑC}
is the nodal displacements,[Z] is the shape function matrix which is derived with
Hermite shape functions.
{Ñ} = {V ! "}(
(6.7)
{ÑC} = {V$ !$ "$ V% !% "%} (6.8)
[N] = {ZD ZE ZH}( (6.9)
Shape function components are determined as,
Z$ = 1 −3âÖ%
%
+2âÖ/
/
(6.10)
Z% = â −2âÖ
%
+âÖ%
/ (6.11)
Z/ =3âÖ%
%
−2âÖ/
/
(6.12)
Z8 = −âÖ
%+âÖ%
/ (6.13)
Z? = 1 −
âÖ
(6.14)
Z& =
âÖ
(6.15)
25
Shape functions for displacements are,
ZD = [Z$ Z% 0 Z/ Z8 0] (6.16)
ZE = [\Z$\P
\Z%\P
0\Z/\P
\Z8\P
0] (6.17)
ZH = [0 0 Z? 0 0 Z&] (6.18)
where ZD is the shape function for (w) flapwise bending, ZE is the shape function for
(!) angle due to bending and ZH is the shape function for (") torsion.
Before acquiring Element Stiffness matrix of the beam element, strain energy equation
is given,
û =12q[{:}(
!2
{$} \P \^ (6.19)
For composite beam strain energy equation is given (XuanWang, 2015),
û =12[ü{:}(`!
{:}\P (6.20)
where b is the width of the beam, D is bending stiffness matrix. Stress-strain field is
introduced before,
$ = [Q]ε
Where [Q] is reduced transformed stiffness matrix. Resulting moments in the
laminated composite beam is given in matrix form, for only bending moment and
twisting moment,
•]!]!"
¶=ó $̀$ $̀&
$̀& `&&ò •G!G!"
¶ (6.21)
where G!=Bending curvature, G!"=Twisting derivation. Here :( is strain and refer
to {G!G!"} and strain field takes form of,
26
{:}(`{:} = {G!G!"} ó $̀$ $̀&
$̀& `&&ò•
G!G!"
¶
where G! =9$49:$
and G!" =9H9:
,
û =12[( $̀$ü(G!)% + `&&ü(G!")% + $̀&üG!G!" + $̀&üG!"G!)!
\P (6.22)
It gives,
! = 12%&'
(")($(*%′′$&
[-'']/*%′′ + *(′[-))]/*(′$
+ [-')]/*%′′$*(′ + [-')]/*(′$*%′′)&'(")(23 (6.23)
Element stiffness matrix equation is obtained as follows,
[°C] = [( $̀$ü
6
'
ZDII%ZD′′ + `&&üZH
I%ZH′ + $̀&üZHI%ZD
II
+ $̀&üZDII(ZH)\P
(6.24)
Substituting shape functions and obtaining element stiffness matrix,
["!]
=
⎣⎢⎢⎢⎢⎢⎡ 12(+"",)//
# 6(+"",)//$ 0 −12(+"",)//# 6(+"",)//$ 06(+"",)//$ 4(+"",)// (+"%,)// −6(+"",)//$ 2(+"",)// −(+"%,)//
0 (+"%,)// (+%%,)// 0 −(+"%,)// −(+%%,)//−12(+"",)//# −6(+"",)//$ 0 12(+"",)//# −6(+"",)//$ 06(+"",)//$ 2(+"",)// −(+"%,)// −6(+"",)//$ 4(+"",)// (+"%,)//
0 −(+"%,)// −(+%%,)// 0 (+"%,)// (+%%,)// ⎦⎥⎥⎥⎥⎥⎤
(6.25)
The vector of nodal forces derived for distributed force is obtained as (Ferreira &
Media., 2009),
[yC] = [(£**( + )*H() \P6
'
(6.26)
For a beam element which has q distributed force acting,
[yC] = £
⎣⎢⎢⎢⎢⎡Ö/2Ö%/120Ö/2
−Ö%/120 ⎦
⎥⎥⎥⎥⎤
+ )
⎣⎢⎢⎢⎢⎡00Ö/200Ö/2⎦
⎥⎥⎥⎥⎤
(6.27)
27
Summation gives force vector as,
[yC] =
⎣⎢⎢⎢⎢⎢⎡£Ö/2£Ö%/12)Ö/2£Ö/2
−£Ö%/12)Ö/2 ⎦
⎥⎥⎥⎥⎥⎤
(6.28)
where p is the distributed force and T is the torsion.
