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Abstract

The present study introduces a novel higher-order shear deformation theory for assessing buckling and free vibration characteristics in laminated composite and functionally graded porous beams. The proposed theoretical framework effectively considers three variables and eliminates the need for a shear correction factor. The governing equations are derived from the Lagrange principle, while Legendre-Ritz functions are utilised to solve the resulting problem. Various types of laminated composite beams with arbitrary lay-ups and functionally graded porous beams with symmetric or unsymmetric configurations are analysed. To validate the accuracy and efficiency of the proposed theory, several numerical examples are conducted and compared against the results of existing research endeavours.

Keywords

Laminated composite beams functionally graded porous beams higher-order shear deformation theory Ritz method buckling free vibration

Introduction

Laminated composite (LC) and functionally graded porous (FGP) materials are widely used in engineering fields due to their exceptional properties, such as high strength, low weight, and good heat and sound insulation.^1,2 Researchers have been keenly interested in these materials, developing many theories for analysing their mechanical responses.^3–6 Elasticity theories, which describe LC and FGP structure’s behaviour accurately, have been commonly used in past studies; for example, Pagano’s elasticity theory⁷ was presented for bending analysis of LC beams and plates, Sankar⁸ proposed an elasticity solution for FG beams subjected to transverse load, while Wang et al.⁹ invented an elasticity model for vibration and bending analysis of LC beams. Although the elasticity theory describes the beam’s responses precisely, it is computationally complex and difficult to apply to arbitrary boundary conditions or complicated geometries. As a result, several alternative methods have been proposed, including layer-wise theories,^10,11 equivalent single-layer theories (ESLT),^12–15 zig-zag theories,^16–19 and Carrera’s Unified Formulation.^20,21 Among these theories, the ESLT is widely utilised due to its formulation simplicity and programming ease. It can be classified into four categories: classical beam theory, first-order beam theory, higher-order beam theory, and Quasi-3D theory.

Classical beam theory (CBT) is one of the simplest forms of ESLT. Many studies have applied CBT to analyse the behaviours of beams. Özütok and Madenci²² studied the free vibration of LC beams using the finite element method. Ghasemi et al.²³ examined the nonlinear vibration of LC beams. Baghani et al.²⁴ analysed LC beams’ free vibration and buckling. El Harti et al.²⁵ studied the dynamic control of FGP beams. Mutlak et al.²⁶ examined dynamic response of FGP beams on the foundation under a moving load. Sari et al.²⁷ analysed the free and forced nonlinear vibrations of FGP beams, while Mirjavadi et al.²⁸ analysed buckling and vibration of nano FGP beams. However, due to the neglect of shear deformation in the beam model, the CBT tends to overestimate natural frequencies/critical buckling load and underestimate deflection. To address the limitations of the CBT, first-order beam theory (FBT) was proposed to analyse beam’s responses. Several studies utilised FBT to examine the mechanical behaviour of LC and FGP beams are mentioned here. Jun et al.²⁹ analysed dynamic of LC beams based on FEM, while Zhang et al.³⁰ examined the mechanical analysis of FG graphene oxide-reinforced composite beams. Abramovich and Livshits³¹ investigated the free vibration of cross-ply LC beams, and Fu et al.³² presented LC beams’ thermal buckling and postbuckling behaviours. Bending behaviour of thin-walled LC beams was presented by Vukasović et al.³³ Fridman and Abramovich³⁴ studied the structural behaviour of LC beams under axial compression using piezoelectric layers. Vosoughi et al.³⁵ presented a hybrid numerical method for optimising of LC beams. Chen et al.^36–39 analysed the buckling, free vibration and static behaviours of FGP beams, while Zhao et al.⁴⁰ presented free vibration of FGP straight and curved beams. Noori et al.⁴¹ presented the dynamic analysis of FGP beams based on the complementary functions method in the Laplace domain, and Gao et al.⁴² analysed the dynamic response of FGP beams. Notably, the FBT requires a shear correction factor to correct the strain energy under deformation, but selecting an appropriate value remains a challenging task for researchers. The limitations of FBT have spurred the development of higher-order beam theories (HBT). One such theory is the third-order shear deformation theory proposed for analysing the mechanical behaviour of laminated composite plates.^43,44 Levinson⁴⁵ and Murty⁴⁶ introduced polynomial shear functions to analyse the behaviour of beams, and this approach has been adopted in numerous studies. For example, Nguyen et al.^47–49 examined static, free vibration, and buckling behaviours of FGP beams, while Özütok Madenci⁵⁰ presented the static analysis of LC beams. Shi and Lam⁵¹ analysed the free vibration of composite beams using FEM. Song and Waas⁵² studied LC beams’ free vibration and buckling responses. Khdeir and Reddy^53,54 analysed LC beams’ free vibration and buckling behaviours. Wattanasakulpong et al.⁵⁵ analysed the free vibration of FGP beams. In addition to polynomial functions, other shear functions such as trigonometric,^56–58 hyperbolic functions,^59,60 and exponential functions^61,62 have also been developed for analysing LC and FGP beams. The accuracy of these theories depends on the specific shear functions utilised, making the development of suitable functions a topic of great interest among researchers, as noted in the review study.⁶³ It can be stated that the CBT, FBT, and HBT ignore normal transverse strain, which plays an essential role in thick beams. Therefore, to account for this effect, the Quasi-3D theory (Quasi-3D), which approximates axial and transverse displacements as high-order variations through the beam thickness, was proposed.^64–68 Although Quasi-3D can predict the behaviour of LC and FGP beams more accurately than CBT, FBT, and HBT, it involves more unknowns, making it more complex.

