STRESS ANALYSIS OF THIN-WALLED LAMINATED COMPOSITE BEAMS

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STRESS ANALYSIS OF THIN-WALLED LAMINATED COMPOSITE BEAMS

BY

MASTURAH MOHAMAD

A dissertation submitted in fulfilment of the requirement for the degree of Master of Science (Mechanical Engineering)

Kulliyyah of Engineering

International Islamic University Malaysia

JUNE 2017

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ABSTRACT

An educational software for stress analysis of thin-walled I, T and C-sections has been developed which can aid students, lecturers and researchers in stress analysis of thin- walled open sections made of composite material. The software is based on the beam theory developed by Kollar and Springer which is limited for linear static analysis with linear stress-strain relationship. The software enables users to select type of load viz.: axial load, bending moment, torsional load and shear force. The ply stress and average stress of thin-walled open section will be instantly calculated by the software. The software also is able to calculate centroid location, shear center, equivalent axial stiffness, equivalent bending stiffness, rate of twist and torsional stiffness of the beam. Results obtained from the present software were validated against the FEA results using ANSYS and they were found to be in good agreement.

Results are presented in tabular and graphical form for various different symmetrical and unsymmetrical thin walled sections with different layup for web and flanges.

Parametric study can also be carried out with the aid of the software which can be useful in the design of thin-walled section made of composite material. The software is intended to be used as a resourceful tool for effective teaching and learning process on thin-walled structures related courses.

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ثحبلا ةصلاخ

ةساردلا هذه في تم

عضو يميلعت جمانرب لم

ينثحابلاو نيرضالمحاو بلاطلا ةدعاس في

ليلتح ةقيقر لكايلها تاداهجا

لكش يلع ةضرعتسم عطاقم اله تيلا نرادلجا

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و

)

ةبكرم داوم نم ةعونصلما .

جمانبرلا اذه دنتسي

ةيرظن ىلع

لا ضراع تا تيلا نالماعلا ةطساوب اهعضو تم رلاوك

و رغنيبرس لماو روصق ة ليلحتلا يلع طقف يطلخا

لا نيوكس تاقلاعل

.لاعفنلااو داهجلاا مدختسملل حمسي جمانبرلا

ب رايتخا لملحا عون : ةيتلاا لاحملاا نم

و يرولمحا لملحا مزع

ءاننحلاا

مزعو صقلا لاحملأاو ءاوتلا ةي

. جمانبرلا موقي كلذ دعب ةرشابم

ب باستحا في تاداهجلاا

لا قبط تا داهجلااو طسوتلما

ةضرعتسلما عطاقملل لما

حوتف لكايهلل ة .ناردلجا ةقيقر

جمانبرلا اق رد باسح ىلع عقوم

لما زكر لما طسوت و زكرم صقلا

ولا ءاننحلاا ةبلاصو ةيرولمحا ةبلاص ليا ةفاضلااب

ةبلاص مزع لاا تاو ء.

ةنراقم تتم لا

عم جمانبرلا نم اهيلع لصحتلما جئاتن

جئاتن ةدودلمحا رصانعلا ليلتح جمانرب مادختساب

ANSYS

دقو لما ترهظا ةنراق

نأ جئاتن جمانبرلا اماتم قفتت جئاتن عم

جمانرب

ANSYS

جئاتنلا ضرع تم

.

لكايله ناردلجا ةقيقر ةرظانتم يرغو ةرظانتم

و فلتمخ ة لاكشلاا و داعبلاا في

لكش

رو لوادج موس

.ةينايب ذه ةطساوب نكيم ت نا جمانبرلا ا

مت سارد ةيليصفت تا لل

ةرثؤلما لماوع و

في ةديفم نوكت نأ نكيم تيلا

ميمصت لأا نم ةعونصلما ناردلجا ةقيقر ءازج لما

ةبكرم داو . جمانبرلا اذه نكيم

ةادأك همادختسا ةيكذ سيردت

و لاعف في ة

ةيلمع

سيردت

ليلتح

.ناردلجا ةقيقر لكايلها

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APPROVAL PAGE

I certify that I have supervised and read this study and that in my opinion, it conforms to acceptable standard of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Master of Science (Mechanical

Engineering).

