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THESIS INTRODUCTION

1              Introduction

 1.1       Introduction to Composites

The construction of composite materials has had a long and extensive history dating back thousands of years. By intuitively combining different materials a new, enhanced material that exhibits characteristics of the mixed constituents can be produced. In general, ‘composites’ refers to the combination of two or more materials; but, more specifically, the word ‘composites’ refers to (or is often associated with) a combination of highly engineered resins and reinforcing materials.

More than most disciplines of engineering, aerospace designs are rigorously optimized for weight. With this in mind, the industry has been drawn towards the attractive characteristics of composite materials. Composites provide many advantages over homogenous materials, some of which are found in their mechanical properties, such as weight, strength, stiffness, corrosion resistance and fatigue life.

Composites are widely used in control surfaces such as ailerons, flaps, stabilizers, rudders, as well as rotary props and fixed wings. These fibrous composites can provide a high strength for control surfaces at a fraction of the weight of homogeneous metallics. The versatility in the design of fibrous composites is especially important. The stiffness of a particular composite beam, for example, can be altered in design by simply adjusting the fiber orientation and stacking sequence.

1.2       Flutter and divergence instabilities of composites

Aeroelastic flutter is one of the most critical safety concerns in aerospace designs. The design and optimization of wing structures requires a strong emphasis on aeroelastic tailoring, so the predicted instabilities can be avoided. Aeroelasticity, the study of the mutual interaction between fluid flow and structure, embraces various disciplines, including aerodynamics, solid mechanics and vibrations. 

Many different types of flutter exist: namely bending torsion flutter, binary flutter, stall flutter, panel flutter and single degree of freedom flutter. The current research is focused on bending-torsion flutter, other types of flutter may be considered for future endeavours, however, they will not be this studied in this paper.

Coalescence-type flutter is another type of flutter that occurs when the bending and twisting frequencies merge at a specific dynamic pressure. This is generally observed to occur when aerodynamic damping is omitted. There is a distinct difference in the coalescence flutter between homogeneous wings and composite wings. Coalescence flutter of homogeneous wings composed normally of metal, usually occurs with the merging of the first two fundamental frequencies (Hashemi and Alighanbari, 2002). By contrast, in composite wings this merging of frequencies between bending and twisting modes generally occurs at higher frequencies.

Generally a wing can undergo bending, torsion and coupled bending-torsion displacements if there is an offset of the mass axis (CG) from the elastic axis (EA). For composite wings, a material coupling exists due to the anisotropy of composite material. A bending-torsion material coupling is created from symmetric laminates or, in the case of thin-walled box-wing structures, Circumferentially Asymmetric Stiffness (CAS) configurations (Armanios et al, 1995). This coupling between bending and torsion, whether it derives from the physical geometry of the wing or as a result of material coupling, can lead to the flutter phenomenon. Bending-torsion flutter is a pitch-plunge instability generated at a specific applied dynamic pressure.

1.3       Free Vibration of Laminated Fibrous Composites

The accuracy of an aeroelastic flutter or divergence analysis depends greatly on the reliability of the free vibration analysis used to describe the natural modes of the system. There are various well-established methods to formulate and extract these natural frequencies, namely, Finite Elements (FE), Dynamic Stiffness Matrix (DSM) and Dynamic Finite Elements (DFE).

Finite Elements has a long, well established history and is the most commonly used numerical method for analyzing structures. Finite Elements is the most general and systematic approach to formulate the element, mass and stiffness matrices for a given system and it is easily adaptable to complex systems involving variations in geometry or loading. The use of fixed polynomial shape (interpolation) functions leads to constant mass and stiffness matrices, and the natural frequencies can be readily found by solving the resulting linear eigenvalue problem. The Finite Element method has been widely used by Teh and Huang, 1979; Chandra et al, 1990; Wu and Sun, 1991; Jaehong and Kim, 2002; for the analysis of beam-type structures.

Finite elements are also commonly used in commercial software packages such as ANSYS® and ABAQUS®. In the last decade, composite elements have been available in these software packages and used by a number of authors to analyze the vibration of composite wings.

