Nghiên cứu ứng xử tĩnh, ổn định và dao động dầm composite với tiết diện khác nhau

Nghiên cứu ứng xử tĩnh, ổn định và dao động dầm composite với tiết diện khác nhau Nghiên cứu ứng xử tĩnh, ổn định và dao động dầm composite với tiết diện khác nhau Nghiên cứu ứng xử tĩnh, ổn định và dao động dầm composite với tiết diện khác nhau

INTRODUCTION

Necessity of the thesis

1.1.1 Composite material - Fiber and matrix

Composite materials, formed by combining two or more components, enhance properties like stiffness, strength, weight reduction, corrosion resistance, and thermal performance compared to the individual materials Typically composed of a reinforcing fiber and a matrix base, composites are categorized into three primary types.

(1) Fibrous materials: fibers of one material and matrix of another one (Fig 1.1a)

(2) Particulate composites: macro size particles of one material and matrix of another one (Fig 1.1b)

(3) Laminate composites: made of several layers of different materials, including of the first two types

Fiber and particle reinforcements enhance the strength and stiffness of composite materials, surpassing that of the matrix The matrix itself can be categorized based on its strength and stiffness, including polymers (low strength), metals (intermediate strength), and ceramics (high strength but brittle) It plays a crucial role in maintaining the proper alignment and spacing of the fibers while also protecting them from environmental factors and wear Composite materials are primarily classified into two types: fiber composites and particulate composites.

1.1.2 Composite material - Lamina and laminate

A fiber-reinforced lamina consists of many fiber embedded in a matrix material The fiber can be continuous or discontinuous, woven, unidirectional, bidirectional or randomly distributed (Fig.1.2)

Figure 1.2 Various types of fiber-reinforced composite lamina [1]

A laminate consists of multiple laminae arranged in different orientations, stacked to achieve specific stiffness and thickness These layers are typically bonded using the same matrix material found in a single lamina By carefully choosing the lamination scheme and the material properties of each lamina, the stiffness and strength of the laminate can be customized to meet specific requirements.

Figure 1.3 A laminate made up of laminae with different fiber orientations [1]

In recent years, composite materials have gained popularity across various engineering fields due to their exceptional stiffness-to-weight and strength-to-weight ratios, as well as low thermal expansion, improved fatigue life, and excellent corrosion resistance Notably, laminated composite beams with diverse cross-sections are increasingly utilized in multi-physics environments, including aircraft, spacecraft, robotic arms, bridges, buildings, and other advanced technological systems.

The development of composite beam theories is essential for accurately predicting the behavior of composite materials in engineering applications Various theories, including layer-wise theories (LWT), equivalent single-layer theories (ESLT), zigzag theories (ZZT), and Carrera’s Unified Formulation (CUF), have been explored, with ESLT being favored for its simplicity in formulation and programming ESLT encompasses classical beam theory, first-order beam theory, higher-order beam theory, and quasi-3D beam theory, highlighting that the accuracy of composite beam responses is contingent on the selected beam theory Consequently, creating suitable models for analyzing composite beam behavior is crucial Numerous computational methods have been developed to predict the responses of composite structures through both analytical and numerical approaches While numerical methods are increasingly utilized, analytical techniques remain effective, with the Ritz method being the most general However, the accuracy and efficiency of the Ritz method rely heavily on the choice of approximation functions, as poor selections can lead to slow convergence and numerical instabilities Notably, the Ritz method is underutilized in researching composite beams with diverse cross-sections, indicating a need for further investigation in this area.

In addition, the development of science and technology in recent years led to the trends in which structural elements become smaller and smaller in their dimension

Small-scale composite beams are increasingly utilized in diverse applications, including shape memory alloys and micro- and nano-electro-mechanical system actuators Experimental studies have highlighted the importance of size effects, prompting advancements in Eringen’s nonlocal elasticity theory, strain gradient theory, and modified couple stress theory, each achieving varying levels of success Consequently, it is essential to investigate the behaviors of small-scale laminated composite beams.

Composite beams made from standard structural fiber reinforced plastic pultruded profiles can be manufactured in various thin-walled cross-sectional shapes, such as I-sections and channels, which are extensively used in engineering applications However, analyzing the behavior of laminated composite thin-walled beams with both open and closed sections poses challenges due to the anisotropic properties of the composite materials in the web and flanges While several analytical and numerical methods have been developed with varying success, there is a growing interest in creating a computational method to effectively assess static, buckling, and vibration responses in this area.

From the short literature review on the laminated composite beams showed that the following researches need to investigate further:

- Ritz solutions for behaviours of composite beams with various cross-sections

- Free vibration, buckling and static responses of laminated composite beams based on different beam theories and loads

- Free vibration, buckling and static responses of micro-laminated composite beams based on micro beam theory

- Free vibration, buckling and static responses of thin-walled laminated composite beams

Figure 1.4 Composite material applied in engineering field 1

Figure 1.5 Material used in Boeing 787 2

Review

This section provides a comprehensive literature review on composite beams, followed by an outline of the thesis objectives and scope Additionally, it establishes the theoretical formulations and constitutive laws governing laminated composite beams.

1 https://www.slideshare.net/NAACO/vat-lieu-composite-frp-trong-xay-dung

2 https://www.1001crash.com/index-page-composite-lg-2.html

Numerous studies have been conducted to analyze the structural responses of composite materials, as indicated by available literature Various beam theories, constitutive laws, and methods have been proposed to predict the vibration, buckling, and bending behaviors of beams This section provides a brief review of laminated composite beams, with more detailed reviews available at the beginning of each chapter.

A literature review by Ghugal and Shimpi highlights various composite beam theories, categorizing them into classical beam theory, first order beam theory, higher-order beam theory, and quasi-3D beam theory Classical beam theory is limited to thin beams as it neglects transverse shear strain effects, while first order shear deformation theory includes these effects but requires a shear correction factor for accurate stress distribution To address this limitation, higher-order beam theory utilizes distribution functions for transverse shear stresses, yet it overlooks the impact of transverse normal strain Consequently, quasi-3D beam theory emerges as the most comprehensive option, demonstrating that the accuracy of beam responses is contingent on the selected theories Additionally, Ghugal and Shimpi's review notes a lack of focus on Poisson’s effect in the behavior of laminated composite beams Furthermore, experimental studies indicate that size effects are significant at smaller scales, prompting the development of Eringen’s nonlocal elasticity theory and strain gradient theory.

Numerous studies have explored the analysis of composite microbeams, primarily concentrating on cross-ply configurations However, there is a significant need to investigate micro general laminated composite beams with arbitrary lay-ups to expand the current understanding in this field.

Composite materials are increasingly utilized in thin-walled structures, leading to numerous studies grounded in thin-walled structure theory Vlasov initially proposed a theory for isotropic materials, which was later expanded by Bauld and Tzeng to include buckling analysis of laminated composite thin-walled rods, examining both linear and nonlinear responses of laminated composite beams with open sections Additionally, Song and Librescu explored the dynamic responses of laminated composite beams with arbitrary sections Lee and Kim developed a general model using classical beam theory to predict vibration and buckling responses of laminated composite thin-walled beams with I-sections However, the impact of shear deformation on the behavior of thin-walled composite beams remains underexplored.

Numerous computational methods have been developed to accurately predict the responses of composite structures, including analytical, numerical, and semi-numerical approaches These methods can be categorized as either exact or approximate Among the numerical methods, the finite element method (FEM) is the most commonly utilized for analyzing free vibration, buckling, and bending of composite beams Other methods such as the finite difference method, Chebyshev collocation technique, and dynamic stiffness matrix method are also employed for beam behavior predictions Recently, isogeometric analysis has gained attention from researchers This thesis emphasizes analytical solutions, particularly the Navier procedure, which, despite being limited to simply supported boundary conditions, is favored for its simplicity Additional analytical methods explored include the differential quadrature method, Galerkin method, and differential transform method The Ritz method, which is the primary focus of this thesis, is a versatile variational approach that accommodates various boundary conditions by selecting appropriate shape functions to approximate unknown displacement fields.

In the analysis of laminated composite beams, inappropriate selection of unknown functions can lead to numerical instabilities and slow convergence rates While polynomial and orthogonal polynomial functions are commonly employed, traditional polynomial functions often fail to meet boundary conditions, necessitating the use of penalty or Lagrange multiplier methods, which increase the dimensions of stiffness and mass matrices and consequently raise computational costs Orthogonal polynomial functions address this issue by satisfying specific boundary conditions, yet they remain underutilized in bending behavior analysis of laminated composite beams Recent work by Mantari and Canales introduced hybrid polynomial-trigonometric functions for examining free vibration and buckling behaviors, but these too are associated with slow convergence rates and high computational expenses Thus, the Ritz approximation functions show limitations, highlighting the need for further research into the Ritz method for beam problems.