This article introduces a novel seventh-order shear deformation theory for analysing the free vibration and buckling responses of LC and FGP beams, considering both symmetric and unsymmetric distributions. The displacement fields are described using Legendre-Ritz functions, which facilitate solving problems with typical three boundary conditions. The governing equations are derived using Lagrange’s principle. Numerical examples are presented to evaluate the accuracy and efficiency of the proposed method. Furthermore, the impacts of span-to-height ratio, distribution types, and fibre orientation on the critical buckling load and fundamental frequency of beams are considered in greater detail.

Theoretical formulation

Constitutive relations

Laminated composite beam

The strain and stress relation of $k^{th}$ -layer for an orthotropic ply in global coordinates is given by⁵⁰:

{\begin{matrix} σ_{x} \\ σ_{xz} \end{matrix}}^{(k)} = (\begin{matrix} {\bar{Q}}_{11}^{(k)} & 0 \\ 0 & {\bar{Q}}_{55}^{(k)} \end{matrix}) {\begin{matrix} ε_{x} \\ γ_{xz} \end{matrix}}^{(k)}

(1)

where the ${\bar{Q}}_{11}^{(k)}$ and ${\bar{Q}}_{55}^{(k)}$ are the plane stress-reduced stiffness of $k^{th}$ -layer in global coordinates as follows:

\begin{matrix} {\bar{Q}}_{11}^{(k)} = Q_{11}^{(k)} \cos^{4} θ^{(k)} + 2 (Q_{12}^{(k)} + 2 Q_{66}^{(k)}) \sin^{2} θ^{(k)} \cos^{2} θ^{(k)} \\ + Q_{22}^{(k)} \sin^{4} θ^{(k)} \end{matrix}

(2a)

{\bar{Q}}_{55}^{(k)} = Q_{44}^{(k)} \sin^{2} θ^{(k)} + Q_{55}^{(k)} \cos^{2} θ^{(k)}

(2b)

where $Q_{11}^{(k)} = \frac{E_{1}^{(k)}}{1 - ν_{12}^{(k)} ν_{21}^{(k)}}$ , $Q_{12}^{(k)} = \frac{ν_{12}^{(k)} E_{2}^{(k)}}{1 - ν_{12}^{(k)} ν_{21}^{(k)}}$ , $Q_{22}^{(k)} = \frac{E_{2}^{(k)}}{1 - ν_{12}^{(k)} ν_{21}^{(k)}}$ , $Q_{44}^{(k)} = G_{23}^{(k)}$ , $Q_{55}^{(k)} = G_{13}^{(k)}$ , $Q_{66}^{(k)} = G_{12}^{(k)}$ ; $θ^{(k)}$ is the angle between the fibre direction and x-axis; $E_{1}^{(k)}$ and $E_{2}^{(k)}$ are Young’s modulus of the material in $x_{1}$ and $x_{2}$ directions, respectively; $ν_{12}^{(k)}$ and $ν_{21}^{(k)}$ are Poisson ratio; $G_{23}^{(k)}$ , $G_{13}^{(k)}$ , and $G_{12}^{(k)}$ are the shear moduli in the $x_{2} x_{3}$ , $x_{1} x_{3}$ , and $x_{1} x_{2}$ plane, respectively.