Alternatively to Finite elements, the Dynamic Stiffness Matrix (DSM) method can be employed to evaluate the natural frequencies and modes of a structure. The DSM was first developed by Kolousek in 1940 for homogeneous Euler-Bernoulli beams and later extended to a number of structural elements by other researchers. In fact, the DSM has since been refined continuously. In the last decade, Banerjee and his colleagues have published a large number of papers pertaining to the vibration analysis of many different homogeneous and composite beam models. Banerjee and Williams (1995) developed the DSM for a uniform Euler-Bernoulli beam, Banerjee and Williams (1996) extended that to the Timoshenko beam theory and then later this model was further extended to include an axial force (Banerjee, 1998). Based on the exact member theory, the DSM produces exact natural frequencies for a uniform beam element using generally only one element.

Other analytical, semi-analytical, numerical and hybrid methods have also been researched. Based in part, on the DSM, the Dynamic Finite Element (DFE) method is a hybrid ‘dynamic’ numerical method. Dynamic refers to the frequency dependency of the trigonometric shape functions used to approximate the stiffness matrix of a beam-type system. The other building block of this hybrid method is based on Finite Elements, by providing a general systematic procedure (i.e., Integral Formulation based on the Weighted Residual Method (WRM)). The DFE technique follows the same typical procedure as the Finite elements by formulating the element equations discretized to a local (element) domain, where element stiffness matrices are constructed and assembled into a single global matrix. As a result, the DFE can be easily extended to elements with a higher degree of complexity.

The Dynamic Finite Element (DFE) formulation was first proposed by Hashemi (1996) and has since been well established for the free vibration analysis of homogeneous beams, blades and beam-like structures (Hashemi et al,1997; Hashemi, 1998; Hashemi and Richard, 1999; Hashemi, 2002). The DFE has been shown to converge faster to the exact natural frequencies than both the FE and DSM when such higher complexities as tapered geometries are involved (Borneman, 2004; Hashemi and Borneman, 2005). The DFE has the advantage of incorporating added terms known as ‘refining terms’ or more commonly called ‘deviators terms’ to enhance or refine the stiffness matrix. This consequently results in much more accurate natural frequencies of the analyzed structure. This enhanced stiffness matrix is sometimes referred to as Refined Dynamic Finite Elements (RDFE).  The DFE is validated by its faster convergence to the natural frequencies than other existing methods (Borneman and Hashemi, 2003; Hashemi and Borneman, 2003; Hashemi and Borneman, 2004).

1.4       Damaged Composites

The free vibration analysis of composite beam-type structures is well established using various techniques. There are also a number of publications that extend this study to incorporate damages such as delamination, fiber matrix cracking, de-bonding between matrix and fibers or fatigue damage into a free vibration analysis.

The analysis of cracked beams has been conducted by a number of authors, including Mujumdar and Suryanarayan (1988), where an analytical model for the free vibration analysis is developed for beams with delamination. Later Shen and Pierre (1994) conducted the free vibration analysis of a homogenous cracked Euler-Bernoulli beam using Galerkin and Rayleigh-Ritz methods. The illustrated mode plots show distinct slope changes at the crack location. Cracked fibrous composites have been studied by numerous authors in the past decade.

Boa (1992) developed a function to correct the stress intensity factors which describe the energy release rate of a cracked composite beam initially developed by Tada et al (2000), to take into account the anisotropy of composite material. He found that this function produced consistent results for 4 different cracked-typed specimens. Ghoneam (1995) studied the vibration characteristics of an open cracked composite beam with various end boundaries, (C-C, C-F, C-S, S-S) but didn’t account for material anisotropy in the stress intensity factors. Krawczuk published a number of papers involving the vibration characteristics of damaged beams. Krawczuk (1994) observed that the natural frequencies are highest when the composite fibers are perpendicular to the crack. Krawczuk et al (1995) studied the modes of damaged beams containing different crack depths and positions as well as the influence of fiber volume fraction. It is observed that for low fiber angles the influence of fiber volume fraction is of great significance. Krawczuk et al (1997) extended this vibration analysis to include damages such as delaminations. Kisa et al (1998) developed a free vibration model for damaged composite beams using finite elements and extended it to include Timoshenko beam assumptions.

More recently a paper was published by Zheng and Kessissoglou (2003) who proposed a new approach to formulating a cracked beam by using a ‘total flexibility’ approach instead of the local flexibility used previously. The development of new interpolation functions were designed specifically for this cracked member. By using this total flexibility technique the extracted frequencies tend to be more accurate.

Also Nayfeh and Abdelrahman (2003) developed a new micromechanical model to study a damaged composite beam that can be adapted for beams with a single fiber crack, single matrix crack, or multiple damage scenarios.