In Vietnam, research on the behavior analysis of composite structures has gained significant attention, with notable contributions from Nguyen-Xuan et al and Nguyen et al., who focus on advanced numerical methods such as the Finite Element Method (FEM), mesh-less methods, and isogeometric analysis for composite plates Additionally, Nguyen et al have developed analytical methods for various geometric shapes and loading conditions in composite plates and shells Experimental studies on composite structures have been conducted by Tran et al., while Hoang et al investigated the responses of functionally graded plates and shells under thermo-mechanical loads Further, Nguyen et al explored the behaviors of functionally graded beams using FEM under diverse geometric and loading scenarios, and Tran et al utilized Higher Order Beam Theory (HOBT) for similar studies Nguyen et al also analyzed the buckling and vibration behaviors of functionally graded beams subjected to hygro-thermal loads, highlighting the dynamic research landscape in Vietnam's composite structure analysis.

9 have not paid much attention to laminated composite beam problems, especially composite thin-walled beams yet

1.2.2 Objectives and scopes of the thesis

This study aims to address the limited application of the Ritz method in analyzing the behaviors of composite beams under various boundary conditions by developing simple, effective, and accurate approximation functions Existing literature highlights a gap in understanding the impacts of Poisson ratio, normal strain, and size dependence on the behaviors of composite beams Consequently, the objectives of this thesis include proposing Ritz's approximation functions to effectively analyze the free vibration, buckling, and bending behaviors of macro/micro-laminated composite beams using techniques such as FOBT, HOBT, quasi-3D, and MCST.

Numerous beam theories have been developed for the analysis of laminated composite beams, categorized into layer-wise theories (LWT), equivalent single layer theories (ESLT), zig-zag theories (ZZT), and Carrera’s Unified Formulation (CUF) Among these, ESLT are particularly favored for their simplicity in both formulation and programming To effectively describe these beam theories, a coordinate system is introduced, with the x-axis representing the length, the y-axis the width, and the z-axis the height of the beam.

In the study of a rectangular laminated composite beam, the geometry and coordinates are crucial for understanding its behavior General beam theory describes axial and transverse displacements as products of two distinct sets of unknown functions, specifically focusing on variations in the z-direction and x-direction.

( , , ) ( , ) ( , ) ( , ) w x z t  w x t  zw x t  z w x t  (1.2) where u i and w i are are undetermined functions of x and t

The most simple and commonly used beam theory is the Euler-Bernoulli or classical beam theory (CBT) [78], which is based on the displacement field:

The equation \( w(x, t) = 0 \) describes the axial and vertical displacements at the mid-plane of a beam, represented by \( u_0 \) and \( w_0 \), respectively The Classical Beam Theory (CBT) is commonly applied to thin beams as it disregards shear deformation effects To account for these effects, Timoshenko's Beam Theory or the First Order Beam Theory (FOBT) is introduced, providing a more accurate analysis of beam behavior under various loading conditions.

( , , ) 0( , ) w x z t w x t (1.6) where u 1 denotes rotation of a transverse normal about the y-axis FOBT requires a shear correction factor, which depends not only on the material and geometric

Determining the shear correction factor is a complex challenge influenced by various parameters, boundary conditions, and loading To address this difficulty, higher-order beam theories (HOBT) have been developed, with numerous researchers proposing a third-order beam theory based on the displacement field.

By imposing traction-free boundary condition on the top and bottom face of the beam, Eqs (1.7) and (1.8) lead to:

Reddy’s third-order beam theory, as outlined in Eqs (1.9) and (1.10), is a significant framework in structural analysis Additionally, numerous researchers have contributed to the development of refined high-order theories, which enhance the representation of higher-order variations in axial displacement through the function f z ( ).

The rotational angle at a point on the neutral line is denoted as u 1 a ( , )x t, and the shear functions must satisfy the traction-free boundary conditions of beams, as outlined in Table 1.1.

Organization

There are eight chapters in this thesis as followings:

- Chapter One describes the purpose, scope and organization as well as the review study on laminated composite beams

Chapter Two of the HOBT focuses on the buckling, bending, and free vibration behaviors of laminated composite beams It introduces innovative trigonometric functions tailored for beams subjected to different boundary conditions.

Chapter Three builds upon the theoretical framework established in Chapter Two by examining the impact of mechanical and thermal loads on the buckling and free vibration behaviors of composite beams It introduces innovative hybrid functions that integrate admissible and exponential functions tailored for different boundary conditions.

Chapter Four examines how normal strain and Poisson's ratio influence the buckling, bending, and free vibration behaviors of beams Utilizing both Higher Order Beam Theory (HOBT) and quasi-3D theories, the study integrates these factors into the constitutive equations to provide a comprehensive analysis of their effects.

Chapter Five utilizes the modified couple stress theory to analyze micro-laminated composite beams through higher-order beam theory (HOBT) The study introduces exponential functions as a solution method, examining the impact of Poisson's effect on the buckling, bending, and free vibration behaviors of these composite beams.

Chapter Six examines thin-walled laminated composite and functionally graded thin-walled beams using the Finite Element Method (FOBT) It presents numerical results related to the frequency, critical buckling load, and displacement of both channel and I-beams.

- Study on new approximation functions is carried in Chapter Seven Convergence, computational cost and numerical stability are investigated to verify efficiency of proposed functions

- In the last Chapter, the main conclusions and recommendations for future research are presented

ANALYSIS OF LAMINATED COMPOSITE BEAMS BASED ON A HIGH-ORDER BEAM THEORY

Introduction

To accurately predict the structural responses of composite laminated beams, various beam theories employing different methodologies have been established A comprehensive review and evaluation of these theories related to composite beams is available in the literature.

108] It should be noted that HOBT has been increasingly applied in predicting responses of composite beams

The finite element method is extensively utilized for analyzing composite beams, while the Navier solution serves as the simplest analytical approach, applicable solely to simply supported boundary conditions To address arbitrary boundary conditions, various researchers have developed alternative methods, with the Ritz-type method being a commonly employed solution.

Khdeir and Reddy developed a state-space approach to accurately determine the natural frequencies and critical buckling loads of cross-ply composite beams Similarly, Chen et al proposed an analytical solution utilizing state-space differential quadrature for the vibration analysis of composite beams Additionally, Jun et al employed the dynamic stiffness matrix method to compute the natural frequencies based on third-order beam theory Despite the efficiency of the Ritz procedure in addressing static, buckling, and vibration issues of composite beams under various boundary conditions, a literature review indicates that research in this area remains limited.

This chapter aims to establish a novel trigonometric-series solution for analyzing composite beams, utilizing a higher-order beam theory that incorporates a more complex variation of axial displacement The governing equations of motion are derived through Lagrange equations, leading to the development of a Ritz-type analytical solution featuring the new trigonometric series, applicable to beams with diverse boundary conditions Additionally, convergence and verification studies are conducted to ensure the accuracy and reliability of the proposed solution.

3 A slightly different version of this chapter has been published in Composite Structures in 2017

The study validates the proposed solution through numerical results, examining how the length-to-height ratio, fiber angle, and material anisotropy influence the deflections, stresses, natural frequencies, and critical buckling loads of composite beams.

Beam model based on the HOBT

A laminated composite beam with a rectangular cross-section (width b and height h) and length L is analyzed, consisting of n layers of orthotropic materials arranged at various fiber angles relative to the x-axis.

Figure 2.1 Geometry and coordinate of a laminated composite beam

2.2.1 Kinetic, strain and stress relations

In this Chapter, the displacement ﬁeld of composite beams is based on the HOBT

The equation \( w_{,x} = w_{,t} \) describes the relationship between partial differentiation with respect to the corresponding subscript coordinates Here, \( u_{x,t}(0) \) and \( w_{x,t}(0) \) denote the axial and vertical displacements at the mid-plane of the beam, while \( u_{1,a}(x,t) \) represents the rotational angle at a specific point on the mid-plane Additionally, \( f(z) \) accounts for the higher-order variation of axial displacement The non-zero strain in beams can be derived from these parameters.

   , x s u 1 , a x , g  f ,z (2.5) The elastic strain and stress relations of k th -layer in global coordinates is given by:

(2.6) where the Q ( ) ij k are showed in Appendix A

The strain energy  E of beam is given by:

(2.7) where the stiffness coefficients of the beam are determined as follows:

The work done  W by the compression load N 0 and transverse load q is given by:

The kinetic energy  K of beam is expressed as:

           (2.11) where dot-superscript denotes the differentiation with respect to the time t;  is the mass density of each layer, and I 0 , I 1 ,I 2 ,J 1 ,J 2 ,K 2 are the inertia coefficients determined by:

The total energy  of beam is obtained as:

In this Chapter, Ritz solution is used to approximate the displacement field as:

The equation (2.16) incorporates frequency (ω) and the imaginary unit (i² = -1), with values u₀j, w₀j, and u₁j to be determined The approximation functions ψₑ(x), φₑ(x), and ξₑ(x) are designed for different boundary conditions: simply-supported (S-S), clamped-free (C-F), and clamped-clamped (C-C), as outlined in Table 2.1 These functions effectively meet the boundary conditions specified in Table 2.2 It is important to note that using inappropriate approximation functions can lead to slow convergence rates and numerical instabilities, particularly when the functions do not adhere to the required boundary conditions.