Functionally graded porous beam

The strain and stress relation of FGP beams are written as follows^37,48:

{\begin{matrix} σ_{x} \\ σ_{xz} \end{matrix}} = (\begin{matrix} {\bar{Q}}_{11} & 0 \\ 0 & {\bar{Q}}_{55} \end{matrix}) {\begin{matrix} ε_{x} \\ γ_{xz} \end{matrix}}

(3a)

{\bar{Q}}_{11} = \frac{E (z)}{1 - ν^{2}}

(3b)

{\bar{Q}}_{55} = \frac{E (z)}{2 (1 + v)}

(3c)

where $v$ is Poisson’s ratio; $E (z)$ is Young’s modulus. In this study, the symmetric and unsymmetric distributions of beams displayed in Figure 1 are investigated as follows³⁹:

Figure 1.

Two porous beams: (a) SPD: Porosity is symmetric distribution and (b) APD: Porosity is asymmetric distribution.⁴⁷

Symmetric porosity distribution (SPD)

E (z) = E_{1} [1 - r_{0} \cos (\frac{π z}{h})]

(4a)

ρ (z) = ρ_{1} [1 - r_{m} \cos (\frac{π z}{h})]

(4b)

Asymmetric porosity distribution (APD)

E (z) = E_{1} [1 - r_{0} \cos (\frac{π z}{2 h} + \frac{π}{4})]

(5a)

ρ (z) = ρ_{1} [1 - r_{m} \cos (\frac{π z}{2 h} + \frac{π}{4})]

(5b)

where: $ρ (z)$ is mass density; $ρ_{1}$ and $E_{1}$ are mass density and Young’s modulus’ maximum values; $r_{m} = 1 - \sqrt{1 - r_{0}}$ and $r_{0}$ are the porosity coefficient for mass density and porosity parameters.

Kinematics

Consider LC and FGP beams with height h, width b, and length L as shown in Figures 2 and 3. The displacement fields of beams based on the higher-order shear deformation theory are introduced as follows⁴⁴:

\begin{matrix} u_{1} (x, z, t) = u (x, t) - z \frac{\partial w (x, t)}{\partial x} + f (z) φ (x, t) = u (x, t) \\ - z w_{, x} (x, t) + f (z) φ (x, t) \end{matrix}

(6a)

u_{3} (x, z, t) = w (x, t)

(6b)

where $u (x, t)$ , $w (x, t)$ , and $φ (x, t)$ denote the displacement along x-, z-axis and the rotational angle at the point in the mid-plane. The comma indicates partial differentiation with respect to the coordinate subscript that follows. $f (z) = z - \frac{z^{3}}{h^{2}} - \frac{z^{5}}{h^{4}} + \frac{4 z^{7}}{7 h^{6}}$ is the proposed shear shape function, which determines the distribution of transverse shear strains and stresses throughout the beam’s thickness. It can be seen that the shear shape function is the odd power polynomial. The first derivative of the shear shape function must give a parabolic response in the thickness direction of the beam and satisfy the free-traction boundary conditions on the top and bottom surfaces of the beam, $σ_{xz} (\pm h / 2) = 0$ .^60,61

Figure 2.

Geometry of laminated composite beams.

Figure 3.

Geometry of functionally graded porous beams.⁴⁸

The non-zero strain of beams is written as follows:

ε_{x} = \frac{\partial u_{1}}{\partial x} = u_{, x} - z w_{, xx} + f φ_{, x}

(7a)

γ_{xz} = \frac{\partial u_{1}}{\partial z} + \frac{\partial u_{3}}{\partial x} = f_{, z} φ

(7b)

Variational formulation

Strain energy

The strain energy of the beam is introduced⁶⁹:

\begin{matrix} Π_{E} = \frac{1}{2} \int_{V} (σ_{xx} ε_{xx} + σ_{xz} γ_{xz}) dV \\ = \frac{1}{2} \int_{0}^{L} [A {(u_{, x})}^{2} - 2 B u_{, x} w_{, xx} + 2 B^{s} u_{, x} φ_{, x} + D {(w_{, xx})}^{2} \\ - 2 D^{s} w_{, xx} φ_{, x} + H^{s} {(φ_{, x})}^{2} + A^{s} {(φ)}^{2}] dx \end{matrix}