Wang et al (2005) used the Dynamic Stiffness Matrix (DSM) method to model a through thickness cracked beam. The DSM provides accurate modes for a bending-torsion coupled uniform composite beam using a relatively coarse mesh, compared to the other methods. Wang (2004) had also studied how a through thickness crack will change the free vibration modes and aeroelastic flutter and divergence of a composite wing. A method was devised to detect a crack in a wing using Cawley-Adams Criterion (CAC) by a different ordering of modes.

1.5       Structural Health Monitoring and Damage Diagnosis

A method to detect small cracks in a metal or homogeneous material in a non-destructive way by means of natural frequencies was first proposed by Adams et al (1978). He proposed that the crack size and location on a bar specimen could be found using natural frequencies of free vibration ignoring the natural modes all together.

Non-Destructive Testing (NDT) is normally used for the detection of defects in structures, for example an aircraft wing. Liquid penetrate, magnetic particles, radiography and eddy current techniques usually require long periods of time to localize and access the damage of the structures. Alternatively, a technique exists, often referred to as Structural Health Monitoring (SHM), which has been growing in attention in recent years. As being a subsidiary of NDT, SHM provides the same non-destructive testing but without the offline inspections normally required by NDT. With a number of sensors imbedded into a structure such as a wing, the wing can be monitored continuously for defects without offline inspection.

This type of detection is advantageous for cost saving, but also for early detection of possible critical defects. For example, a wing with a crack in flight could be detected immediately using SHM, whereas, the crack could go unnoticed for a period of time before being detected using classical NDT techniques.

1.6       Key Objectives

The focus of this research is geared towards the vibrational behaviour, aeroelasticity and detection of damaged composite wings. The free vibration modes are evaluated using Dynamic Finite Elements (DFE), Finite Elements (FE) and Dynamic Stiffness Matrix (DSM). The wing damage is considered as a through thickness edge crack for all proposed formulations and numerical tests.

Wang (2004) used a Dynamic Stiffness Matrix (DSM), which is well suited for this analysis as it provides exact fundamental natural frequencies for a uniform wing. The motivation of implementing the DFE methodology in this research is that the technique has shown to generally have higher accuracy and convergence rates in the calculated natural frequencies and modes of beam structures, when compared to other existing methods and more specifically for cases with a higher degree of complexity (e.g., tapered wings). The DFE has proven to be an excellent preliminary tool in the free vibration analysis of homogeneous metallic and composite beam and blades (Hashemi, 1998; Borneman, 2004).

1.7       Contributions

The following contributions are new concepts and strategies for the analysis of cracked composite wings:

1)      The development of various formulations is accomplished for the free vibration of wings with specific geometry and loading, and,

2)       A new strategy is devised and tested for detection of single and multiple cracks with an accurate method to capture the natural frequencies of a defective wing.  

1.7.1       A New Composite Dynamic Finite Cracked Element (DFCE)

The Dynamic Finite Element (DFE) method is well established. It is also known to produce accurate free vibration frequencies and modes for various structures. The development of a new DFCE is accomplished and numerically tested for various laminated composite defective wing configurations. The motivation for constructing a DFCE is based on the enhanced convergence and accuracy of the technique. Dynamic Finite Elements (DFE) have been shown to provide excellent accuracy involving the calculations and convergence of the natural frequencies for homogeneous and laminated fibrous composite beam and wing structures. This dynamic element can be applied readily to models that consist of variations in geometry or loading, implementing refined terms to improve the convergence of the calculated frequencies. A DFCE provides the necessary precision in the natural frequencies with the intention that it will be used in a free vibratory detection methodology.

1.7.2       A new Cracked Dynamic Stiffness Timoshenko Beam Element

A Dynamic Stiffness Matrix for a cracked composite beam element using Euler-Bernoulli beam bending and St. Venant torsion is well established by Wang (2004). A natural extension to this development would be to investigate thicker beams where shear deformation must be acknowledged. The free vibration of slender beams is well established using finite elements. Shear deformation is avoided by many researchers that rely on finite elements due to the shear locking phenomenon. A Dynamic Stiffness Matrix formulated by Banerjee (1998) for composite intact Timoshenko beams does not fail from shear locking.  It provides generally the exact natural frequencies for a uniform beam. The development of a cracked beam model including shear deformation is well established for homogeneous structures. Takahashi constructed a non-uniform cracked Timoshenko beam model and extended this analysis to include an aeroelastic flutter investigation. Takahashi also applied shear correction to the appropriate stress intensity factors. Alsaid compared the natural frequencies of a uniform cracked Timoshenko beam with the same defective beam modeled with Euler-Bernoulli bending. The cracked beam modeled with Timoshenko beam theory provided a greater reduction in the frequencies than the beam approximated with the Euler-Bernoulli beam theory.