..………

Jaffar Syed Mohamed.Ali Supervisor

………

Meftah Hrairi Co-Supervisor

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Master of Science (Mechanical Engineering).

………..

Name:

Examiner

………..

Name:

Examiner

………..

Name:

Examiner

This dissertation was submitted to the Department of Mechanical Engineering and is acceptable as a fulfillment of the requirement for the degree of Master of Science (Mechanical Engineering).

……….…….

Waqar Asrar

Head, Department of Mechanical Engineering This dissertation was submitted to the Kulliyyah of Engineering and is acceptable as a fulfillment of the requirement for the degree of Master of Science (Mechanical Engineering).

………….…….…………

Erry Yulian Triblas Adesta

Dean, Kulliyyah of Engineering

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DECLARATION

I hereby declare that this dissertation is the result of my own investigations, except where otherwise stated. I also declare that it has not been previously or concurrently submitted as a whole for any other degrees at IIUM or other institutions.

Masturah Mohamad

Signature: ……… Date: ………

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INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA DECLARATION OF COPYRIGHT AND AFFIRMATION OF

FAIR USE OF UNPUBLISHED RESEARCH

STRESS ANALYSIS OF THIN-WALLED LAMINATED COMPOSITE BEAMS

I declare that the copyright holder of this dissertation are jointly owned by the student and IIUM.

No part of this unpublished research may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without prior written permission of the copyright holder except as provided below

1. Any material contained in or derived from this unpublished research may be used by others in their writing with due acknowledgement.

2. IIUM or its library will have the right to make and transmit copies (print or electronic) for institutional and academic purpose.

3. The IIUM library will have the right to make, store in a retrieved system and supply copies of this unpublished research if requested by other universities and research libraries.

By signing this form, I acknowledged that I have read and understand the IIUM Intellectual Property Right and Commercialization policy.

Affirmed by Masturah Mohamad

…..……….. ……….

Signature Date

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ACKNOWLEDGEMENTS

All glory is due to Allah, the Almighty, whose Grace and Mercies have been with me throughout my whole life. He gave me patience and guided me whenever I faced difficulties in completing my master dissertation.

I am so grateful to my supervisor, Dr. J.S. Mohamed Ali, whose enduring kindness, thoroughness and friendship have facilitated the successful completion of my work. I also would like to thank my co-supervisor, Dr. Meftah Hrairi for his useful advice and co-operation contributed on my work. May their lives will always be in Allah’s and bless and care.

Not forgotten, I would like to thank my parents for their blessings throughout the journey of my life. Lastly, I would like to prolong my gratitude to my beloved husband and daughter for their support, patience, and understanding throughout the completion of my dissertation.

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LIST OF TABLES

Table No. Page No.

Table 6.1 Geometrical dimension of I-beam 41

Table 6.2 Geometrical dimension of C-beam 42

Table 6.3 Geometrical dimension of T-beam 42

Table 6.4 Tensile and Bending Stiffness 44

Table 6.5 Stresses due to axial load on the upper flange for I-beam Case 2 46 Table 6.6 Stresses due to axial load on the lower flange for I-beam Case 2 47 Table 6.7 Stresses due to axial load on the web for I-beam Case 2 48 Table 6.8 Stresses on the web due to axial load for C-beam Case 1 50 Table 6.9 Stresses on the upper flange due to axial load for C-beam Case 1 51 Table 6.10 Stresses on the lower flange due to axial load for C-beam Case 1 52 Table 6.11 Stresses due to axial load on the top flange of T-beam 53 Table 6.12 Stresses due to axial load on the web of T-beam 54 Table 6.13 Stresses on the upper flange due to bending moment my for