Lagrangian multipliers method can be used to impose boundary conditions [101,

Table 2.1 Approximation functions of the beams

Table 2.2 Kinematic BCs of the beams

C-C x=0 u 00,w 0 0, w 0, x  0,u 1 a 0 x=L u 00,w 0 0, w 0, x  0,u 1 a 0 The governing equations of motion can be obtained by substituting Eqs (2.14, 2.15 and 2.16) into Eq (2.13) and using Lagrange’s equations:

  (2.17) with p j representing the values of ( u 0j , w 0j , u 1 j ), the bending, buckling and vibration responses of beams can be obtained from the following equations:

(2.18) where the components of stiffness matrix K, mass matrix M and vector F are given by:

The deflection, stresses, critical buckling loads and natural frequencies of composite beams can be determined by solving Eq (2.18).

Numerical examples

This section presents convergence and verification studies to validate the accuracy of the proposed solution while examining the behavior of composite beams under different boundary conditions for bending, vibration, and buckling issues The static analysis involves applying a uniformly distributed load with density \( q \) to the beam, which consists of laminates of equal thickness made from the same orthotropic materials (MAT).

- Material III.2 E 1 144.9GPa, E 2 9.65GPa,G 12 G 13 4.14GPa, G 23 3.45GPa

For convenience, the following normalized terms are used:

 E bh for Material I.2 and II.2,

In order to evaluate the convergence and reliability of the proposed solution,

The study examines composite beams with a length-to-height ratio of 5, utilizing MAT I.2 and a modulus ratio of E1/E2 equal to 40 Key findings on mid-span displacements, fundamental natural frequencies, and critical buckling loads for various boundary conditions are summarized in Table 2.3 It is noted that the responses exhibit rapid convergence for three specific boundary conditions: m = 2 for buckling, m = 12 for vibration, and m = 14 for deflection These series numbers will be consistently applied in subsequent numerical analyses Notably, the trigonometric solution demonstrates faster convergence compared to the polynomial series solution, particularly in buckling analysis.

Table 2.3 Convergence studies for the non-dimensional fundamental frequencies, critical buckling loads and mid-span displacements of (0 0 /90 0 /0 0 ) composite beams (MAT I.2, /L h5, E 1 /E 2 = 40)

As the first example, (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams with MAT II.2 and

The mid-span displacements of beams with various boundary conditions and length-to-height ratios (L/h = 5, 10, 20, 50) are presented in Tables 2.4 and 2.5, demonstrating excellent alignment with results from higher-order beam theories [20, 22, 24, 36, 72] Additionally, the axial and transverse shear stresses for beams with L/h ratios of 5 and 10 are detailed in Table 2.6, showing strong correlation with the findings of Vo and Thai [22] and Zenkour [36], further validating the accuracy of the current solutions.

Table 2.4 Normalized mid-span displacements of (0 0 /90 0 /0 0 ) composite beam under a uniformly distributed load (MAT II.2, E 1 /E 2 = 25)

Table 2.5 Normalized mid-span displacements of (0 0 /90 0 ) composite beam under a uniformly distributed load (MAT II.2, E 1 /E 2 = 25)

Murthy et al [24] 15.334 12.398 - 11.392 Khdeir and Reddy [72] 15.279 12.343 - 11.337

Vo and Thai (HOBT) [22] 15.305 12.369 11.588 11.363 Mantari and Canales [20] - 12.475 - -

Table 2.6 Normalized stresses of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams with simply-supported boundary conditions (MAT II.2, E 1 /E 2 = 25)

Lay-up Theory  xx  xz

Figure 2.2 illustrates the variation of axial and shear stress across the depth of composite beams with simply-supported boundary conditions The shear stress exhibits a parabolic distribution, while the traction-free boundary conditions are evident The normalized stresses (σxx, σxz) are analyzed for the configurations (0°/90°/0°) and (0°/90°), with a material ratio of E1/E2 = 25.

The impact of fiber angle variations on the mid-span displacements of (θ/-θ)S composite beams (L/h = 10) with MAT II.2 and E1/E2 = 25 is illustrated in Fig 2.3 The findings indicate that as the fiber angle increases, the mid-span transverse displacement also rises, with the lowest displacement observed in C-F beams and the highest in C-C beams.

Figure 2.3 Effects of the ﬁbre angle change on the normalized transverse displacement of   /    s composite beams ( /L h10, MAT II.2, E 1 /E 2 = 25)

Table 2.7 Normalized critical buckling loads of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams (MAT I.2, E1/E2 = 40)

Mantari and Canales [48] 8.585 18.796 - - Khdeir and Reddy [113] 8.613 18.832 - -

Tables 2.7-2.9 report the fundamental frequencies and critical buckling loads of

(0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams with different boundary conditions The present solutions are validated by comparison with those derived from HOBTs [21,

24, 47, 48, 103, 104, 113] Excellent agreements between solutions from the present model and previous ones are observed while a slight deviation with those from Mantari and Canales [48] is found for L/h = 5

Table 2.8 Normalized critical buckling loads of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams with simply-supported boundary conditions (MAT I.2 and II.2, E 1 /E 2 = 10)

Vo and Thai [21] 1.910 2.156 2.228 2.249 MAT II.2

The first three mode shapes of the composite beams (0 0 /90 0 /0 0) and (0 0 /90 0) with a length-to-height ratio of 10, using MAT I.2 and a ratio of E1/E2 equal to 40, are illustrated in Figure 2.4 The symmetric beam demonstrates double coupled vibration modes (w0, u1), while the anti-symmetric beam exhibits triply coupled vibration modes (u0, w0, u1).

Table 2.9 Normalized fundamental frequencies of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams (MAT I.2, E 1 /E 2 = 25)

Murthy et al [24] 9.207 13.611 - - Khdeir and Reddy [104] 9.208 13.614 - -

Vo and Thai [21] 9.206 13.607 16.327 - Mantari and Canales [48] 9.208 13.610 - -

Vo and Thai [21] 6.058 6.909 7.204 7.296 Mantari and Canales [48] 6.109 6.913 - - C-F (0 0 /90 0 /0 0 ) Present 4.234 5.498 6.070 6.267

Figure 2.4.The first three mode shapes of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams with simply-supported boundary conditions (L/h = 10, MAT I.2, E 1 /E 2 = 40) x/L -1

The effect of the ratio of material anisotropy on the fundamental frequencies and critical buckling loads is plotted in Fig 2.5 Obviously, the results increase with

Figure 2.5 Effects of material anisotropy on the normalized fundamental frequencies and critical buckling loads of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams with simply- supported boundary conditions (L h/ 10, MAT I.2)

The analysis of   /    S composite beams (L/h = 15) with MAT III.2 reveals that variations in fiber angle significantly impact fundamental frequencies and critical buckling loads, as shown in Table 2.10 and Fig 2.6, with results decreasing as the fiber angle increases A close correlation is noted between the current findings and those reported by Chandrashekhara et al [27], although minor discrepancies exist when compared to earlier studies [21, 46, 87] Additionally, the investigation of  30 / 30 0  0  S composite beams under S-S, C-F, and C-C boundary conditions demonstrates that an increase in the length-to-height ratio (L/h) leads to higher fundamental frequencies and critical buckling loads, particularly under the C-C boundary condition when L/h is less than 20, as illustrated in Fig 2.7.

Table 2.10 Normalized fundamental frequencies of   /    s composite beams with respect to the ﬁbre angle change (L h/ 15MAT III.2)

Vo and Thai [21] 4.8969 3.2355 1.6309 1.6152 Chen et al [87] 4.8575 2.3445 1.6711 1.6237 a b

Figure 2.6 Effects of the ﬁbre angle change on the normalized fundamental frequencies and critical buckling loads of   /    s composite beams (L h/ 15, MAT III.2)

Figure 2.7 Effects of the length-to-height ratio on the normalized fundamental frequencies and critical buckling loads of  30 / 30   s composite beams (L h/ 15, MAT III.2)

Conclusion

A new analytical solution utilizing a Higher Order Beam Theory (HOBT) is introduced for the static, buckling, and vibration analysis of laminated composite beams, accommodating various boundary conditions through a trigonometric series approach This study presents numerical results that are compared with prior research, examining the influence of fiber angle and material anisotropy on key parameters such as deflections, stresses, natural frequencies, critical buckling loads, and corresponding mode shapes The findings reveal significant insights into these effects.