Viola (2001) devised a detection scheme using modal data for a cracked Timoshenko beam and later developed a Timoshenko Dynamic Stiffness Matrix (DSM) element for homogeneous crack beams. Krawczuk (2003) performed a free vibration analysis of defective Timoshenko beam using a spectral element. Shear correction is used to adjust the stress intensity factors associated with shear loading. Loya (2006) investigated the vibration of a cracked Timoshenko simply supported beam. Darpe (2004) investigated the free vibration of a coupled bending-torsion cracked rotor with shear deformation. Darpe also corrected the appropriate stress intensity factors taking into account shear deformation. A thick cracked beam, where shear deformation significantly alters the free vibration, is investigated. A new cracked Timoshenko composite DSM beam element is formulated and tested. 

1.7.3       Aeroelastic Analysis using a New DFCE

The aeroelastic instabilities of a cracked wing are analyzed through using the natural modes provided by a new DFCE. The natural modes calculated using a DFCE provides excellent flutter and divergence speeds. For intact tapered wings, the DFE modes are compared with modes extracted through existing methods. The DFE modes are observed to provide the best calculated aeroelastic instabilities.

1.7.4       Multi-crack detection methodology

Applying the Cawley-Adams Criterion for the single crack analysis of a defective wing has been shown by a number of researchers to be successful. Using natural frequency data to detect defects in structures is highly attractive since frequency data can be easily measured. Multiple cracks can be caused by impact damage or material defects in manufacturing. The propagation of two cracks simultaneously is impractical for cantilevered structures. The crack with the higher stress concentration would propagate before the second crack. With this in mind, a second crack near the tip of the wing where the stress is less intense may not propagate significantly;  however the detection of this crack is essential.

A modest number of research papers have been published in the area of multiple crack detection using frequency data. Patil (2005) uses experimental verification of multiple cracks in homogenous cantilevered beams using frequency measurements. The detection scheme is extended to multiple cracks by implementing a new strategy. This new strategy involves the implementation of a second indicator. The technique provides highly sensitive results and successfully detects more than one crack for location and size with uncertainty in measured frequency data.

REFERENCES

[1]    Abd El-Hamid Hamada, A. (1997). An Investigation into the Eigen-Nature of Cracked Composite Beams. Composite Structures. 38(1-4), pp. 45-55.

[2]    Armanios, E. A. and Badir, A. M. (1995). Free Vibration Analysis of Anisotropic Thin-Walled Closed-Section Beams. AIAA Journal, 33(10), pp.1905-1910.

[3]    Balakrishnan, A. V. (2001). Subsonic Flutter Suppression Using Self-Straining Actuators. Journal of the Franklin Institute. 338, pp. 149-170.

[4]    Bao, G. and Ho, S. and Suo, Z. (1992). The Role of Material Orthotropy in Fracture Specimens for Composites. International Journal of Solids and Structures. 29, pp. 1105-1116.

[5]    Berthelot, J. M.  (1999).  Composite Materials Mechanical Behaviour and Structural Analysis. Springer-Verlag New York Inc.

[6]    Bhaskar, K. and Librescu, L. (1995). A Geometrically Non-Linear Theory for Laminated Anisotropic Thin-Walled Beams. International Journal of Engineering Science. 33(9), pp. 1331-1344.

[7]    Borneman, S. and Hashemi, S. M. (2003). Free Vibration Analysis of Laminated Composite Wings: A New Dynamic Finite Element (DFE) Formulation. Proceeding International Conference for Upcoming Engineers (ICUE), Maxwell session. Toronto, May 2003.

[8]    Buchholz, F.-G., Rikards, R. and Wang, H. (1997). Computational Analysis of Interlaminar Fracture of Laminated Composites. International Journal of Fracture. 86, pp. 37-57.

[9]    Cattarius, J. (1999). Numerical Wing/Store Interaction Analysis of a Parametric F16 Wing. Ph.D Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University.