28
7. DETERMINING COMPOSITE BEAM PROPERTIES
7.1 Determining Loads acting on wing spar
The properties of the unmanned aerial vehicle which is used in this study are shown
below,
• Length: 3.4 m
• Wing Span: 4 m
• Chord length: 0.5 m
• Wing area: 2 m2
• Weight: 100 kg
Considering level, flight lift and the pitching moment acting to the wing of the UAV
is calculated. For level flight lift will be equal to the weight of the aircraft.
Therefore, lift distribution can be calculated as,
Ö =12ßx%+®6 (7.1)
Lift will be equal to,
Ö = 100 ∗ 9.81 = 981Z
Distributed lift will be,
981/4 = 245.25Z/|
Distributed load used in study will be taken as P=250N/m. In the program which is
called XFLR5 Naca 4415 airfoil is analysed. In analysis wing element which has
L=4m and c=0.5 m is used. ®6 = 0.4 and ®J = −0.208 is obtained from analysis. So,
obtaining these values pitching moment can be calculated as,
] =12ßx%+%®J (7. 2)
Substituting values into the moment equation gives,
] =121.225(40)%(0.5)%(−0.208) = −50.96Z
29
The distributed buckling moment is taken as M=-50N along the x axis. The calculated
values are used in the analysis as distributed load and the distributed moment to
calculate bending and twisting of the wing during cruise flight.
7.2 Determining Composite Beam Properties
The composite wing spar is designed considering properties of an unmanned air
vehicle, and wing spar is assumed reaching to the end of the wing tip. So considering
an UAV which has 4m wing span and 0.5m chord length, the wing spar that is used
for analysis will be 2m cantilever composite beam. The length of the composite beam
is shown below,
• Length: 2 m
• Width: 10 cm
• Height: 3.6 cm
The fiber angles and the number of lamina of the beam is given below,
• [6(15)/6(-15)]
• [6(30)/6(-30)]
• [6(45)/6(-45)]
which means there are 12 plies and the fiber angles of firs laminate that analyzed are
15°, 15°, 15°, 15°, 15°, 15° , -15°, -15°,-15°, -15°,-15°, -15°from top ply to the bottom
ply of the composite beam respectively. Three different materials are analyzed for the
symmetric beam and every material analyzed for different fiber orientation showed
above. Every material has been compared by their fiber orientation. Material
properties of analyzed composite materials are shown in Table 1.
Table 1: Material Properties of Analyzed composite materials.
Material Properties of a ply
Materials Graphite-
epoxy Hmcf fabric
Std ud cf
p$(GPa) 181 85 135 p%(GPa) 10.3 85 10 á$%(GPa) 7.17 5 5 Q 0.28 0.1 0.3
30
8. RESULTS AND DISCUSSION
Results of analysis for beam material std ud cf with different fiber orientation is given
below.
Figure 8: Transverse displacement and buckling angle of std ud cf composite beam.
Figure 9: Angle due to bending (rad), beam material std ud cf(uncoupling).
0 0.5 1 1.5 2X axis of the beam (m)
0
0.02
0.04
0.06
0.08
0.1
0.12
w(x
) Tra
nsve
rse
disp
lace
men
t (m
)
Transverse Displacement of Composite Beam
0 0.5 1 1.5 2X axis of the beam (m)
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
phi(x
) Buc
klin
g an
gle(
rad)
Twisting angle of Composite Beam
[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
thet
a(x)
Ang
le d
ue to
ben
ding
(rad
)
Bending Slope of Composite Beam
[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]
31
Results of analysis for beam material Hmcf fabric with different fiber orientation is given below.
Figure 10: Transverse displacement (m) and buckling angle(rad), beam material
Hmcf fabric(uncoupling).
Figure 11: Angle due to bending (rad), beam material Hmcf fabric(uncoupling).