- Beam model is suitable for free vibration, buckling and bending analysis of laminated composite beams

- The proposed series solution converges quickly for buckling analysis

- The present solution is found to simple and efficient in analysis of laminated composite beams with various boundary conditions

VIBRATION AND BUCKLING ANALYSIS OF LAMINATED

Introduction

Laminated composite beams are extensively utilized in multi-physics environments such as construction, transportation, and nuclear applications, prompting significant research into their thermal and mechanical behaviors Various analytical and numerical methods have been employed to predict these behaviors, with the finite element method being the most prevalent Researchers like Mathew et al and Lee have explored thermal buckling using first-order shear deformation and layerwise theories, respectively Additionally, Murthy et al and Vo and Thai applied higher-order shear deformation beam theory to assess bending, mechanical buckling, and vibration Analytical studies by Kant et al focused on dynamic responses through refined higher-order beam theory, while Emam and Eltaher examined buckling behaviors in hygrothermal conditions Khdeir and Reddy utilized a state-space approach to analyze vibration and buckling across different boundary conditions, and Khdeir further investigated thermal buckling using various beam theories Abramovich predicted thermal buckling loads based on the first-order beam theory, and Aydogdu employed the Ritz method alongside higher-order beam theory to study free vibration and mechanical buckling.

Recent studies have explored various methods for analyzing the thermal buckling and vibration of laminated composite beams under different boundary conditions Wattanasakulpong et al utilized polynomial functions to assess thermal buckling loads and free vibrations in functionally graded beams Similarly, Mantari and Canales employed the Ritz method to predict mechanical buckling and vibration responses in laminated composite beams Asadi et al focused on the nonlinear vibration and thermal stability of shape memory alloy hybrid laminated composite beams using the Galerkin method Warminska et al examined the vibration of composite beams subjected to thermal and mechanical loads through the First Order Beam Theory (FOBT) Jun et al applied the dynamic stiffness method for analyzing the vibration and buckling behavior of laminated composite beams Vosoughi et al utilized the differential quadrature method alongside FOBT to investigate the thermal buckling and postbuckling behaviors of laminated composite beams with temperature-dependent properties Notably, the Ritz method has not been extensively applied in the analysis of laminated composite beams under thermal and mechanical loading conditions.

This chapter aims to develop a Ritz solution for the thermo-mechanical buckling and vibration of laminated composite beams, introducing new approximation functions The displacement field is grounded in Higher Order Beam Theory (HOBT) A verification study is conducted through numerical examples to demonstrate the accuracy of the proposed solution Additionally, a parametric study investigates how factors such as length-to-height ratio, boundary conditions, material anisotropy, and temperature variations affect the buckling and vibration behavior of laminated composite beams under mechanical and thermal loads.

Theoretical formulation

A laminated composite beam, which is defined in Chapter Two (Fig 2.1), is supposed to be embedded in thermal environment with a uniform temperature rise through the beam thickness as:

T T 0 T (3.1) where T 0 is reference temperature which is supposed to be one at the bottom surface of the beam

3.2.1 Beam model based on the HOBT

The displacement field and constitutive equations of composite beams are derived from the Higher-Order Beam Theory (HOBT), as discussed in Chapter Two Consequently, the strain energy (Π_E) and kinetic energy (Π_K) of the beams are defined by Equations (2.7) and (2.11) Additionally, the work done (Π_W) by thermal and mechanical loads can be expressed accordingly.

      (3.2) where N x 0m and N x 0t are the axial mechanical load and axial thermal stress resultant, respectively For a temperature rise  TT 0 , the thermal axial stress resultant is given by:

      (3.3) where  x , y and  xy are the transformed thermal expansion coefficients in global coordinates (see Appendix A for more details)

The total energy  of beams is obtained as:

Ritz solution is used to approximate the displacement field as:

This chapter introduces new hybrid approximation functions, as detailed in Table 3.1, which are combinations of exponential and admissible functions These functions are designed to meet various boundary conditions, including simply-supported (S-S), hinged-hinged (H-H), clamped-free (C-F), clamped-simply supported (C-S), clamped-hinged (C-H), and clamped-clamped (C-C) The frequency is represented by ω, with the imaginary unit defined as i² = -1, while values such as u₀j, w₀j, and u₁j remain to be determined, alongside the approximation functions ψj(x), φj(x), and ξj(x).

Table 3.1 Approximation functions and kinematic BC of the beams

By substituting Eqs (3.5, 3.6 and 3.7) into Eq (3.4) and using Lagrange’s equations Eq (3.8)

38 with p j representing the values of ( u 0j , w 0j , u 1j ), the thermo-mechanical buckling and vibration responses of laminated composite beams can be obtained from the following equations

(3.9) where the components of stiffness matrix K and mass matrix M are given by:

The critical buckling loads and natural frequencies of composite beams can be determined by solving Eq.(3.9).

Numerical results

Laminated composite beams consist of layers with uniform thickness, constructed from orthotropic materials as specified in Table 3.2 These beams are designed to operate in thermal environments, experiencing a uniform temperature increase throughout their depth For the purposes of the examples discussed, it is assumed that the initial temperature (T₀) is 0°C Additionally, several nondimensional terms are introduced for convenience in analysis.

Table 3.2 Material properties of laminated composite beams

To test the convergence of the current solutions, laminated composite beams (MAT I.3, 0 0 /90 0 /0 0 , L h/ = 5, E 1 /E 2 = 40) with varying boundary conditions under mechanical loads were analyzed Table 3.3 presents the variations of nondimensional fundamental frequencies and critical buckling loads concerning the series number m The findings indicate that m = 12 serves as the convergence point for both natural frequency and buckling load across all boundary conditions Consequently, this series term count will be utilized in subsequent numerical examples.

Table 3.3 Convergence study of nondimensional critical buckling load and fundamental frequency of (0 0 /90 0 /0 0 ) beams (MAT I.3, L h /  5, E 1 /E 2 = 40)

This study examines the free vibration characteristics of symmetric (0 0 /90 0 /0 0) and unsymmetric (0 0 /90 0) cross-ply beams (MAT I.3) across various length-to-height ratios and boundary conditions (BCs) The nondimensional fundamental frequencies are compared with previous findings, as shown in Tables 3.4 and 3.5, demonstrating strong agreement with earlier research Notably, for symmetric composite beams, the nondimensional fundamental frequencies for simply supported-simply supported (S-S) and clamped-simply supported (C-S) boundary conditions closely resemble those of fixed-fixed (H-H) and clamped-clamped conditions.

Table 3.4 Nondimensional fundamental frequency of (0 0 /90 0 /0 0 ) beams (MAT I.3,

5 Present 9.208 9.208 4.232 10.239 10.239 11.605 Murthy et al [24] 9.207 - 4.230 10.238 - 11.602 Khdeir and Reddy

Table 3.5 Nondimensional fundamental frequency of (0 0 /90 0 ) beams (MAT I.3, E 1 /E 2 = 40)

Murthy et al [24] 6.045 - 2.378 8.033 - 10.011 Khdeir and Reddy

The study examines the fundamental frequencies of composite beams (0 0 /90 0 /0 0) and (0 0 /90 0) with MAT II.3 under uniform temperature rise (UTR), as detailed in Table 3.6 for various boundary conditions (BCs) The findings align closely with the results presented by Jun et al [127] Figures 3.2a and 3.2b illustrate the impact of UTR on the fundamental frequency of the beams, indicating a decrease in frequency with increasing temperature change (ΔT) until reaching critical temperatures, at which point the fundamental frequencies disappear.

Table 3.6 The fundamental frequency (Hz) of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) beams with various boundary conditions (MAT II.3)

Jun et al [127] 702.5 819.7 1030.0 a 0 0 /90 0 /0 0 b 0 0 /90 0 Figure 3.1 Variation of fundamental frequency of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) beams

(MAT II.3) with respect to uniform temperature rise ∆T

This article examines the buckling behavior of laminated composite beams subjected to mechanical loads It highlights critical buckling loads for cross-ply configurations (0°/90°/0° and 0°/90°) and angle-ply composite beams, as detailed in Tables 3.7, 3.8, and 7.9.

F un da m e nt al f re qu en c y( H z )

44 length-to-height ratios and BCs The present results are very close with published results

Table 3.7 Nondimensional critical buckling load of (0 0 /90 0 /0 0 ) beams (MAT I.3,

Mantari and Canales [48] 8.585 - 4.673 10.192 - 11.502 Khdeir and Reddy [120] 8.613 - 4.708 9.814 - 11.602

10 Present 18.832 18.832 6.772 25.857 25.857 34.453 Mantari and Canales [48] 18.796 - 6.757 27.090 - 34.365 Khdeir and Reddy [120] 18.832 - 6.772 25.857 - 34.453

Table 3.8 Nondimensional critical buckling load of (0 0 /90 0 ) beams (MAT I.3,

Table 3.9 Nondimensional critical buckling load of angle-ply beams (MAT I.3, E 1 /E 2 = 40)

Mantari and Canales [48] 9.0718 4.8633 10.8326 12.2267 Canales and Mantari [129] 9.0658 4.5551 10.2878 12.0767

This study investigates the critical buckling temperature of symmetric and unsymmetric composite beams using MAT I.3, with a ratio of E1/E2 set at 10 and α*2/α1* at 3 The findings, presented in Tables 3.10 and 3.11, demonstrate strong alignment with the results reported by Aydogdu [123] and Wattanasakulpong et al [124].