[10]  Cesnik, C. E. S., Hodges, D. H. and Patil, M. J. (1996). Aeroelastic Analysis of Composite Wings. 37^th Structures, Structural Dynamics and Materials Conference, Salt Lake City Utah, April 15-17, 1996.

[11]  Charonenphan, S., Bank, L. C. and Plesha, M. E. (2004). Progressive Tearing Failure in Pultruded Composite Material Tubes. Composite Structures. 63, pp. 45-52.

[12]  Chaudhari, T. D.  and Maiti. (1999). Modelling of Transverse Vibration of Beam of Linearly Variable Depth with Edge Crack. Engineering Fracture Mechanics. 63, pp. 425-445.

[13]  Cui, W., Wisnom, M. R. and Jones, M. (1995). New Model to Predict Static Strength of Tapered Laminates. Composites. 26, pp. 141-146.

[14]  Dancila, D.S. and Armanios, E. A. (1998). The Influence of Coupling on the Free Vibration of Anisotropic Thin-Walled Closed-Section Beams. International Journal of Solids and Structures. 35(23), pp. 3105-3119.

[15]  Davidson, B. D., Kruger, R. and Konig, M. (1995). Three-Dimensional Analysis of Center-Delaminated Unidirectional and Multidirectional Single-Leg Bending Specimens. Composites Science and Technology. 54, pp. 385-394.

[16]  Dey, A., Tryon, R., Mahadevan, S., Zou, T. and Rousseau, C. (2000). Reliability Prediction of Tapered Composites Subjected to Fatigue Delamination. 8^th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability. PMC2000-107.

[17]  Dirikolu, M. H.  and Aktas, A. (2000). Analytical and Finite Element Comparisons of Stress Intensity Factors of Composite Materials. 50, pp. 99-102.

[18]  Ferrero, J. F., Barrau, J. J., Segura, J. M., Sudre, M. and Castanie, B. (2001). Analytical Theory for an Approach Calculation of Non-Balanced Composite Box Beams. Thin-Walled Structures. 39, pp. 709-729.

[19]  Fung, Y. C. (1993). An Introduction to the Theory of Aeroelasticity. Dover Publications, Inc. New York.

[20]  Gao, B., Kim, J. K. and Leung, C. K. Y. (2004). Experimental Investigation of Taper Ended FRP Strips in FRP Strengthened RC Beams. 4^th International Conference on Advanced Composite Materials in Bridges and Structures. Calgary , Alberta, July 20-23, 2004.

[21]  Ghoneam, S. M. (1995). Dynamic Analysis of Open Cracked Laminated Composite Beams. Composite Structures. 32, pp. 3-11.

[22]  Green, I. and Casey, C. (2005). Crack Detection in a Rotor Dynamic System by Vibration Monitoring-Part I: Analysis. Journal of Engineering for Gas Turbines and Power, ASME. April, 127, pp. 425-436.

[23]  Hallauer, W. L. and Liu, R. Y. L. (1982). Beam Bending-Torsion Dynamic Stiffness Method for Calculation of Exact Vibration Modes. Journal of Sound and Vibration. 85(1), pp. 105-113.

[24]  Hashemi, S. M. and Borneman, S. (2003). Application of Frequency Dependent Trigonometric Shape Functions in the Vibration Analysis of Laminated Composite Beams. The Fourth Canadian-International Composites Conference (cancom), proceedings. August 19-22, Ottawa congress centre, Ottawa, Canada.

[25]  Hashemi, S. M. and Borneman, S., (2004). Vibration Analysis of Composite Wings Undergoing Material and Geometrical Couplings: A Dynamic Finite Element Formulation, Submitted to the 2004 ASME International Mechanical Engineering Congress (ICMECE2004), Aerospace Division, November 13-19, 2004, Anaheim, California, Paper#: IMECE2004-60090.

[26]  Hashemi, Seyed Mohammed, (1998). “Free Vibrational Analysis of Rotating Beam-like Structures: A Dynamic Finite Element Approach”, Ph.D. Thesis, Department of Mechanical Engineering, Laval University, Quebec, Canada.

[27]  Hashemi, S. M. and Richard, M. J. (1999). A New Dynamic Finite Element (DFE) Formulation on Lateral Free Vibrations of Euler-Bernoulli Spinning Beams Using Trigonometric Shape Functions. Journal of Sound and Vibration, 220(4), pp. 601-624.