I-beam Case 2 56

Table 6.14 Stresses on the lower flange due to bending moment my for

I-beam Case 2 57

Table 6.15 Stresses on the web due to bending moment my for I-beam Case 2 58 Table 6.16 Stresses on the upper flange due to moment my for C-beam Case 1 59 Table 6.17 Stresses on the lower flange due to moment my for C-beam Case 1 60 Table 6.18 Stresses on the web due to moment my for C-beam Case 1 at z=0 61 Table 6.19 Stresses on the upper flange on T-beam due to moment my 63 Table 6.20 Stresses on the web on T-beam due to moment my at z=0.21 64 Table 6.21 Stresses on the upper flange due to moment my and mz for

I-beam Case2 66

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Table 6.22 Stresses on the lower flange due to moment my and mz

for I-beam Case2 67

Table 6.23 Stresses on the web due to moment my and mz for I-beam Case 2 68 Table 6.24 Stresses on the lower flange due to moment my and mz for C-beam

Case1 69

Table 6.25 Stresses on the upper flange due to moment my and mz for C-beam

Case1 70

Table 6.26 Stresses on the web due to moment my and mz for C-beam Case1 71 Table 6.27 Stresses on the web on T-beam due to moment my and mz 72 Table 6.28 Stresses on the upper flange on T-beam due to moment my and

mz 73

Table 6.29 Stress on the flanges of I-beam Case 2 under torsional load 75 Table 6.30 Stress on the web of I-beam Case 2 under torsional load 76 Table 6.31 Stress on the flanges of C-beam Case 1 under

torsional load 77

Table 6.32 Stress on the web of C-beam Case 1 under torsional load 78 Table 6.33 Stress on the upper flange and web of T-beam under

torsional load 79

Table 6.34 Average shear stress for the lower and upper flanges of I-beam

Case2 at y=0 82

Table 6.35 Average shear stress for the web of I-beam Case2 at z=0.25 83 Table 6.36 Average shear stress for the upper and lower flanges of C-beam

Case1 at y=0 83

Table 6.37 Average shear stress for the web of C-beam Case1 at z=0.5 84 Table 6.38 Average shear stress for the upper flange and web of T-beam 84 Table 6.39 T-beam with modified ply orientation on upper flange under my 85

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LIST OF FIGURES

Figure No. Page No.

Figure 3.1 Material axes 15

Figure 3.2 Top view of an orthotropic material 16

Figure 3.3 Specially orthotropic ply 16

Figure 3.4 Generally orthotropic ply 17

Figure 3.5 A layered laminate 22

Figure 4.1 I, C and T-sections 26

Figure 4.2 Radius of curvature and strain distribution under moment

about y-axis 30

Figure 4.3 zp location as per ANSYS 32

Figure 5.1 Flowchart for development of the software 36 Figure 5.2 A cantilevered beam under axial load at the centroid 38 Figure 5.3 I-beam subjected to an equivalent axial load at one end 38 Figure 5.4 Percentage error versus meshing size-NDIV 39

Figure 5.5 C-beam with meshing denser at tips 40

Figure 6.1 C-beam Case1 model 42

Figure 6.2 T-beam model 43

Figure 6.3 I-beam Case2 model 43

Figure 6.4 I-Beam Case 2 under axial load for layer 3 49 Figure 6.5 Stress SX on layer 1 of C-beam Case1 under my 62 Figure 6.6 Stress SX on layer 8 of C-beam Case1 under my 62 Figure 6.7 Shear stress SXY for T-beam on layer 2 under torsion 80 Figure 6.8 Shear stress SXY for T-beam on layer 6 under torsion 80

Figure A.1 Defining lines 95

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Figure A.2 Locating lines for T-beam 95