Table 3.10 Nondimensional critical buckling temperature of (0 0 /90 0 /0 0 ) beams (MAT I.3, E 1 /E 2 = 40,   2 * / 1 * 3)

Table 3.11 Nondimensional critical buckling temperature of unsymmetric C-C beams (MAT I.3, E 1 /E 2 = 20,   2 * / 1 * 3)

The impact of the ratios α*2/α*1 and E1/E2 on the critical buckling temperature is examined, with the nondimensional critical buckling temperatures for (0°/90°/0°) and (0°/90°) composite beams presented in Tables 3.12 and 3.13, respectively The results demonstrate a strong correlation between the findings.

Table 3.12 Nondimensional critical buckling temperature of (0 0 /90 0 ) composite beams (MAT I.3, L h /  10)

Figure 3.3 plots the variation of nondimensional critical buckling temperature with respect to   2 * / 1 * of (0 0 /90 0 /0 0 ) laminated composite beams with MAT I.3,

E E  , L h/ 10 It is found that the results decrease with the increase of   2 * / 1 *

Table 3.13 Nondimensional critical buckling temperature of (0 0 /90 0 /0 0 ) composite beams (MAT I.3, L h/ 10)

Figure 3.2 Effect of   2 * / 1 * ratio on nondimensional critical buckling temperature of

Conclusions

The study introduces new hybrid approximation functions to analyze the mechanical and thermal buckling, along with free vibration behaviors of laminated composite beams utilizing Higher Order Beam Theory (HOBT) Numerical results are derived under various boundary conditions to investigate the influence of factors such as material anisotropy, length-to-height ratio, thermal expansion ratio, and temperature variations on the buckling load, buckling temperature, and natural frequencies of composite beams The findings reveal significant insights into these effects.

Thermal load plays a crucial role in the buckling and vibration behaviors of laminated composite beams As temperatures rise, these beams exhibit increased flexibility, leading to a decrease in both fundamental frequency and critical buckling load.

- The proposed solution is not only good convergence but also simple and effecient for the buckling and vibration analysis of laminated composite beams under thermal and mechanical loadings

N o rm a liz e d c ri ti c a l b u c k lin g t h e rm a l

EFFECT OF TRANSVERSE NORMAL STRAIN ON BEHAVIOURS

Introduction

In Chapters Two and Three, Higher-Order Beam Theory (HOBT) is employed to examine the bending, buckling, and vibration characteristics of laminated composite beams subjected to mechanical and thermal loads The significant interest in HOBT among researchers highlights its relevance and applicability in the study of composite materials.

85], it neglected transverse normal strain For this reason, quasi-3D theories [36, 86-

88] were developed based on higher-order variations of both in-plane and out-of- plane displacements

Numerous analytical and numerical methods have been proposed to analyze the behavior of laminated composite beams Notably, Zenkour employed the Navier solution for the bending analysis of cross-ply laminated and sandwich beams Aydogdu advanced the Ritz method for buckling and vibration analysis, a technique also utilized by Mantari and Canales Khdeir and Reddy introduced a state space approach for analyzing the vibration and buckling of cross-ply laminated beams The finite element method has been extensively applied for static, dynamic, and buckling analyses of composite beams Additionally, Jun et al utilized the dynamic stiffness matrix method for vibration analysis, while Shao et al applied the reverberation ray matrix method to study free vibrations under various boundary conditions Experimental investigations have also been conducted to examine the natural frequency of laminated composite beams.

The Ritz method is an effective approach for analyzing the behavior of composite beams under various boundary conditions; however, existing literature shows that research utilizing this method remains limited Notably, only a handful of studies have explored the impact of transverse normal strain on the static response, vibration characteristics, and buckling behavior of these beams.

5 A slightly different version of this chapter has been published in International Journal of Structural Stability and Dynamics in 2018

While previous research primarily concentrated on cross-ply composite beams, which represent a specific category of laminated composite beams, it is essential to explore angle-ply laminated beams due to their unique coupling of out-of-plane stresses and strains This highlights the necessity and practical significance of studying composite beams with arbitrary lay-ups, as they offer a broader understanding of their mechanical behavior.

This chapter aims to introduce innovative Ritz-method-based analytical solutions for the static, vibration, and buckling analysis of laminated composite beams, utilizing a quasi-3D theory that incorporates higher-order variations of axial and transverse displacements The governing equations of motion are established through Lagrange’s equations, and convergence and verification studies are conducted to validate the accuracy of the proposed methods Numerical results are provided to explore the impact of factors such as transverse normal strain, length-to-height ratio, fiber angle, Poisson’s ratio, and material anisotropy on the deflections, stresses, natural frequencies, and buckling loads of laminated composite beams.

4.2.1 Kinetic, strain and stress relations

Consider a laminated composite beam, which is defined in Chapter Two (Fig 2.1) The displacement ﬁeld of composite beams is given by [36]:

( , , ) ( , ) ( , ) ( , ) w x z t  w x t  zw x t  z w x t (4.2) where u x t 0 ( , )and w x t 0 ( , ) are the axial and transverse displacements of mid-plan of the beam, respectively; u x t 1 ( , ) is the rotation of a transverse normal about the y-axis;

1( , ) w x t and w x t 2 ( , ) are additional higher-order terms The present theory therefore holds five variables to be determined

The strain ﬁeld of beams is given by:

The elastic strain and stress relations of k th -layer of quasi-3D theory are given by:

(4.8) where C 11 , C 13 , C 33 , C 55 are indicated in Appendix A If the transverse normal stress is omitted ( z 0), the strain and stress relations of HOBT are recovered as:

(4.9) where C * 11 and C * 55 are showed in Appendix A It should be noted that for HOBT, the higher-order terms (w x t 1 ( , ) and w x t 2 ( , )) will be vanished

The strain energy  E of beam is given by:

3 3 9 x xx xx xx xx xx

3 F w xx w 4 F w xx D w xx w A w 3 G w xx w xx E w xx w

 (4.10) where the stiffness coefficients of the beam are determined as follows:

The work done  W by compression load N 0 and transverse load q applied on the beam bottom surface is given by:

The kinetic energy  K of beam is written by:

54 where dot-superscript denotes the differentiation with respect to the time t;  is the mass density of each layer, and I 1 ,I 3 ,I 5 ,I 7 are the inertia coefficients defined by:

The total potential energy of system is expressed by:

Based on the Ritz method, the displacement field in Eq (4.1 and 4.2) is approximated in the following forms:

In this chapter, we introduce new approximation functions that integrate polynomial and exponential functions, designed to meet the essential boundary conditions of the Ritz method The functions, which are essential for accurately solving problems involving frequency (ω) and the imaginary unit (i² = -1), are detailed in Table 4.1, specifically for S-S, C-C, and C-F boundary conditions These innovative approximation functions aim to enhance the effectiveness of the Ritz method by ensuring compliance with specified boundary conditions.

Table 4.1 Approximation functions and kinematic BCs of beams

The governing equations of motion can be obtained by substituting Eqs (4.19- 4.23) into Eq (4.18) and using Lagrange’s equations:

  (4.24) with q j representing the values of ( u 0 j , u 1 j , w 0 j , w 1 j , w 2 j ), that leads to:

(4.25) where the components of stiffness matrix K, mass matrix M and load vector F are given by:

L L L L ij i xx j xx i xx j i j xx i j

Finally, the static, vibration and buckling responses of composite beams can be determined by solving Eq (4.25).

Numerical results

This section presents convergence and verification studies to validate the accuracy of the current research The static analysis involves a beam subjected to a uniformly distributed load with density q, applied on the surface at z = -h/2 in the z-direction The laminates are assumed to have equal thicknesses and are constructed from identical orthotropic materials, as detailed in Table 4.2 To streamline the analysis, specific nondimensional terms are utilized.

Table 4.2 Material properties of laminated composite beams

The study evaluates the convergence of composite beams (MAT I.4, 0 0 /90 0 , L h/ 5,E 1 /E 2 40) under various boundary conditions Table 4.3 presents the nondimensional fundamental frequencies, critical buckling loads, and mid-span displacements in relation to the series number m The findings indicate that convergence is achieved at m = 12 for both natural frequency and critical buckling load, while m = 14 is necessary for deflection analysis Consequently, these series term values are selected for further analysis.

Table 4.3 Convergence studies for the nondimensional fundamental frequencies, critical buckling loads and mid-span displacements of (0 0 /90 0 ) composite beams (MAT I.4, L h/ 5, E1/E2 = 40)

The symmetric (0 0 /90 0 /0 0 )andun-symmetric (0 0 /90 0 ) beams with different length- to-height ratios and BCs are considered in this example The nondimensional fundamental frequencies, critical buckling loads and mid-span displacements

 xL / 2, z0  of beams are presented in Tables 4.4, 4.5 and 4.6, respectively The present results are compared with those from previous works using the HOBT [21,

The current findings align with previous research on both higher-order beam theory (HOBT) and quasi-3D theories Notably, discrepancies arise between the predictions of HOBT and quasi-3D for unsymmetric and thick beams with a length-to-height ratio of 5 Conversely, for slender beams with a length-to-height ratio of 50, the predictions from both theories are closely matched.