[28]  Hashemi, S. M. (2002). The Use of Frequency Dependent Trigonometric Shape Functions in Vibration Analysis of Beam Structures- Bridging Gap Between FEM and Exact DSM Formulations. Asian Journal of Civil Engineering. 3(3 & 4), pp. 33-56.

[29]  Hashemi, S. M. and Borneman S. R. (2005). A Dynamic Finite Element Formulation for the vibration Analysis of Laminated Tapered Composite Beams. CD proceedings of the Sixth Canadian-International Composites Conference (cancom), Vancouver, Canada, August 16-19, 2005, 1-13.

[30]  Hayes, M. D. (2003). Structural Analysis of a Pultruded Composite Beam: Shear Stiffness Determination and Strength and Fatigue Life Predictions. Ph.D Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University.

[31]  He, K., Hoa, S. V. and Ganesan, R. (2000). The Study of Tapered Laminated Composite Structures: A Review. Composites Science and Technology. 60, pp. 2643-2657.

[32]  Imregun, M. (1995). Prediction of Flutter Stability Using Aeroelastic Frequency Response Functions. Journal of Fluids and Structures. 9, pp. 419-434.

[33]  Jones, I. S. (1998). A Wide Range Weight Function for a Single Edge Cracked Geometry with Clamped Ends. International Journal of Fracture. 89, pp. 1-18.

[34]  Jones, R. M. (1998). Mechanics of Composite Materials Second Edition. Taylor and Francis Inc.

[35]  Jun, L. and Xianding, J. (2005). Response of Flexure-Torsion Coupled Composite Thin-Walled Beams with Closed Cross-Sections to Random Loads. Mathematics Research Communications. 32, pp. 25-41.

[36]  Ju., S. H.  (1996). Simulating Stress Intensity Factors for Anisotropic Materials by the Least-Squares Method. International Journal of Fracture. 81, pp. 283-297.

[37]  Jyoti, A., Gibson, R. F. and Newaz, G. M. (2005). Experimental Studies of Mode I Energy Release Rate in Adhesively Bonded Width Tapered Composite DCB Specimens. Composite Science and Technology. 65, pp. 9-18.

[38]  Kim, J-T. (2003). Crack Detection in Beam-Type Structures Using Frequency Data. Journal of Sound and Vibration. 259(1), pp. 145-160.

[39]  Kisa, M. and Brandon, J. (2000). The Effects of Closure of Cracks on the Dynamics of a Cracked Cantilever Beam. Journal of Sound and vibration. 238(1), pp. 1-18.

[40]  Kisa, M., Brandon, J. and Topcu, M. (1998). Free Vibration Analysis of Cracked Beams by a Combination of Finite Elements and Component Mode Synthesis Methods. Computer and Structures. 67, pp. 215-223.

[41]  Krawczuk, M. (1994). A New Finite Element for the Static and Dynamic Analysis of Cracked Composite Beams.  Computer and Structures. 52(3), pp. 551-561.

[42]  Krawczuk, M.  and Ostachowicz, W. M. (1995). Modeling and Vibration Analysis of a Cantilever Composite Beam with a Transverse Open Crack. Journal of Sound and Vibration. 183(1), pp. 69-89.

[43]  Krawczuk, M., Ostachowicz, W. and Zak, A. (1997). Dynamics of Cracked Composite Material Structures. Computational mechanics. 20, pp. 79-83.

[44]  Krawczuk, M., Ostachowicz, M. and Zak, A. (1997). Modal Analysis of Cracked, Unidirectional Composite Beam. Composites part B28 B. pp. 641-650.

[45]  Krueger, R., Minguet, P. J. and O’Brien, T. K. (1999). A Method for Calculating Strain Energy Release Rates in Preliminary Design of Composite Skin/Stringer Debonding Under Multi-Axial Loading. NASA Centre for AeroSpace Information (CASI). NASA/TM-1999-209365 ARL-TR-2012.

[46]  Leissa, A. W., McGee, O. G. and Huang, C. S. (1993). Vibrations of Circular Plates Having V-notches or Sharp Radial Cracks.  Journal of sound and vibration. 161(2), pp. 227-239.

[47]  Lillico, M., Butler, R., Guo, S., Banerjee, J. R. (1997) Aeroelastic Optimisation of Composite Wings Using the Dynamic Stiffness Method. The Aeronautical Journal, pp.77-86.