Figure A.3 Locating lines for C-beam 96

Figure A.4 Locating lines for I-beam 96

Figure C.1 MATLAB command window 123

Figure C.2 Typical display of the software 124

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LIST OF SYMBOLS

Qij Reduced stiffness term Sij Compliance stiffness term E1, E2 Young’s Moduli in 1,2- direction G12 Shear Modulus in 12- plane υ12 Poisson ratio in 12- plane

σx, σy Normal stresses in x, y coordinate system τxy Shear stress in x, y coordinate system

ϵx, ϵy Engineering normal strain in x, y coordinate system γxy Engineering shear strain in x, y coordinate system

u0 Displacement of reference surface in x-direction w0 Deflection of reference surface in z-direction kx, ky, kxy Curvatures of the reference surface

Nx, Ny, Nxy In-plane forces per unit length acting on a laminate

Mx, My, Mxy Bending and twist moment per unit length acting on a laminate Aij Tensile stiffness of a laminate

Bij Coupling stiffness of a laminate Dij Bending stiffness of a laminate aij, bij, dij Inverse of A, B, and D

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zc ,yc Centroid coordinates of the cross section EA Tensile stiffness of a beam

EI Bending stiffness of a beam t Thickness of a laminate N Axial force acting on a beam my,mz Bending moments acting on a beam

[Rij] Compliance matrix under plain-strain condition in the x, y coordinate system

T Torque acting on a beam [W] Compliance matrix on a beam GIt Torsional stiffness of a beam dυ/dL Rate of twist

q Shear flow

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LIST OF ABBREVIATION

et al. (et alia): and others

eq./eqn. equation

%Diff percentage difference FEA finite element analysis

lb pound (unit)

lb.in pound inch (unit)

NDIV number of division

t thickness

vol. volume

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CHAPTER ONE INTRODUCTION

1.1 BACKGROUND OF THE STUDY

History had recorded that composites were used since ancient time by the Mesopotamians around 3400 B.C. Composites underwent development from natural sources like bamboo, wood, and natural pine resin to plastics made of synthetic components which has been used for electrical components, radio and telephone casings. In 1930’s, first fiber reinforced polymer was introduced which is glass fiber and it became the leading choice for many manufacturing industries. In 1960’s, carbon fiber was introduced which possessing better mechanical properties than glass fiber which led to more applications in automotive, aerospace, and sports. Nowadays, fiber reinforced polymer can also be made of aramid or Kevlar, boron and silicon carbide (Daniel, 2006).

Composite materials are well known to have excellent fatigue resistance, high specific strength and stiffness, good corrosion resistance, excellent fire resistance and lower thermal expansion and many more. Composite materials attract researchers to study their behavior due to their complexity and up until now, complete composite behavior is still unknown. Research and development in finding new fibers and resins also will create more applications for composites.

In aerospace industry, most aircraft structures are thin-walled in nature and open section beams are used in aircrafts to stiffen the thin skins of the cellular

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components and provide support for internal loads from floors, engine mountings, etc.

Moreover the focus of this study is on the composite thin-walled open sections since more aircraft structures today are being fabricated from composite materials as they have the advantage of higher strength and stiffness but lighter weight (Megson, 2007).

In early years of development of composite materials in aerospace application, they are only used for secondary and tertiary structure of the aircraft. However, recently for some small and big aircraft like Boeing, the airframe and primary structures are made entirely of composite. Other than possessing good mechanical properties, composite materials require less scheduled maintenance than the non- composite materials. This will reduce the maintenance cost and indirectly increase the profit of the airline operators. However, there are some minor aircraft accidents caused by the less scheduled maintenance procedures in aircraft operating company and therefore it should be revised in order to avoid any aircraft accidents in the future.

Due to complex material properties of composite materials, they give headache not only to people in academic sector, but also to industrial people like engineers and non-destructive (NDT) specialists. This is because it is not only difficult in predicting the structural behavior of composite materials but it is also difficult to detect early crack propagation in the composite laminates. For metals, there are many non- destructive techniques to detect the unseen defects in metals like eddy current, magnetic particles, radiography and liquid penetration techniques. However, for composite materials, there are two inexpensive techniques utilized widely in aerospace industries which are visual inspection and tapping coin inspection. The two techniques may be inaccurate due to man’s fault. Radiography technique is rarely used because it is more expensive and only can be done by skilled NDT specialists.