Table 4.4 Nondimensional fundamental frequencies of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams (MAT I.4, E1/E2 = 40)

Vo and Thai [21] 9.206 17.449 6.058 7.296 Murthy et al [24] 9.207 - 6.045 -

Khdeir and Reddy [104] 4.234 - 2.386 - Murthy et al [24] 4.230 - 2.378 -

Mantari and Canales [48] 4.221 - 2.375 - c C-C boundary condition

Khdeir and Reddy [104] 11.603 - 10.026 - Murthy et al [24] 11.602 - 10.011 -

Table 4.5 Nondimensional critical buckling loads of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams (MAT I.4, E1/E2 = 40)

Mantari and Canales [48] 8.585 - 3.856 - b C-F boundary condition

Mantari and Canales [48] 4.673 - 1.221 - c C-C boundary condition

Table 4.6 Nondimensional mid-span displacements of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams under a uniformly distributed load (MAT II.4)

Murthy et al [24] 2.398 1.090 0.661 4.750 3.668 3.318 Khdeir and Reddy

Murthy et al [24] 6.836 3.466 2.262 15.334 12.398 11.392 Khdeir and Reddy

Table 4.7 presents the nondimensional axial, transverse shear, and transverse normal stresses of simply supported beams with a length-to-height ratio (L/h) of 5 and 50 These results are compared to the solutions obtained by Vo and Thai [22] using the Higher Order Beam Theory (HOBT) as well as those by Mantari and Canales.

[20], Zenkour [36] using quasi-3D theory Good agreements with the previous models are also found

Table 4.7 Nondimensional stresses of (0 0 /90 0 /0 0 ) and (0 0 /90 0 ) composite beams with S-S boundary condition under a uniformly distributed load (MAT II.4)

Vo and Thai [22] 0.4057 - 0.9187 - Quasi-3D Present 0.4013 0.4521 0.9052 0.9869

The distribution of nondimensional transverse displacements across the thickness direction for L h /  5,10, 50 is displayed in Figs 4.1-4.3 It is observed that

The nonlinear variation of transverse displacement is prominently observed in thick beams with a length-to-height ratio of 5, regardless of boundary conditions Figure 4.1 illustrates the distribution of nondimensional transverse displacement across the thickness of composite beams with specific boundary conditions, namely (0°/90°) and (0°/90°/0°), under simply supported (S-S) conditions.

Figure 4.2 Distribution of nondimensional transverse displacement through the thickness of (0 0 /90 0 ) and (0 0 /90 0 /0 0 ) composite beams with C-F boundary condition

64 a 0 0 /90 0 b 0 0 /90 0 /0 0 Figure 4.3 Distribution of nondimensional transverse displacement through the thickness of (0 0 /90 0 ) and (0 0 /90 0 /0 0 ) composite beams with C-C boundary condition

This example builds upon previous findings, focusing on the (0 / 0 / 0 0) and (0 / 0 ) beams Tables 4.8 through 4.11 illustrate the variations in nondimensional fundamental frequencies, critical buckling loads, mid-span displacements (at x = L / 2, z = 0), and stresses of beams analyzed using Quasi-3D theory relative to the angle-ply configuration Notably, the results presented in Tables 4.10 and 4.11 closely align with those reported by Vo et al [19].

Table 4.8 Nondimensional fundamental frequencies of (0 0 / /0 0 ) and (0 0 / ) composite beams (MAT I.4, E1/E2 = 40)

Lay-up BC L/h Angle-ply ()

Table 4.9 Nondimensional critical buckling loads of (0 0 / /0 0 ) and (0 0 / ) composite beams (MAT I.4, E1/E2 = 40)

Lay-up BC L/h Angle-ply ()

Table 4.10 Nondimensional mid-span displacements of (0 0 / /0 0 ) and (0 0 / ) composite beams under a uniformly distributed load (MAT II.4)

Lay-up BC L/h Reference Angle-ply ()

Table 4.11 Nondimensional stresses of (0 0 / /0 0 ) and (0 0 / ) composite beams with

S-S boundary condition under a uniformly distributed load (MAT II.4)

Lay-up L/h Reference Angle-ply ()

Vo et al [19] 0.7523 0.3633 0.2449 0.2358 b Shear stress

Figs 4.4-4.6 show the displacements of the (0 / 0 / 0 0 ) and (0 / 0 ) thick beams

 L h / 3  increase with the increase of angle-ply There are significant differences between the results of HOBT and quasi-3D solutions

Figure 4.4 The nondimensional mid-span transverse displacement with respect to the fiber angle change of composite beams with S-S boundary condition (L h/ 3,

N o n d im e n s io n a l t ra n s v e rs e d is p la c e m e n t

Figure 4.5 The nondimensional mid-span transverse displacement with respect to the fiber angle change of composite beams with C-F boundary condition (L h/ 3, MAT

 o N o n d im e n s io n a l t ra n s v e rs e d is p la c e m e n t HOBT, 0 o / o /0 o

Figure 4.6 The nondimensional mid-span transverse displacement with respect to the fiber angle change of composite beams with C-C boundary condition (L h/ 3, MAT

This study analyzes the behavior of composite beams with arbitrary ply configurations, focusing on symmetric single-layered C-F beams with 15° and 30° ply orientations (MAT III.4) The first four natural frequencies of these beams are presented in Table 4.12 and compared with findings from Chen et al [87] and experimental results by Abarcar and Cunniff [131] The results demonstrate a strong consistency, particularly in the first mode of vibration, between the current study and the referenced works.

N o n d im e n s io n a l t ra n s v e rs e d is p la c e m e n t

Table 4.12 Fundamental frequencies (Hz) of single-layer composite beam with C-F boundary condition (MAT III.4)

Chen et al [87] 82.55 515.68 1437.02 - Experiment Abarcar and Cunniff [131] 82.50 511.30 1423.40 1526.90 *

Chen et al [87] 52.73 330.04 922.45 1803.01 Experiment Abarcar and Cunniff [131] 52.70 331.80 924.70 1766.90

Note: ‘*’ denotes: the results are the torsional mode

Table 4.13 Nondimensional fundamental frequencies of arbitrary-ply laminated composite beams (MAT IV.4)

Lay-up Theory Reference BC

Chen et al [87] report the analysis of un-symmetric beams with various boundary conditions, specifically the (45°/-45°/45°/-45°) and (30°/-50°/50°/-30°) configurations The responses of these beams on fundamental frequencies are detailed in Table 4.13, demonstrating a strong correlation between the current theoretical findings and previous studies.

74 previous studies are again found Finally, the symmetric   /    s composite beams (MAT IV.4) are considered

Table 4.14 illustrates the impact of angle-ply variation on frequency, buckling, and displacement, while Fig 4.7 presents the nondimensional fundamental frequencies The results indicate that the current frequencies align more closely with those from references [46, 87] and are lower than those from [81, 133], which did not account for the Poisson effect, particularly in beams with arbitrary-ply configurations This discrepancy arises because the present study incorporates the Poisson effect into the constitutive equations by assuming that σy = σxy = σyz = 0, indicating that the strains (εy) are considered in the analysis.

The Poisson's effect significantly influences the stiffness of arbitrary-ply laminated beams, making them more flexible compared to when this effect is ignored This finding highlights the importance of considering the Poisson's effect, particularly for arbitrary-ply configurations, while its neglect may be acceptable for cross-ply laminated beams Additionally, there is a noted discrepancy between the critical buckling loads reported in this study and those from Wang et al [33], attributed to differences in the displacement field, specifically regarding the rotation of the normal to the mid-plane in the y-direction.

Table 4.14 Nondimensional fundamental frequencies, critical buckling loads and mid-span displacements of   /    s composite beams (MAT IV.4)

BC Theory Reference Angle-ply ()

Nguyen et al [133] 2.656 2.103 1.012 0.732 FOBT Chandrashekhara et al [81] 2.656 2.103 1.012 0.732

BC Theory Reference Angle-ply ()

Chen et al [87] 4.858 2.345 1.671 1.623 b Critical buckling load

FOBT Wang et al [33] 30.592 10.008 3.187 3.136 c Mid-span displacement

Figure 4.7 Effects of the fiber angle change on the nondimensional fundamental frequency of   /    s composite beams (MAT IV.4).

Conclusions

This article introduces new approximation functions that integrate polynomial and exponential models to analyze the free vibration, buckling, and static behaviors of laminated composite beams Utilizing a quasi-3D theory, the displacement field accounts for higher-order variations in both axial and transverse displacements, while also incorporating Poisson’s effect into the beam model Numerical results are presented for various boundary conditions to compare with existing studies and to explore the impact of material anisotropy, Poisson’s ratio, and angle-ply configurations on the natural frequencies, buckling loads, displacements, and stresses of composite beams The findings reveal significant insights into these dynamic behaviors.