[48]  Mahajan, A. J. and Kaza, K. R. V. (1993). Semi-Empirical Model for Prediction of Unsteady Forces on an Airfoil with Application to Flutter. Journal of Fluids and Structures. 7, pp. 87-103.

[49]  McCarthy, T. R. and Chattopadhyay, A. (1997). A Refined Higher-Order Composite Box Beam Theory. Composites Part B. 28 B. pp. 523-534.

[50]  Meshii, T.  and Watanabe, K. (1999). Stress Intensity Factor of an Arbitrarily Located Circumferential Crack in a Thin-Walled Cylinder with Axissymmetrically Loaded Ends. Engineering Fracture Mechannics. 62, pp. 371-382.

[51]  Moosavi, M. R., Oskouei, A. R. N. and A. Khelil. (2005). Flutter of a Subsonic Wing. Thin-Walled Structures. 43, pp. 617-627.

[52]  Mujumdar, P. M.  and Suryanarayan S. (1988). Flexural Vibrations of Beans with Delaminations.  Journal of sound and vibration. 125(3), pp. 441-461.

[53]  Murri, G. B. and Schaff, J. R. (2006). Fatigue Life Methodology for Tapered Hybrid Composite Flexbeams. Composite Science and Technology. 66, pp. 499-508.

[54]  Nandwana, B. P. and Maiti, S. K.  (1997). Modelling of Vibration of Beam in Presence of Inclined Edge or Internal Crack for its Possible Detection Based on Frequency Measurements. Engineering Fracture Mechanics. 58(3), pp. 193-205.

[55]  Nayfeh, A. H. and Abdelrahman, W. G. (2003). Vibration of Fibrous Composites with Damage. Journal of Vibration and Control. 9, pp. 709-729.

[56]  Nikpour, K., Dimarogonas, A.D. Local compliance of composite cracked bodies. Composites Science and Technology, Vol. 32, 1988, pp. 209–223.

[57]  Palacz, M. and Krawczuk. (2002). Vibration parameters for damage detection in structures.  Journal of sound and vibration. 249(5), pp 999-1010.

[58]  Pandey, A. K. and Biswas, M. (1994). Damage Detection in Structures Using Changes and Flexibility.  Journal of sound and vibration.  169(1), pp. 3-17.

[59]  Pidaparti, R. M. V., Chang, C. C. (1998). Finite Element Supersonic Flutter Analysis of Skewed and Cracked Composite Panels. Computers and Structures. 69, pp. 265-270.

[60]  Qin, Z. P., Marzocca and Librescu, L. (2002). Aeroelastic Instability and Response of Advanced Aircraft Wings at Subsonic Flight Speeds. Aerospace Science and Technology. 6, pp. 195-208.

[61]  Qin, Z. (2001). Vibration and Aeroelasticity of Advanced Aircraft Wings Modeled as Thin-Walled Beams-Dynamics, Stability and Control. Ph.D Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University.

[62]  Raju, I. S. (1987). Calculation of Strain-Energy Release Rates with Higher Order and Singular Finite Elements. Engineering Fracture Mechanics. 28(3), pp. 251-274.

[63]  Rand, O. (2000). Bending and Extension of Thin-Walled Composite Beams of Open Cross-Sectional Geometry. Journal of Applied Mechanics, ASME. 67, pp. 813-818.

[64]  Rand, O. (1997). Fundamental Closed-Form Solutions for Solid and Thin-Walled Composite Beams Including a Complete Out-of-Plane Warping Model. International Journal of Solids and Structures. 35(21), pp. 2775-2793.

[65]  Runyan, H. L. and Watkins, C. E. Flutter of a Uniform Wing with an Arbitrary Placed Mass According to a Differential-Equation Analysis and a Comparison with Experiment. Report 966-National Advisory committee for aeronautics. 339-355.

[66]  Sergeev, B., Madenci, E.  and Ambur, D. R.. (1999). Stress Intensity Factors for an Arbitrarily Oriented Crack Near a Hole in Longeron Web. Theorectical and Applied Fracture Mechanics. 31, pp. 213-222.

[67]  Shahrzad, P. and Mahzoon, M. (2002). Limit Cycle Flutter of Airfoils in Steady and Unsteady Flows. Journal of Sound and Vibration. 256(2), pp. 213-225.

[68]  Sheinman, I. and Kardonmateas, G. A. (1997). Energy Release Rate and Stress Intensity Factors for Delaminated Composite Laminates. International Journal of Solids and Structures. 34(4), pp. 451-459.