(8)

For laminated composite beam:

(A, B, D, B^{s}, D^{s}, H^{s}) = \sum_{k = 1}^{n} \int_{z_{k}}^{z_{k + 1}} (1, z, z^{2}, f, zf, f^{2}) b \bar{Q}_{11}^{(k)} dz

(9a)

A^{s} = \sum_{k = 1}^{n} \int_{z_{k}}^{z_{k + 1}} {(f_{, z})}^{2} b \bar{Q}_{55}^{(k)} dz

(9b)

n is the number of laminates, $z_{k}$ and $z_{k + 1}$ are coordinate at bottom and top of every laminate.

For functionally graded porous beam:

(A, B, D, B^{s}, D^{s}, H^{s}) = \int_{- h / 2}^{h / 2} (1, z, z^{2}, f, zf, f^{2}) b {\bar{Q}}_{11} dz

(10a)

A^{s} = \int_{- h / 2}^{h / 2} {(f_{, z})}^{2} b {\bar{Q}}_{55} dz

(10b)

Work done by axial compression

The work done by axial compression force $N_{0}$ is determined⁶⁹:

Π_{V} = \frac{1}{2} \int_{0}^{L} N_{0} {(w_{, x})}^{2} dx

(11)

Kinetic energy

The kinetic energy of the beam is written⁶⁹:

\begin{matrix} Π_{K} = \frac{1}{2} \int_{V} ρ (z) [{(\frac{\partial u_{1}}{\partial t})}^{2} + {(\frac{\partial u_{3}}{\partial t})}^{2}] dV \\ = \frac{1}{2} \int_{0}^{L} [I_{0} {(\overset{\cdot}{u})}^{2} - 2 I_{1} \overset{\cdot}{u} {\overset{\cdot}{w}}_{, x} + I_{2} {({\overset{\cdot}{w}}_{, x})}^{2} + 2 J_{1} \overset{\cdot}{φ} \overset{\cdot}{u} - 2 J_{2} \overset{\cdot}{φ} {\overset{\cdot}{w}}_{, x} + K_{2} {(\overset{\cdot}{u})}^{2} + I_{0} {(\overset{\cdot}{w})}^{2}] dx \end{matrix}

(12)

where (.) denotes the time derivative

For laminated composite beam:

(I_{0}, I_{1}, I_{2}, J_{1}, J_{2}, K_{2}) = \sum_{k = 1}^{n} \int_{z_{k}}^{z_{k + 1}} ρ^{(k)} (1, z, z^{2}, f, zf, f^{2}) bdz

(13)

For functionally graded porous beam:

(I_{0}, I_{1}, I_{2}, J_{1}, J_{2}, K_{2}) = \int_{- h / 2}^{h / 2} ρ (z) (1, z, z^{2}, f, zf, f^{2}) bdz

(14)

Total energy

The beam’s total energy is written⁶⁹:

Π = Π_{E} + Π_{V} - Π_{K}

(14)

\begin{matrix} Π = \frac{1}{2} \int_{0}^{L} [A {(u_{, x})}^{2} - 2 B u_{, x} w_{, xx} + 2 B^{s} u_{, x} φ_{, x} + D {(w_{, xx})}^{2} - 2 D^{s} w_{, xx} φ_{, x} \\ + H^{s} {(φ_{, x})}^{2} + A^{s} {(φ)}^{2}] dx + \frac{1}{2} \int_{0}^{L} N_{0} {(w_{, x})}^{2} dx \\ - \frac{1}{2} \int_{0}^{L} [I_{0} {(\overset{\cdot}{u})}^{2} - 2 I_{1} \overset{\cdot}{u} {\overset{\cdot}{w}}_{, x} + I_{2} {({\overset{\cdot}{w}}_{, x})}^{2} + 2 J_{1} \overset{\cdot}{φ} \overset{\cdot}{u} - 2 J_{2} \overset{\cdot}{φ} {\overset{\cdot}{w}}_{, x} + K_{2} {(\overset{\cdot}{u})}^{2} + I_{0} {(\overset{\cdot}{w})}^{2}] dx \end{matrix}

(15)

Ritz solution

Ritz method is used to describe the displacement fields of the beam as follows^47,69:

u (x, t) = \sum_{j = 1}^{m} ς_{j, x} (x) u_{0 j} e^{i ω t}

(16a)

w (x, t) = \sum_{j = 1}^{m} ς_{j} (x) w_{0 j} e^{i ω t}

(16b)

