Login | Register

Stress analysis of straight and initially curved composite tubular beams using an analytical meshless dimensional reduction method


Stress analysis of straight and initially curved composite tubular beams using an analytical meshless dimensional reduction method

Khadem Moshir, Saeid ORCID: https://orcid.org/0000-0002-3145-7527 (2021) Stress analysis of straight and initially curved composite tubular beams using an analytical meshless dimensional reduction method. PhD thesis, Concordia University.

[thumbnail of Khadem Moshir_PhD_S2022.pdf]
Text (application/pdf)
Khadem Moshir_PhD_S2022.pdf - Accepted Version


Due to high strength to weight ratio of thermoplastic composite structures, their use attracted industrial interests, especially in the aerospace industry. The application of composite curved tubes for landing gear of helicopters instead of aluminum is the subject of interest for the aerospace industry. The focus of the present work is to study mechanical behavior of composite straight and curved composite tubes subjected to the bending loading which is similar to the loading conditions of cross tubes for landing gear of helicopters during landing. The analysis of these structures with a large number of layers may be computationally expensive. The finite element method is widely used for stress analysis of initially curved composite tubes. This method is computationally expensive for design and optimization process. The main objective of the present thesis is to develop an efficient, simple-input method for stress analysis of initially curved and straight composite tubes under four-point bending loadings. To do so, different methodologies for stress analysis of straight and initially curve tubes are introduced. The advantages and disadvantages of the methods are presented. The first part of this thesis focuses on the stress analysis of straight composite cylinders under pure bending moment. The 3D elasticity of Lekhnitskii is employed to carry out the stress analysis of an anisotropic cylinders with many layers. In the second part, an analytical meshless polynomial based method in conjunction with dimensional reduction method are presented to carry out cross-sectional analysis as well as determining strain distribution in straight and curved composite tubes under bending loading (four-point bending loading) in which the effect of shear strains is taken into consideration. In order to obtain the stiffness constants of the cross-section and strains, the powerful mathematical Variational Asymptotic Method (VAM) is employed to decompose a three-dimensional elasticity problem into a two-dimensional cross-sectional analysis and a one-dimensional analysis along the length. The VAM presents Classical and Timoshenko beam models and cross-sectional stiffness matrices. In the Classical beam model cross-sectional stiffness matrix, the effect of shear is not considered. In order to achieve a more precise beam model for the analysis of initially curved and straight tubular beams, the Timoshenko-like beam model and the effect of shear strain is taken into consideration. VABS is a commercial finite element-based software for cross-sectional analysis of composite beams having complex cross-section shape. For the case of simple cross-sections such as rectangular, circular and elliptical shape, the modeling process of VABS can be eliminated by employing meshless method. The presented polynomial meshless dimensional reduction method is employed for tubes with circular and elliptical cross-sections which is the main novelty of this work. In the present work, the utilization of the Pascal polynomials for the cross-sectional analysis of the beams takes advantage of the meshless method compared with the three-dimensional finite element method or VABS. Moreover, a one-dimensional finite element solution is provided for straight and initially curved composite tubes. The presented method for the analysis of initially curved and straight tubes is computationally more efficient and simple-input compared to the 3D finite element method. In addition, it eliminates generating two-dimensional cross-sectional mesh and dividing the cross-section into different segments for straight and curved tubes compared to Variational Asymptotic Beam Sectional Analysis (VABS) software. So, the present method can be more efficient and straightforward in terms of modeling procedure compared to VABS. The parametric study such as determining cross-sectional stiffness constants, strains and displacements for design and optimization will be more straightforward.
The experimental tests are carried out to validate the present method of solution. Test setups for four-point bending test of the straight and initially curved tubes are provided by a teamwork at Concordia Center for Composites (CONCOM). The obtained strains at different spots of the tubes as well as the transverse displacement are compared with the theoretical solution. The effect of lay-up sequence, initial curvature on the mechanical behavior of the composite tubes are studied.

Divisions:Concordia University > Gina Cody School of Engineering and Computer Science > Mechanical, Industrial and Aerospace Engineering
Item Type:Thesis (PhD)
Authors:Khadem Moshir, Saeid
Institution:Concordia University
Degree Name:Ph. D.