[69]  Shen M.-H. H. and Pierre, C. (1990). Natural Modes of Bernoulli-Euler Beams with Symmetric Cracks.  Journal of sound and vibration. 138(1), pp. 115-134.

[70]  Shen, M.-H. H. and Pierre, C. (1994). Free Vibrations of Beams with Single-Edge Crack. Journal of Sound and Vibration. 170(2), pp. 237-259.

[71]  Tada, H., Paris, P. C. and Irwin, G. R., (2000). The stress analysis of cracks handbook,

3^rd  edition, ASME Press.

[72]  Tang, R-J. and Tang, X.-Y. (2001). Method to Calculate Bending Center and Stress Intensity Factors of Cracked Cylinder Under Saint-Venant Bending. Applied Mathmatics and Mechanics. 22(1).

[73]  Tuzcu, I. (2001). Dynamics and Control of Flexible Aircraft. Ph.D Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University. December 2001.

[74]  Theodorsen, T. (1935). General Theory of Aerodynamic Instability and Mechanism of Flutter, NACA TR 496.

[75]  Volovoi, V. V. and Hodges, D. H. (2001). Assessment of Beam Modeling Methods for Rotor Blade Applications. Mathematical and Computer Modelling. 33, pp. 1099-1112.

[76]  Volovoi, V. V. and Hodges, D. H. (2002). Single-and Multi-Celled Composite Thin-Walled Beams. AIAA Journal. 40(5).

[77]  Volovoi, V. V. and Hodges, D. H. (2000). Theory of Anisotropic Thin-walled Beams. Journal of Applied Mechanics. ASME. 67, pp. 453-459.

[78]  Wang, K., Inman, D. J. and Farrar, C. R. (2005). Crack-Induced Changes in Divergence and Flutter of Cantilever Composite Panels. Structural Health Monitoring. 4(4).

[79]  Wang, K., Inman, D. J. and Farrar, C. R. (2005). Modeling and Analysis of Cracked Composite Cantilever Beam Vibrating in Coupled Bending and Torsion. Journal of Sound and Vibration. 284, pp. 23-49

[80]  Wang, K.. (2004). Vibration Analysis of Cracked Composite Bending-Torsion Beams for Damage Diagnosis. Ph.D Thesis Submitted to the Faculty of the Virginia Polytechnic Institute and State University

[81]  Wittrick, W. H. and Williams, F. W. (1971). A General Algorithm for Computing the Natural Frequencies of Elastic Structures. Quart. Journ. Mech. and Applied Math., Vol. XXIV, Pt. 3, pp.263-284.

[82]  Wittrick, W. H. and Williams, F. W. (1983). Exact Buckling and Frequency Calculations Surveyed. J. Struct. Eng. ASCE, 109, pp. 169-187

[83]  Wittrick, W. H. and Williams, F. W., (1982). On The Free Vibration Analysis of Spinning Structures by Using Discrete or Distributed Mass Models. Journal of Sound and Vibration, 82(1), pp 1-15.

[84]  Xie, Y. J., Wang, X. H., Lin, Y.C. (2004). Stress Intensity Factors for Cracked Rectangular Cross Section Thin Walled Tubes. Engineering fracture mechanics. 71, pp. 1501-1513.

[85]  Xu, Y. G., Liu, G. R., Wu, Z. P., Huang, X. M. (2001). Adaptive Multilayer Perceptron Networks for Detection of Cracks in Anisotropic Laminated Plates. International Journal of Solids and Structures. 38, pp. 5625-5645.

[86]  Yu, W., Hodges, D. H., Volovoi, V. and Cesnik, C. E. S. (2002). On Timoshenko-Like Modeling of Initially Curved and Twisted Composite Beams. International Journal of Solids and Structures. 39, pp. 5101-5121.

[87]  Zhang, S. Y. (1998). Debonding and Cracking Energy Release Rate of the Fiber/Matrix Interface. Composites Science and Technology. 58, pp. 331-335.

[88]  Zheng, D. Y. and Kessissoglou, N. J. (2004). Free Vibration Analysis of a Cracked Beam by Finite Element Method. Journal of Sound and Vibration. 273, pp. 457-475.

[89]  Zuo, Q. H. and Schreyer, H. L. (1996). Flutter and Divergence Instability of Non-            Conservative Beams and Plates. International Journal of Solids Structures. 33(9), pp. 1355-1367.