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3 1.2 PROBLEM STATEMENT

Thin-walled composite beam theory that are taught in university level involves calculations that are lengthy, tedious and hence time consuming. An available software such as MDSolids has its limitation because it is deals only with basic Mechanics of Materials, and the students will not be able to learn advanced topic on structures using this software. An extensive literature shows that a software named TWProfile is the only commercial software available in the market that can perform analysis of thin-walled sections made of composite materials. The software determines structural properties and stresses as defined by Vlasov theory (Ali, J.S.M.

et al, 2015).

Advance finite element software like ANSYS and ABAQUS are very helpful to understand the behavior of either isotropic or composite beams under various loading. However, the original software are very expensive and the students should be able to do processes like modelling, meshing and applying loads correctly in order to get good results and understand the results thoroughly. The process to familiarize themselves with the finite element software itself is very challenging and stressful. It requires many trial and error, especially for beginner level. Thus, there is a need of an educational software which is user friendly in thin-walled composite stress analysis as an alternative tools for education.

In order to develop a good software, other scholars’ work can be used as references to ensure that method used in this work is correct. However a thorough exhaustive literature survey shows that there are very meagre results available for ply stress analysis for thin-walled composite beams especially for torsion and shear force.

Other previous works were just focused on I-beam, laminated plate and box beam.

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Thus, validation process becomes difficult. Therefore, it is hoped that this work will be a guidance for other scholars who are interested in this field.

1.3 RESEARCH OBJECTIVES

The objectives of this research are as follows:

1. To develop a closed form analytical solution for stress analysis of open sections such as C-section, T-section and I-section composite laminate beams under different loading conditions.

2. To develop software based on the above analytical method which is useful for parametric analysis.

3. To develop a Finite Element Analysis model of the above problem to validate the calculated analytical results.

1.4 SIGNIFICANCE OF THE STUDY

In this work, a software is developed using MATLAB which is capable of evaluating ply stresses and average stress for I, C and T- section composite beams under axial, bending loads, torsional load and shear force. The software is developed so that it is user friendly and will display results instantly. Ply stresses and average stresses of thin- walled laminated beam will be instantly calculated by the software. Sadly, there is no such advance software is available in the market. Therefore, other than efficiently used as educational tool for students, lecturers and researchers, the developed software is hopefully can be a baseline for other students or researchers so that more advance software for thin-walled composite analysis can be developed in the future.

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5 1.5 LIMITATION OF THE SOFTWARE Limitation of the software includes:

i) The software is limited for linear static analysis with linear stress-strain relationship for I, C and T- section beams only which means this analysis obeys Hooke’s Law. Any form of non-linearity is not considered in this study.

ii) The software utilizes a beam theory developed by Kollar, L.P. and Pluszik, A.(2002) in which this theory neglects the effect of shear deformation and restrained warping.

iii) This software is not valid for closed section beams. It also may not be used for L and Z section beams due to variation in the calculation of centroid location.

iv) The analysis for this software is valid only for long, un-tapered, and slender beam with ratio of the cross sectional dimension to the length should be at least 1/10. The ratio of the laminate thickness to the cross sectional dimension should be 1/10.

1.6 GENERAL OUTLINES

Chapter Two describes the literature surveys from various scholars and critical reviews from two previous works related to this research. Chapter Three describes definition of composite laminates and general equations for laminated beam. Chapter Four describes steps for stress analysis of thin-walled composite beams under four loading conditions.

Chapter Five describes the methodology in the development of present software.

Chapter Six summarizes the results for selected cases under four loading conditions.

Chapter Seven concludes the overall findings of this research.