Program:Mechanical Engineering
Date:8 November 2021
Thesis Supervisor(s):Hoa, Suong Van and Shadmehri, Farjad
Keywords:Composite tube; Initially curved; Variational Asymptotic Method; Pascal polynomials; Meshless;
ID Code:990151
Deposited By: Saeid Khadem Moshir
Deposited On:16 Jun 2022 15:16
Last Modified:27 Oct 2022 15:09


[1] Derisi B. Development of thermoplastic composite tubes for large deformation, Ph.D thesis: Concordia University, 2008.
[2] Moshir SK, Hoa SV, Shadmehri F, Rosca D, Ahmed A. Mechanical behavior of thick composite tubes under four-point bending. Composite structures. 2020;242:112097.
[3] https://www.bellflight.com/products/bell-412.
[4] Derisi B, Hoa SV, Xu D, Hojjati M, Fews R. Mechanical Behavior of Carbon/PEKK Thermoplastic Composite Tube Under Bending Load. Journal of Thermoplastic Composite Materials. 2010;24:29-49.
[5] Lekhnitskii S, Fern P, Brandstatter JJ, Dill E. Theory of elasticity of an anisotropic elastic body. Physics Today. 1964;17:84.
[6] Davies GC, Bruce DM. A stress analysis model for composite coaxial cylinders. Journal of Materials Science. 1997;32:5425-37.
[7] Hu G, Bai J, Demianouchko E, Bompard P. Mechanical behaviour of±55° filament-wound glass-fibre/epoxy-resin tubes—III. Macromechanical model of the macroscopic behaviour of tubular structures with damage and failure envelope prediction. Composites science and technology. 1998;58:19-29.
[8] Bai J, Hu G, Bompard P. Mechanical behaviour of±55° filament-wound glass-fibre/epoxy-resin tubes: II. Micromechanical model of damage initiation and the competition between different mechanisms. Composites science and technology. 1997;57:155-64.
[9] Bouhafs M, Sereir Z, Chateauneuf A. Probabilistic analysis of the mechanical response of thick composite pipes under internal pressure. International Journal of Pressure Vessels and Piping. 2012;95:7-15.
[10] Aksoy Ş, Kurşun A, Çetin E, Haboğlu MR. Stress Analysis of Laminated Cylinders Subject to the Thermomechanical Loads. World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering. 2014;8:244-9.
[11] Yuan F. Exact solutions for laminated composite cylindrical shells in cylindrical bending. Journal of Reinforced Plastics and Composites. 1992;11:340-71.
[12] Jolicoeur C, Cardou A. Analytical solution for bending of coaxial orthotropic cylinders. Journal of Engineering Mechanics. 1994;120:2556-74.
[13] Tarn J-Q, Wang Y-M. Laminated composite tubes under extension, torsion, bending, shearing and pressuring: a state space approach. International Journal of Solids and Structures. 2001;38:9053-75.
[14] Xia M, Takayanagi H, Kemmochi K. Bending behavior of filament-wound fiber-reinforced sandwich pipes. Composite Structures. 2002;56:201-10.
[15] Zhang C, Hoa SV. A limit-based approach to the stress analysis of cylindrically orthotropic composite cylinders (0/90) subjected to pure bending. Composite Structures. 2012;94:2610-9.
[16] Zhang C, Hoa SV, Liu P. A method to analyze the pure bending of tubes of cylindrically anisotropic layers with arbitrary angles including 0 or 90. Composite structures. 2014;109:57-67.
[17] Ahmad MG, Hoa S. Flexural stiffness of thick walled composite tubes. Composite Structures. 2016;149:125-33.
[18] Sun X, Tan V, Chen Y, Tan L, Jaiman R, Tay T. Stress analysis of multi-layered hollow anisotropic composite cylindrical structures using the homogenization method. Acta Mechanica. 2014;225:1649-72.
[19] Menshykova M, Guz IA. Stress analysis of layered thick-walled composite pipes subjected to bending loading. International Journal of Mechanical Sciences. 2014;88:289-99.
[20] Blom A, Stickler P, Rassaian M, Gürdal Z. Bending Test of a Variable-Stiffness Fiber Reinforced Composite Cylinder. 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 18th AIAA/ASME/AHS Adaptive Structures Conference 12th2010. p. 2600.