- The transverse normal strain effects are significant for un-symmetric and thick beams o

S-S, HOBT S-S, Quasi-3D C-F, Nguyen et al [133]

C-F, HOBT C-F, Quasi-3D C-C, Nguyen et al [133]

- The Poisson’s effect is quite significant to the laminated beams with arbitrary lay-up, and the neglecting of this effect is only suitable for the cross-ply laminated beams

- The present model is found to be appropriate for vibration, buckling and bending analysis of cross-ply and arbitrary-ply composite beams

SIZE DEPENDENT BEHAVIOURS OF MICRO GENERAL

Introduction

Chapters Two, Three, and Four analyze macro composite beams through classical continuum mechanics theories However, the recent application of composite materials with microstructures in micro-electro-mechanical systems, like microswitches and microrobots, has prompted researchers to investigate material behavior at micron and sub-micron scales Findings indicate that classical continuum mechanics theories are inadequate for accurately describing the behavior of these micro-structures due to their size-dependent characteristics.

A comprehensive review of non-classical continuum mechanics models for size-dependent analysis of small-scale structures is available in [137] These models are categorized into three primary groups: nonlocal elasticity theory, micro continuum theory, and the strain gradient family Nonlocal elasticity theory, introduced by Eringen [3, 138] and further developed with Eringen and Edelen [139], captures size effects through constitutive equations that relate the stress at a reference point to the strain field across the entire body, utilizing a nonlocal parameter The micro continuum theory, also developed by Eringen [144-146], allows each particle to rotate and deform independently of its centroid's motion The strain gradient family includes strain gradient theory [136, 147], modified strain gradient theory [134], couple stress theories [4-6], and the modified couple stress theory (MCST) [148], where both strains and strain gradients are incorporated into the strain energy, with size effects represented by material length scale parameters (MLSP) Notably, the MCST simplifies the model by requiring only one MLSP instead of two, ensuring the couple stress tensor remains symmetric through an equilibrium condition of moments of couples.

The Modified Continuum Strain Theory (MCST) simplifies the analysis of size effects in materials by effectively addressing the complexities associated with determining the Material Length Scale Parameter (MLSP), making it a more favorable choice among various theoretical approaches.

Chen et al developed Timoshenko and Reddy beam models to examine the static behaviors of cross-ply simply supported microbeams Additionally, Chen and Si proposed an anisotropic constitutive relation for the MCST and employed global-local theory to analyze Reddy beams through Navier solutions Roque et al utilized a meshless method to assess the static bending response of micro laminated Timoshenko beams Furthermore, Yang et al introduced a size-dependent zigzag model for the bending analysis of cross-ply microbeams.

Recent studies have examined the buckling and dynamic behaviors of micro composite beams using various models, including Euler-Bernoulli and Timoshenko approaches For instance, Mohammadabadi et al investigated the thermal effects on size-dependent buckling behavior, employing the generalized differential quadrature method under different boundary conditions Similarly, Chen and Li focused on the dynamic responses of micro laminated Timoshenko beams, while Mohammad-Abadi and Daneshmehr analyzed the free vibration of cross-ply microbeams using multiple beam models Additionally, Ghadiri et al explored the thermal impacts on the dynamics of both thin and thick microbeams with varying boundary conditions However, most research has predominantly centered on cross-ply microbeams, highlighting the need for further investigation into micro general laminated composite beams (MGLCB) with arbitrary lay-ups.

Despite the fact that numerical approaches are used increasingly [10, 21, 50, 140,

141, 154], Ritz method is still efficient to analyse structural behaviours of beams [44,

The accuracy and efficiency of the Ritz method are heavily influenced by the selection of approximation functions Choosing inappropriate functions can lead to slow convergence rates and numerical instabilities It is crucial that these approximation functions meet the specified essential boundary conditions; failure to do so may result in complications such as the need for Lagrangian multipliers and penalties.

The 80 method is effective for addressing arbitrary boundary conditions, but it results in larger stiffness and mass matrices, leading to increased computational costs This chapter aims to introduce approximation functions for Ritz-type solutions that ensure rapid convergence, numerical stability, and compliance with the specified boundary conditions.

This chapter introduces innovative exponential approximation functions for the size-dependent analysis of micro-glass fiber composite beams (MGLCB) utilizing a refined shear deformation theory based on the Modified Classical Shear Theory (MCST) The governing equations of motion are derived using Lagrange’s equations, ensuring a solid theoretical foundation The model's accuracy is validated through comprehensive verification studies Numerical results are provided to explore the impact of various parameters, including micro-length scale parameter (MLSP), length-to-height ratio, and fiber angle, on the deflections, stresses, natural frequencies, and critical buckling loads of micro composite beams with diverse lay-up configurations.

The article discusses a rectangular cross-section MGLCB, defined by its length (L), width (b), and thickness (h) It consists of multiple plies made from orthotropic materials, each oriented at varying fiber angles relative to the x-axis The x'-axis aligns with the fiber orientation, while the y'-axis is perpendicular to it, providing a comprehensive understanding of the material's structural properties.

Figure 5.1 Geometry and coordinate of a laminated composite beam

This chapter employs Higher-Order Beam Theory (HOBT) to analyze beam behavior, incorporating a displacement field that allows for higher-order variations in axial displacement This approach effectively satisfies the traction-free boundary conditions for transverse shear stress on both the top and bottom surfaces of the beams.

The equation \( w(x, t) = 0 \) describes the axial and transverse displacements of a point on the mid-plane of a beam, represented by \( u(x, t) \) and \( w(x, t) \) in the x- and z-directions, respectively Additionally, \( u_1(x, t) \) indicates the rotational angle at a specific point along the mid-plane of the beam.

( ) 4 3 z z f z   h is the shape function The comma indicates a partial differentiation with respect to the corresponding subscript coordinate

Based on the MCST [148, 152], the rotation displacement about the x-, y -, z -axes are determined by:

    (5.6) a y’-z plane fiber cross-section b Fiber within x’-z plane

Figure 5.2 Rotation displacement about the x’-, y’-axes

The strain and curvature-strain fields of beams are obtained as [37]:

The stress-strain relations for the k th -ply of a laminated beam in global coordinate system are expressed as [88]:

The couple stress moments-curvature relation for the k th -ply of a laminated beam can be given by [37]:

In the above formulas, C ij are elastic coefficients of orthotropic material [1]; m xy and m zy are the couple stress moments which are caused by rotation displacement;  kb ,

The micro-scale material parameter, denoted as  kb, characterizes the behavior of fibers rotating in the y-z plane, highlighting the interaction between the fiber's cross-section and the surrounding matrix This parameter plays a crucial role in understanding how fibers, considered impurities, influence the rotational equilibrium in the system Additionally, the parameters  km 1 and  km 2 represent the minimum local stability parameters (MLSPs) in the x, y, and z directions, further elucidating the dynamics at play.

83 represent the micro-scale material parameter within the matrix rotating about the impurity in the x ,  z and x , y , plane, respectively [6, 152]; m  cos k , n  sin k ,

 k is a fiber angle with respect to the x-axis

The strain energy, work done and kinetic energy are denoted by  E ,  W and  K respectively The strain energy  E of the beam is given by:

E 2 V   x x   xz xz m xy  xy m zy  zy dV

(5.15) where the stiffness coefficients of the beam are determined as follows:

The work done  W by axially compressive load N 0 and transverse load q is given by:

The kinetic energy  K of the beam is written by:

           (5.21) where the dot-superscript denotes the differentiation with respect to the time t;  is the mass density of each layer; I 0 ,I 1 ,I 2 ,J 1 ,J 2 ,K 2 are the inertia coefficients determined by:

The total potential energy of system is expressed by:

By using Ritz method, the displacement field in Eq (5.23) is approximated by:

This chapter introduces new exponential approximation functions for the Ritz solution, as detailed in Table 5.1, specifically designed for three typical boundary conditions: simply supported (S-S), cantilevered (C-F), and clamped (C-C) The variables involved include frequency (ω), the imaginary unit (i² = -1), and unknowns u₀j, w₀j, and u₁j, which need to be determined, along with the approximation functions φj(x).

By substituting Eq (5.24-5.26) into Eq (5.23) and using Lagrange’s equations:

  (5.27) with p j representing the values of  u 0 j , w 0 j , u 1 j , the static, vibration and buckling behaviour of MGLCB can be obtained by solving the following equations:

Table 5.1 Approximation functions and essential BCs of beams

1 a 0 u  , w 0, x  0 where the components of stiffness matrix K , mass matrix M and load vector F are given by:

Numerical results

Convergence and verification studies are essential for confirming the accuracy of the current research The laminates analyzed consist of identical orthotropic materials with uniform thicknesses, as detailed in Table 5.2 The beam is subjected to either a uniformly distributed load (q = q₀) or a sinusoidal load (q = q₀ sin x).