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CHAPTER TWO LITERATURE REVIEW

2.1 LITERATURE REVIEW

Research done by many scholars since year 1985 until recently in various field of thin- walled composite beams are summarized in this chapter. Bauchau (1985) in his work titled ‘A Beam Theory for Anisotropic Materials’ developed a beam theory by improving the Bernoulli and Saint-Venant approximations, consisting of a series of expansion of the axial displacements. The theory developed was applicable for thin- walled closed and open cross section beams. Hjelmstad (1987) in his work, analyzed thin-walled short beams for elastic and elastoplastic warping due to transverse shear by reproducing the Saint Venant’s theory.

Rakesh and Stefano (1989) studied the shear effect, buckling and post- buckling behavior of geometrically no defect and defective laminated plates using analytical, numerical and experimental techniques. They also discussed about delamination and its growth due to buckling in their paper. Ramesh et al. (1990) presented an experimental analysis of symmetric and un-symmetric laminate of rectangular cross section composite beams subjected to bending moment, torsion and extensional loads. The angle of twist results were compared to the results obtained from a simple beam analysis and FEA.

Ramesh and Inderjit (1991) studied the static structural response of composite I-beam with elastic couplings subjected to bending and torsional load by neglecting the shear deformation and an analytical solution developed on the basis of Vlasov’s

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theory. Frassine (1992) presented experimental results of intra-laminar fracture behavior of unidirectional carbon/epoxy composite laminates by applying double torsion tests on the specimens.

Wu and Sun (1992) simplified the Vlasov’s theory of seven differential equations to four coupled ordinary differential equations. For verification, they analyzed the deformation of channel beams under axial load, shear force and torsion and compared to FEA and other existing theories.

Subrahmanyam (1993) carried out an experimental and theoretical analysis of symmetrical and unsymmetrical thin-walled closed beams under tip bending load and tip torque using potential energy method. Barbero et al (1993) carried out a research for open and closed sections of thin-walled laminated beams under axial load and bending moment. They also derived the expression for shear correction factor from energy equivalence method.

Chattopadhyay et al. (1993) presented an analysis of blade-stiffened composite plate under transverse loads using FEA method. They utilized an eight-node iso- parametric quadratic elements in the analysis. A detailed parametric study carried out, which is very important in designing stiffened plates made of thin-walled composite materials.

Salim, Davalos et al (1996) analyzed the pultruded fiber-reinforced plastic beams under bending using mechanics of thin-walled laminated beam (MLB) approach. Based on experimental results, they concluded that the MLB approach give an accurate prediction to the ply stiffness of various type of composite laminate such

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as fiber-bundles or roving, continuous strand mats, cross and angle ply fabric laminates.

Massa and Barbero (1998) in their work developed a simple approach to analyze thin-walled open and closed composite beams under application of bending, torsion, shear and axial forces. They introduced a new concept to derive the equivalent geometrical properties which can be used in the matrix structural analysis.

Chan and Demirhan (2000) developed two new approaches to determine bending stiffness of circular tube of fiber reinforced composites. These approaches were based on laminated plate and shell theories. For verification purpose, the results using new approaches were compared to FEA analysis and the conventional method using the laminate smear property. Jaehong (2001) also had done his work to determine the centroid location and shear center but for thin-walled open section composite beams.

Ferrero et al. (2001) claimed that the stress field from a twisting moment is not accurately modeled by the classical theories. Thus, they presented an alternative way to determine stress and stiffness of composite laminate beam which is more accurate than the classical theory.

In research done by Camomile and Swanson (2002), they studied stiffness and shear strain of thin rectangular rods made of carbon/epoxy laminate under torsional load. To determine torsion stiffness, the specimens were tested at lower level of angle of rotation.

Yang et al. (2002) studied stresses due to bending moment within the thin- walled composite pipe joints consist of three components which are the pipe, coupling

STRESS ANALYSIS OF THIN-WALLED LAMINATED COMPOSITE BEAMS