φ (x, t) = \sum_{j = 1}^{m} ς_{j, x} (x) φ_{0 j} e^{i ω t}

(16c)

where $i^{2} = - 1$ ; $ω$ is the eigen-frequency, $u_{0 j}$ , $w_{0 j}$ , and $φ_{0 j}$ are unknown parameters need to be determined; $ς_{j} (x)$ are the Legendre-Ritz functions which presented in Nguyen et al.⁴⁷ for three typical boundary conditions (BCs): clamped-free (C-F), simply-supported (S-S), and clamped-clamped (C-C) as displayed in Table 1.

Table 1.

Approximation functions and essential BCs of beams.⁴⁷

BC	$ς_{j} (x)$	x = 0	x = L
S-S	$[L_{j} (1 - \frac{x}{L}) - 1] [L_{j} (\frac{x}{L}) - 1]$	$w = 0$	$w = 0$
C-F	${[L_{j} (1 - \frac{x}{L}) - 1]}^{2}$	$u = w = φ = 0$ , $w_{, x} = 0$
C-C	${[L_{j} (1 - \frac{x}{L}) - 1]}^{2} {[L_{j} (\frac{x}{L}) - 1]}^{2}$	$u = w = φ = 0$ , $w_{, x} = 0$	$u = w = φ = 0$ , $w_{, x} = 0$

Substituting equation (16) into equation (15) and using Lagrange’s principle⁶⁹:

\frac{\partial Π}{\partial p_{j}} - \frac{d}{dt} \frac{\partial Π}{\partial {\overset{\cdot}{p}}_{j}} = 0

(17)

with $p_{j}$ representing the values of $(u_{0 j}, w_{0 j}, φ_{0 j})$ , the buckling and free vibration behaviours of the beams can be obtained:

\begin{matrix} ([\begin{matrix} K^{11} & K^{12} & K^{13} \\ ^{T} K^{12} & K^{22} & K^{23} \\ ^{T} K^{13} & ^{T} K^{23} & K^{33} \end{matrix}] - ω^{2} [\begin{matrix} M^{11} & M^{12} & M^{13} \\ ^{T} M^{12} & M^{22} & M^{23} \\ ^{T} M^{13} & ^{T} M^{23} & M^{33} \end{matrix}]) \\ {\begin{matrix} u \\ w \\ φ \end{matrix}} = {\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}} \end{matrix}

(18)

The components of stiffness matrix K and mass matrix M are as follows:

\begin{matrix} K_{ij}^{11} = A \int_{0}^{L} ς_{i, xx} ς_{j, xx} dx, K_{ij}^{12} = - B \int_{0}^{L} ς_{i, xx} ς_{j, xx} dx, \\ K_{ij}^{13} = B^{s} \int_{0}^{L} ς_{i, xx} ς_{j, xx} dx \end{matrix}

(18)

\begin{matrix} K_{ij}^{22} = D \int_{0}^{L} ς_{i, xx} ς_{j, xx} dx + N_{0} \int_{0}^{L} ς_{i, x} ς_{j, x} dx, K_{ij}^{23} \\ = - D^{s} \int_{0}^{L} ς_{i, xx} ς_{j, xx} dx, \end{matrix}

(18)

K_{ij}^{33} = H^{s} \int_{0}^{L} ς_{i, xx} ς_{j, xx} dx + A^{s} \int_{0}^{L} ς_{i, x} ς_{j, x} dx,

(18)

\begin{matrix} M_{ij}^{11} = I_{0} \int_{0}^{L} ς_{i, x} ς_{j, x} dx, M_{ij}^{12} = - I_{1} \int_{0}^{L} ς_{i, x} ς_{j, x} dx, \\ M_{ij}^{13} = J_{1} \int_{0}^{L} ς_{i, x} ς_{j, x} dx, \end{matrix}

(18)

\begin{matrix} M_{ij}^{22} = I_{0} \int_{0}^{L} ς_{i} ς_{j} dx + I_{2} \int_{0}^{L} ς_{i, x} ς_{j, x} dx, \\ M_{ij}^{23} = - J_{2} \int_{0}^{L} ς_{i, x} ς_{j, x} dx, M_{ij}^{33} = K_{2} \int_{0}^{L} ς_{i, x} ς_{j, x} dx \end{matrix}