0 0.5 1 1.5 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09w
(x) T
rans
vers
e di
spla
cem
ent (
m)
Transverse Displacement of Composite Beam
0 0.5 1 1.5 2X axis of the beam (m)
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
phi(x
) Buc
klin
g an
gle(
rad)
Twisting angle of Composite Beam
[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
thet
a(x)
Ang
le d
ue to
ben
ding
(rad
)
Bending Slope of Composite Beam
[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]
32
Results of analysis for beam material graphite-epoxy with different fiber orientation is given below.
Figure 12: Transverse displacement (m) and buckling angle(rad), beam material
graphite-epoxy(uncoupling).
Figure 13: Angle due to bending (rad), beam material graphite-epoxy(uncoupling).
0 0.5 1 1.5 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
w(x
) Tra
nsve
rse
disp
lace
men
t (m
)
Transverse Displacement of Composite Beam
0 0.5 1 1.5 2X axis of the beam (m)
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
phi(x
) Buc
klin
g an
gle(
rad)
Twisting angle of Composite Beam
[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
thet
a(x)
Ang
le d
ue to
ben
ding
(rad
)
Bending Slope of Composite Beam
[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]
33
Table 2: Comparison of analysis results and theoretical results for uncoupling analysis.
Fiber Angles of
Laminates
Material
MATLAB Results Theoretical Results
!(#)(m) %(#)(&'() )(#)(&'() !(#)(m) %(#)(&'() )(#)(&'()
[6(15)/6(-15)] Graphite-
epoxy 0.0270 0.0180 -0.0510 0.0270 0.0180 -0.0510
[6(30)/6(-30)] Graphite-
epoxy 0.0397 0.0267 -0.0236 0.0397 0.0267 -0.0236
[6(45)/6(-45)] Graphite-
epoxy 0.0766 0.0511 -0.0186 0.0766 0.0511 -0.0186
[6(15)/6(-15)] Hmcf
fabric 0.0560 0.0374 -0.0647 0.0560 0.0374 -0.0647
[6(30)/6(-30)] Hmcf
fabric 0.0716 0.0477 -0.0287 0.0716 0.0477 -0.0287
[6(45)/6(-45)] Hmcf
fabric 0.0831 0.0554 -0.0225 0.0831 0.0554 -0.0225
[6(15)/6(-15)] Std ud cf 0.0375 0.0250 -0.0712 0.0375 0.0250 -0.0712
[6(30)/6(-30)] Std ud cf 0.0548 0.0366 -0.0327 0.0548 0.0366 -0.0327
[6(45)/6(-45)] Std ud cf 0.1039 0.0697 -0.0257 0.1039 0.0697 -0.0257
34
In this section for the same materials different fiber orientations are analyzed. Fiber
orientations are adjusted due to bending-twisting coupling can be seen during analysis.
In bending stiffness matrix, !!" which causes coupling is not zero in this section. So,
the difference can be seen easily. Effect of distributed load on the twisting angle and
effect of distributed buckling moment on the bending slope can be recognized by
examining analysis results. Every ply will be 2 mm and summation of12 plies gives
3.6 cm total laminate height.
The fiber orientations are determined as,
• Laminate 1:[30/30/30/45/90/90/90/90/45/-30/-30/-30]
• Laminate 2:[30/60/60/45/90/90/90/90/45/-60/-60/-30]
• Laminate 3:[30/60/60/45/0/0/0/0/45/-60/-60/-30]
• Laminate 4:[30/60/45/60/45/60/45/60/45/60/45/30]
• Laminate 5:[30/60/45/90/45/90/45/90/45/60/45/30]
• Laminate 6:[30/45/60/45/60/90/90/60/45/60/45/30]
35
Results of beam which has hmcf fabric composite material are shown below.
Figure 14: Transverse displacement (m) and buckling angle(rad), beam material
hmcf fabric (coupling).
Figure 15: Angle due to bending (rad), beam material hmcf fabric(coupling).
0 0.5 1 1.5 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07w(
x) T
rans
vers
e dis
place
men
t (m
)
Transverse Displacement of Composite Beam
0 0.5 1 1.5 2X axis of the beam (m)
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
phi(x
) Buc
kling
ang
le(ra
d)
Twisting angle of Composite Beam
Laminate 4Laminate 5Laminate 6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
thet
a(x)
Ang
le d
ue to
ben
ding
(rad
)
Bending Slope of Composite Beam
Laminate 4Laminate 5Laminate 6
36
Results of beam which has std ud cf composite material are shown below.