[21] Sarvestani HY, Hoa SV, Hojjati M. Three-dimensional stress analysis of orthotropic curved tubes-part 1: single-layer solution. European Journal of Mechanics-A/Solids. 2016;60:327-38.
[22] Yazdani Sarvestani H. High-order Simple-input Methods for Thick Laminated Composite Straight and Curved Tubes, Ph.D thesis: Concordia University, 2016.
[23] Yazdani Sarvestani H, Hojjati M. Effects of lay-up sequence in thick composite tubes for helicopter landing gears. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. 2017;231:2098-110.
[24] Fuchs HP, Hyer MW. Bending response of thin-walled laminated composite cylinders. Composite Structures. 1992;22:87-107.
[25] Wang C, Reddy JN, Lee K. Shear deformable beams and plates: Relationships with classical solutions: Elsevier, 2000.
[26] Tatting BF, Gürdal Z, Vasiliev VV. The brazier effect for finite length composite cylinders under bending. International Journal of Solids and Structures. 1997;34:1419-40.
[27] Sanders Jr JL. Nonlinear theories for thin shells. Quarterly of Applied Mathematics. 1963;21:21-36.
[28] Di S, Rothert H. A solution of laminated cylindrical shells using an unconstrained third-order theory. Composite structures. 1995;32:667-80.
[29] Kollár L, Springer GS. Stress analysis of anisotropic laminated cylinders and cylindrical segments. International Journal of Solids and Structures. 1992;29:1499-517.
[30] Love AEH. A treatise on the mathematical theory of elasticity: Cambridge university press, 2013.
[31] Chan WS, Demirhan KC. A Simple Closed-Form Solution of Bending Stiffness for Laminated Composite Tubes. Journal of Reinforced Plastics and Composites. 2000;19:278-91.
[32] Saggar P. Experimental study of laminated composite tubes under bending. 2007.
[33] Kim SJ, Shin JW, Kim H-G, Kim T-U, Kim S. The modified Brazier approach to predict the collapse load of a stiffened circular composite spar under bending load. Aerospace Science and Technology. 2016;55:474-81.
[34] Nayfeh AH, Pai PF. Linear and nonlinear structural mechanics: John Wiley & Sons, 2008.
[35] Librescu L, Song O. Behavior of thin-walled beams made of advanced composite materials and incorporating non-classical effects. Applied Mechanics Reviews. 1991;44:S174-S80.
[36] Rehfield LW, Atilgan AR, Hodges DH. Nonclassical Behavior of Thin‐Walled Composite Beams with Closed Cross Sections. Journal of the American Helicopter Society. 1990;35:42-50.
[37] Kim C, White SR. Thick-walled composite beam theory including 3-D elastic effects and torsional warping. International Journal of Solids and Structures. 1997;34:4237-59.
[38] Shadmehri F, Derisi B, Hoa S. On bending stiffness of composite tubes. Composite structures. 2011;93:2173-9.
[39] Silvestre N. Non-classical effects in FRP composite tubes. Composites Part B: Engineering. 2009;40:681-97.
[40] Hodges DH. Nonlinear composite beam theory: American Institute of Aeronautics and Astronautics, 2006.
[41] Yu W, Volovoi VV, Hodges DH, Hong X. Validation of the Variational Asymptotic Beam Sectional Analysis. AIAA journal. 2002;40:2105-12.
[42] Danielson DA, Hodges DH. Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor. Journal of Applied Mechanics. 1987;54:258-62.
[43] Berdichevskii V. Variational-asymptotic method of constructing a theory of shells: PMM vol. 43, no. 4, 1979, pp. 664–687. Journal of Applied Mathematics and Mechanics. 1979;43:711-36.
[44] Ho J, Yu W, Hodges D. Energy Transformation to Generalized Timoshenko Form by the Variational Asymptotic Beam Sectional Analysis. 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference.
[45] Hodges DH, Atilgan AR, Cesnik CE, Fulton MV. On a simplified strain energy function for geometrically nonlinear behaviour of anisotropic beams. Composites Engineering. 1992;2:513-26.