  Unless otherwise stated, the following non-dimensional terms are used:

The numerical results of marco composite beams can be achieved by setting

Turning a composite laminated beam around the fiber direction is simpler than rotating it around the normal to the fiber direction Additionally, the maximum longitudinal shear stress (MLSP) in the fiber direction exceeds that in other orientations.

    Therefore, only the MLSP in the fiber direction is considered in this study, i.e  kb  b  and  km 1  km 2  0 The value of the MLSP is referred from

Table 5.2 Material properties of laminated composite beams considered in this study

Material properties MAT I.5 [47] MAT II.5 [7] MAT III.5 [81]

The study evaluates the convergence of composite beams (MAT I.5, 0 0 /90 0 /0 0 , L h/ 5) under various boundary conditions by analyzing non-dimensional fundamental frequencies, critical buckling loads, and mid-span displacements, as detailed in Table 5.3 Results show that the convergence point for natural frequency, critical buckling load, and displacement occurs at m = 6, which is adopted for further analysis Notably, the convergence of the proposed solution is faster compared to the solution from reference [155] (m = 12), highlighting the advantages of the proposed approximation function.

Table 5.3 Convergence studies for (0 / 90 / 0 0 0 0 ) composite beams (MAT I.5, L h/ 5)

To validate the current solution, an investigation was conducted on simply supported beams, focusing specifically on cross-ply configurations due to the lack of published data for micro composite beams with arbitrary lay-ups The critical buckling loads and fundamental frequencies of the beams with lay-ups (90/0/90) and (0/90/0) were analyzed and presented in Figures 5.3 and 5.4 These results were compared with the analytical solutions provided by Abadi and Daneshmehr, as well as Mohammad-Abadi and Daneshmehr The findings indicate a strong agreement between the current results and the established outcomes for both Timoshenko and Reddy beam models.

89 a 90 / 0 / 90 0 0 0 b 0 / 90 / 0 0 0 0 Figure 5.4 Comparison of fundamental frequencies of S-S beams (MAT I.5) Further verification is illustrated in Fig 5.5 for  90 / 0 / 90 0 0 0  beams (MAT II.5) with  b h and L=4h, subjected to sinusoidal loads 0 sin x q q

  It should be noted that the non-dimensional forms of the displacement and axial stress are

  , respectively There is a slight discrepancy between the present results and those of Yang et al [7] a Non-dimensional displacement b Non-dimensional normal stress

N o n -d im e n s io n a l d is p la c e m e n t N o n -d im e n s io n a l a x ia l s tr e s s

Figure 5.5 Comparison of displacement and normal stress of  90 / 0 / 90 0 0 0  S-S beams (MAT II.5)

This section examines the static behaviors of the MGLCB with different boundary conditions (BCs) and length-to-height ratios Tables 5.4-5.6 present the non-dimensional mid-span displacements of beams (MAT II.5) subjected to a uniform load (q = q0) at various positions (ξb = 0, h/4, h/2, h) The results indicate that the displacements of the beams decrease as the boundary conditions and length-to-height ratios vary.

The parameter \( \xi_b \) increases across all boundary conditions (BCs) and span-to-thickness ratios When \( \xi_b = 0 \), the findings for macro beams align closely with the results presented by Vo et al [19], which were derived from finite element modeling and higher-order beam theory.

Table 5.4 Displacement of S-S beams (MAT II.5)

L h Lay-ups  Vo et al [19] Present b 0

Table 5.5 Displacement of C-F beams (MAT II.5)

L h Lay-ups  Vo et al [19] Present b 0

Table 5.6 Displacement of C-C beams (MAT II.5)

L h Lay-ups  Vo et al

In the next example, micro composite beams under sinusoidal loads,

The investigation of beams with varying mid-span lengths and boundary conditions reveals that their deflections decrease as the parameter \( \xi_b \) increases Figures 5.6 to 5.8 illustrate this trend, while Figure 5.9 presents the mid-span displacements for beams with different boundary conditions, specifically \( (0/30/0 0 0 0) \) and \( (0/30 0 0) \) Notably, the variation in displacement is influenced by the boundary conditions, with the clamped-free (C-F) beam exhibiting the most significant changes as \( \xi_b/h \) increases.

N o n- di m en s ion al di s p lacem en t

N on -d imensi ona l d ispl a c e m ent

94 c 0 / 45 / 0 0 0 0 d 0 / 45 0 0 Figure 5.6 Effect of MLSP on displacements of S-S beams (MAT II.5, L h/ 4) a 0 / 90 / 0 0 0 0 b 0 / 90 0 0 c 0 / 45 / 0 0 0 0 d 0 / 45 0 0

N o n -d im e nsi o nal d ispl a cem e nt

N on- di m ens iona l d ispl ace m e nt

N on -di m en s ion al di s p la c e m ent

N o n -d im en si o na l di spl a ce m e nt  b = 0

Figure 5.7 Effect of MLSP on displacements of C-F beams (MAT II.5, L h/ 4) a 0 / 90 / 0 0 0 0 b 0 / 90 0 0 c 0 / 45 / 0 0 0 0 d 0 / 45 0 0 Figure 5.8 Effect of MLSP on displacements of C-C beams (MAT II, L h/ 4)

N on -di m e nsi o nal di spl ace m en t

N on- di m e n si o n al di spl acem e n t

N o n -d im e n s io n a l m id -s p a n d is p la c e m e n t

The impact of MLSP on the displacements of beams with different boundary conditions is illustrated in Figure 5.9, specifically for a length-to-height ratio of L/h = 4 Figures 5.10 and 5.11 present the axial and shear stresses for simply supported beams, denoted as (0/60/0) and (0/60) As observed, both displacement and stress levels decrease with an increase in MLSP.

Figure 5.10 Effect of MLSP on through-thickness distribution of stresses of

(0 / 60 / 0 0 0 0 ) S-S beams ( MAT II.5, L h/ 4) a  xx b  xz Figure 5.11 Effect of MLSP on through-thickness distribution of stresses of

Tables 5.7-5.10 present the non-dimensional fundamental frequencies and critical buckling loads of the MGLCB, considering various boundary conditions (BCs) and length-to-height ratios The findings for macro composite beams (where  b = 0) show a strong correlation with previous results, confirming the reliability of the current analysis.

Recent studies by Vo et al and Chen et al present new findings on micro composite beams, establishing benchmarks for future research The results indicate that as the parameter \( \xi_b \) increases, there is a corresponding rise in the beams' stiffness due to the enhanced Multi-Layered Structural Performance (MLSP).

Table 5.7 Fundamental frequencies of (/) beams (MAT III.5)

Table 5.8 Fundamental frequencies of (0 / 0  ) beams (MAT III.5)

Table 5.9 Buckling loads of (/) beams (MAT III.5)

Table 5.10 Buckling loads of (0 / 0  ) beams (MAT III.5)

Figures 5.12 and 5.13 illustrate the relationship between natural frequencies and critical buckling loads in relation to the  b / h ratio for beams designated as (0 / 30 / 0 0 0 0) and (0 / 30 0 0) It is evident that as the  b / h ratio increases, the variation in these parameters is influenced by the boundary conditions, with the C-C beam exhibiting the most significant variation.

Figure 5.12 Effect of MLSP on frequencies of beams with various BC (MAT III.5,

N o n -d im e n s io n a l fu n d a m e n ta l fr e q u e n c y b /h

N o n -d im e n s io n a l fu n d a m e n ta l fr e q u e n c y

102 a 0 / 30 / 0 0 0 0 b 0 / 30 0 0 Figure 5.13 Effect of MLSP on buckling loads of beams with various BCs (MAT

Conclusions

This chapter investigates the size effect on the bending, vibration, and buckling behaviors of micro composite beams with arbitrary lay-ups, incorporating Poisson's effect into the constitutive equations for the first time The governing equations of motion are derived using Lagrange's equations, and new approximation functions are developed to address the problems The analysis yields critical insights into the frequencies, critical buckling loads, displacements, and stresses of micro composite beams under various boundary conditions, highlighting the significance of these findings.

- The size effect is significant for bending, buckling and free vibration analysis of micro laminated composite beams

- Beam model and constitutive behaviours used in this Chapter are suitable for analysis of micro laminated composite beams with arbitrary lay-ups

- The proposed functions are simple and efficient for predicting behaviours of micro laminated composite beams

N o n -d im e n s io n a l c ri ti c a l b u c k lin g l o a d N o n -d im e n s io n a l c ri ti c a l b u c k lin g l o a d

ANALYSIS OF THIN-WALLED LAMINATED COMPOSITE BEAMS

CONVERGENCY, ACCURARY AND NUMERICAL STABILITY OF

CONCLUSIONS AND RECOMMENDATIONS

Tiêu đề	Nghiên Cứu Ứng Xử Tĩnh, Ổn Định Và Dao Động Dầm Composite Với Tiết Diện Khác Nhau
Thể loại	thesis

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