Figure 16 Transverse displacement and buckling angle of std ud cf composite
beam(coupling).
Figure 17: Angle due to bending (rad), beam material std ud cf (coupling).
0 0.5 1 1.5 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1w
(x) T
rans
vers
e di
spla
cem
ent (
m)
Transverse Displacement of Composite Beam
0 0.5 1 1.5 2X axis of the beam (m)
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
phi(x
) Buc
klin
g an
gle(
rad)
Twisting angle of Composite Beam
Laminate 1Laminate 2Laminate 3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
thet
a(x)
Ang
le d
ue to
ben
ding
(rad
)
Bending Slope of Composite Beam
Laminate 1Laminate 2Laminate 3
37
Results of beam which has graphite-epoxy composite material are shown below.
Figure 18: Transverse displacement (m) and buckling angle(rad), beam material
graphite-epoxy (coupling).
Figure 19: Angle due to bending (rad), beam material graphite-epoxy (coupling).
0 0.5 1 1.5 2X axis of the beam (m)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
w(x
) Tra
nsve
rse
disp
lace
men
t (m
)
Transverse Displacement of Composite Beam
0 0.5 1 1.5 2X axis of the beam (m)
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
phi(x
) Buc
klin
g an
gle(
rad)
Twisting angle of Composite Beam
Laminate 1Laminate 2Laminate 3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2X axis of the beam (m)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
thet
a(x)
Ang
le d
ue to
ben
ding
(rad
)
Bending Slope of Composite Beam
Laminate 1Laminate 2Laminate 3
38
Table 3: Comparison of analysis results and theoretical results for coupling analysis.
Fiber Angles of Laminates
Material
MATLAB Results Theoretical Results !(#)(m) %(#)(&'() )(#)(&'() !(#)(m) %(#)(&'() )(#)(&'()
[3(30)/45/4(90)/45/4(-30)] Graphite-epoxy
0.0443 0.0296 -0.0268 0.0429 0.0286 -0.0238
[30/2(60)/45/4(90)/45/2(-60)/-30] Graphite-epoxy
0.0722 0.0483 -0.0288 0.0698 0.0465 -0.0238
[30/2(60)/45/4(90)/45/2(-60)/-30] Graphite-epoxy
0.0655 0.0438 -0.0283 0.0633 0.0422 -0.0238
[30/60/45/60/45/60/45/60/45/60/45/30] Hmcf fabric 0.0759 0.0507 -0.0296 0.0746 0.0497 -0.0266 [30/60/45/90/45/90/45/90/45/60/45/30] Hmcf fabric 0.0747 0.0499 -0.0325 0.0726 0.0484 -0.0279 [30/45/60/45/60/90/90/60/45/60/45/30] Hmcf fabric 0.0776 0.0519 -0.0325 0.0753 0.0502 -0.0261 [3(30)/45/4(90)/45/4(-30)] Std ud cf 0.0610 0.0408 -0.0370 0.0592 0.0395 -0.0329 [30/2(60)/45/4(90)/45/2(-60)/-30] Std ud cf 0.0977 0.0653 -0.0395 0.0946 0.0631 -0.0329 [30/2(60)/45/4(90)/45/2(-60)/-30] Std ud cf 0.0890 0.0595 -0.0389 0.0862 0.0575 -0.0329
39
8.1 Discussion
In this study bending-twisting analysis of composite wing spar has been represented.
Bending-twisting displacements of the composite wing spar of an avearage
reconnaisance UAV, during level flight are calculated. Antisymmetric laminates
which has fiber angle orientations such as [6(15)/6(-15)], has zero coupling stiffness (
!!"). Therefore, theory results fits with MATLAB displacement results. That means
coupling effects of distributed load and buckling moment don’t exist. On the contrary,
when the fiber orientations create coupling stiffness that means ( !!") is not zero,
coupling effect between bending and twisting can be recognized. If a conclusion is
drawn for the three different materials used in this study, the following inferences can
be made. The graphite-epoxy composite beam is the beam that makes the least
displacement, amongst the other materials, for the same fiber orientations. This result
is valid for bending slope, transverse displacement and buckling angle. On the
contrary, the most displaced material is the std ud cf beam. This result can be deduced
by examining the property table of materials as well.
40
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