[46] Cesnik CE, Hodges DH, Sutyrin VG. Cross-sectional analysis of composite beams including large initial twist and curvature effects. AIAA journal. 1996;34:1913-20.
[47] Cesnik CE, Hodges DH. VABS: a new concept for composite rotor blade cross‐sectional modeling. Journal of the American Helicopter Society. 1997;42:27-38.
[48] Yu W, Hodges DH, Volovoi V, Cesnik CE. On Timoshenko-like modeling of initially curved and twisted composite beams. International Journal of Solids and Structures. 2002;39:5101-21.
[49] Yu W, Hodges DH, Ho JC. Variational asymptotic beam sectional analysis – An updated version. International Journal of Engineering Science. 2012;59:40-64.
[50] Ghafari E, Rezaeepazhand J. Isogeometric analysis of composite beams with arbitrary cross-section using dimensional reduction method. Computer Methods in Applied Mechanics and Engineering. 2017;318:594-618.
[51] Rajagopal A. Variational Asymptotic Based Shear Correction Factor for Isotropic Circular Tubes. AIAA journal. 2018:1-7.
[52] Harursampath D, Hodges DH. Asymptotic analysis of the non-linear behavior of long anisotropic tubes. International journal of non-linear mechanics. 1999;34:1003-18.
[53] Jiang F, Yu W. Nonlinear Variational Asymptotic Sectional Analysis of Hyperelastic Beams. AIAA journal. 2016;54:679-90.
[54] Jiang F, Yu W. Damage analysis by physically nonlinear composite beam theory. Composite structures. 2017;182:652-65.
[55] Popescu B, Hodges DH. On asymptotically correct Timoshenko-like anisotropic beam theory. International Journal of Solids and Structures. 2000;37:535-58.
[56] Zupan D, Saje M. The linearized three-dimensional beam theory of naturally curved and twisted beams: The strain vectors formulation. Computer Methods in Applied Mechanics and Engineering. 2006;195:4557-78.
[57] Tabarrok B, Farshad M, Yi H. Finite element formulation of spatially curved and twisted rods. Computer Methods in Applied Mechanics and Engineering. 1988;70:275-99.
[58] Ecsedi I, Dluhi K. A linear model for the static and dynamic analysis of non-homogeneous curved beams. Applied Mathematical Modelling. 2005;29:1211-31.
[59] Hajianmaleki M, Qatu MS. Static and vibration analyses of thick, generally laminated deep curved beams with different boundary conditions. Composites Part B: Engineering. 2012;43:1767-75.
[60] Yu AM, Nie GH. Explicit solutions for shearing and radial stresses in curved beams. Mechanics Research Communications. 2005;32:323-31.
[61] Sheikh AH. New Concept to Include Shear Deformation in a Curved Beam Element. Journal of Structural Engineering. 2002;128:406-10.
[62] Dasgupta S, Sengupta D. Horizontally curved isoparametric beam element with or without elastic foundation including effect of shear deformation. Computers & Structures. 1988;29:967-73.
[63] Yazdani Sarvestani H, Hojjati M. Three-dimensional stress analysis of orthotropic curved tubes-part 2: Laminate solution. European Journal of Mechanics - A/Solids. 2016;60:339-58.
[64] Berdichevsky V, Armanios E, Badir A. Theory of anisotropic thin-walled closed-cross-section beams. Composites Engineering. 1992;2:411-32.
[65] Borri M, Merlini T. A large displacement formulation for anisotropic beam analysis. Meccanica. 1986;21:30-7.
[66] Giavotto V, Borri M, Mantegazza P, Ghiringhelli G, Carmaschi V, Maffioli GC, et al. Anisotropic beam theory and applications. Computers & Structures. 1983;16:403-13.
[67] Borri M, Ghiringhelli GL, Merlini T. Linear analysis of naturally curved and twisted anisotropic beams. Composites Engineering. 1992;2:433-56.
[68] Stemple AD, Lee SW. Finite-element model for composite beams with arbitrary cross-sectional warping. AIAA journal. 1988;26:1512-20.
[69] Meyers CA, Hyer MW. RESPONSE OF ELLIPTICAL COMPOSITE CYLINDERS TO INTERNAL PRESSURE LOADING. Mechanics of Composite Materials and Structures. 1997;4:317-43.
