STATIC FINITE ELEMENT ANALYSIS OF COMPOSITE WING SPAR

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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


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

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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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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


-0.08

-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

0.01

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)



-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

phi(x

) Buc

klin

g an

gle(

rad)


[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]


0

0.01

0.02

0.03

0.04

0.05

0.06

thet

a(x)

Ang

le d

ue to

ben

ding

(rad

)


[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.


graphite-epoxy(uncoupling).

Figure 13: Angle due to bending (rad), beam material graphite-epoxy(uncoupling).


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

)



-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

phi(x

) Buc

klin

g an

gle(

rad)


[6(15)/6(-15)][6(30)/6(30)][6(45)/6(-45]


0

0.01

0.02

0.03

0.04

0.05

0.06

thet

a(x)

Ang

le d

ue to

ben

ding

(rad

)


[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.


hmcf fabric (coupling).

Figure 15: Angle due to bending (rad), beam material hmcf fabric(coupling).


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

)



-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

phi(x

) Buc

kling

ang

le(ra

d)


Laminate 4Laminate 5Laminate 6


0

0.01

0.02

0.03

0.04

0.05

thet

a(x)

Ang

le d

ue to

ben

ding

(rad

)



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

0.01

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)



-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

phi(x

) Buc

klin

g an

gle(

rad)




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

)



37

Results of beam which has graphite-epoxy composite material are shown below.


graphite-epoxy (coupling).

Figure 19: Angle due to bending (rad), beam material graphite-epoxy (coupling).


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

)



-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

phi(x

) Buc

klin

g an

gle(

rad)




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

)



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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finite element analysis of composite beam under static load. AIP Conference Proceedings 2057.

K.Kaw, A. (2006). Mechanics of Composite Materials. London New York: Taylor

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Xiaoshan Lin içinde, Nonlinear Finite Element Analysis of Composite and Reinforced Concrete Beams, (s. 29-45,). Woodhead Publishing,.

Bernoulli, J. (1964). Bernoulli J. Curvatura laminae elasticae. Acta Eruditorum

Lipsiae 1694. Acta Eruditorium Lipsiae, 262-276. Rayleigh, L. (1877). Theory of Sound. Macmillan and Co. Boley, B. (1963). On the accuracy of the Bernoulli-Euler theory for beams of

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theory. Int J Solid Struct, 1105-1112. Levinson, M. (1981). A new rectengular beam theory. J Sound Vib, 74:81-87. E. Carrera, G. G. (2010). Refined Beam theories based on a unified formulation.

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structures. Composite Structures, 375-383. Erochko, J. (2020). 5.2 The Bernoulli-Euler Beam Theory. J. Erochko içinde,

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Ferreira, A. J., & Media., S. S. (2009). Bernoulli Beams. A. J. Ferreira içinde, MATLAB codes for finite element analysis : solids and structures (s. 79-89). [S.l.] : Springer, cop. 2009.

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STATIC FINITE ELEMENT ANALYSIS OF COMPOSITE WING SPAR