[70] Meyers CA. Response of Elliptical Composite Cylinders to Axial Compression Loading. Mechanics of Composite Materials and Structures. 1999;6:169-94.
[71] Lin C, Chan W. Stiffness evaluation of elliptical laminated composite tube under bending. 19th AIAA Applied Aerodynamics Conference2001. p. 1336.
[72] Akgun G, Algul I, Kurtaran H. Nonlinear Static Analysis of Laminated Composite Hollow Beams with Super-Elliptic Cross-Sections. International Journal of Materials and Metallurgical Engineering. 2017;11:1467-72.
[73] Chen D, Sun G, Jin X, Li Q. Quasi-static bending and transverse crushing behaviors for hat-shaped composite tubes made of CFRP, GFRP and their hybrid structures. Composite structures. 2020;239:111842.
[74] Szabó G, Váradi K, Felhős D. Bending analysis of a filament-wound composite tube. Modern Mechanical Engineering. 2018;8:66.
[75] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis: CRC press, 2004.
[76] Hyer MW. Stress analysis of fiber-reinforced composite materials: DEStech Publications, Inc, 2009.
[77] Na S. Control of dynamic response of thin-walled composite beams using structural tailoring and piezoelectric actuation: Virginia Tech, 1997.
[78] Gay D, Hoa SV. Composite Materials: Design and Applications, Second Edition: CRC Press, 2007.
[79] Moshir SK, Hoa SV, Shadmehri F. Structural analysis of composite tubes using a meshless analytical dimensional reduction method. International Journal for Numerical Methods in Engineering.n/a.
[80] Zhao J-M, Song X-X, Liu B. Standardized Compliance Matrices for General Anisotropic Materials and a Simple Measure of Anisotropy Degree Based on Shear-Extension Coupling Coefficient. International Journal of Applied Mechanics. 2016;8:1650076.
[81] Stoer J, Bulirsch R. Introduction to numerical analysis: Springer Science & Business Media, 2013.
[82] Wang HW, Zhou HW, Gui LL, Ji HW, Zhang XC. Analysis of effect of fiber orientation on Young’s modulus for unidirectional fiber reinforced composites. Composites Part B: Engineering. 2014;56:733-9.
[83] Sheikh AH, Thomsen OT. An efficient beam element for the analysis of laminated composite beams of thin-walled open and closed cross sections. Composites science and technology. 2008;68:2273-81.
[84] Sheikh AH, Thomsen OT. An efficient beam element for the analysis of laminated composite beams of thin-walled open and closed cross sections. Composites science and technology. 2008;68:2273-81.
[85] Seli H, Awang M, Ismail AIM, Rachman E, Ahmad ZA. Evaluation of properties and FEM model of the friction welded mild steel-Al6061-alumina. Materials Research. 2013;16:453-67.
[86] Hartmann F, Jantzen R. Apollonius’ ellipse and evolute revisited. Manuscript http://www42 homepage villanova edu/frederick hartmann/resume html. 2003.
[87] Irgens F. Continuum mechanics: Springer Science & Business Media, 2008.
[88] ANSYS. Element Reference www.ANSYS.stuba.sk.
[89] Kovvali RK, Hodges DH. Verification of the variational-asymptotic sectional analysis for initially curved and twisted beams. Journal of Aircraft. 2012;49:861-9.
[90] Timoshenko SP, Goodier JN. Theory of elasticity. 1951.
[91] Borri M, Ghiringhelli GL, Merlini T. Linear analysis of naturally curved and twisted anisotropic beams. Composites Engineering. 1992;2:433-56.
[92] TANG YQ, ZHOU ZH, CHAN SL. AN ACCURATE CURVED BEAM ELEMENT BASED ON TRIGONOMETRICAL MIXED POLYNOMIAL FUNCTION. International Journal of Structural Stability and Dynamics. 2013;13:1250084.
[93] DeWolf JT, Mazurek D, Beer FP, Johnston ER. Mechanics of Materials: McGraw-Hill Education, 2014.
[94] Yu W. VABS Manual for Users. 2013.
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Downloads per month over past year

Research related to the current document (at the CORE website)
- Research related to the current document (at the CORE website)
Back to top Back to top