(19)

Numerical results

This section presents numerical examples to evaluate the efficiency and accuracy of the proposed theory. The LC beams have equal laminates with material properties $E_{1} / E_{2} = open$ , $G_{12} = G_{13} = 0.6 E_{2}$ , $G_{23} = 0.5 E_{2}$ , $ρ$ , and $ν_{12} = 0.25$ , while for FGP beams, the material properties are assumed to be $E_{1}$ , $ρ_{1}$ , and $ν$ . The non-dimensional parameters are employed as follows:

a) LC beam:

\bar{ω} = \frac{ω L^{2}}{h} \sqrt{\frac{ρ}{E_{2}}} and {\bar{N}}_{cr} = N_{0} \frac{L^{2}}{E_{2} b h^{3}}

(20)

b) FGP beam:

\bar{ω} = ω L \sqrt{\frac{ρ_{1} (1 - ν^{2})}{E_{1}}} and {\bar{N}}_{cr} = N_{0} \frac{2 (1 - ν^{2})}{E_{1} h}

(21)

Convergence study

To examine the convergence of the proposed solution, LC beams ( $E_{1} / E_{2} = 40$ , $L / h = 10$ , $0^{0} / 90^{0}$ ) and FGP beams ( $L / h = 10$ , $r_{0} = 0.5$ , APD) with various BCs are considered. The nondimensional fundamental frequency (NFF) and nondimensional critical buckling load (NCBL) of LC and FGP beams are presented in Tables 2 and 3, respectively. As demonstrated, the solution converges at m = 10 for both the fundamental frequency and critical buckling load for all BCs. Therefore, this point is implemented in the numerical examples.

Table 2.

Convergence studies for LC beams ( $E_{1} / E_{2} = 40$ , $L / h = 10$ , $0^{0} / 90^{0}$ ).

BC	m
	2	4	6	8	10	12	14
1. Nondimensional fundamental frequency
S-S	6.9399	6.9380	6.9380	6.9380	6.9380	6.9380	6.9380
C-F	2.5530	2.5428	2.5417	2.5415	2.5415	2.5415	2.5415
C-C	13.6965	13.6222	13.5988	13.5976	13.5974	13.5974	13.5974
2. Nondimensional critical buckling load
S-S	4.9343	4.9318	4.9318	4.9318	4.9318	4.9318	4.9318
C-F	1.3710	1.3256	1.3242	1.3237	1.3236	1.3236	1.3235
C-C	15.5226	15.5193	15.5193	15.5193	15.5193	15.5193	15.5193

Table 3.

Convergence studies for FGP beams ( $L / h = 10$ , $r_{0} = 0.5$ , ASP).

BC	m
	2	4	6	8	10	12	14
1. Nondimensional fundamental frequency
S-S	0.2550	0.2549	0.2549	0.2549	0.2549	0.2549	0.2549
C-F	0.0920	0.0917	0.0917	0.0917	0.0917	0.0917	0.0917
C-C	0.5500	0.5485	0.5478	0.5476	0.5476	0.5475	0.5475
2. Nondimensional critical buckling load
S-S	0.0108	0.0108	0.0108	0.0108	0.0108	0.0108	0.0108
C-F	0.0029	0.0028	0.0028	0.0028	0.0028	0.0028	0.0028
C-C	0.0398	0.0397	0.0397	0.0397	0.0397	0.0397	0.0397

LC beams

To verify the accuracy of the proposed theory, cross-ply ( $0^{0} / 90^{0} / 0^{0}$ and $0^{0} / 90^{0}$ ) LC beams ( $E_{1} / E_{2} = 40$ ) with various BCs and span-to-height ratios are considered. Tables 4 and 5 present NFF and NCBL of LC beams, respectively. The results obtained from the proposed theory are compared to those derived using HBT by Khdeir and Reddy,^53,54 Nguyen et al.,⁴⁹ and Quasi-3D by Mantari and Canales.^65,66 The outcomes demonstrate that the proposed theory agrees well with available results for all BCs and L/h ratios. Moreover, the NFF and NCBL obtained from the proposed theory are closer to those of Quasi-3D^65,66 than those of Nguyen et al.,⁴⁹ and Khdeir and Reddy^53,54 particularly for beams with L/h = 5.

Table 4.

Nondimensional fundamental frequencies of (0°/90°/0°) and (0°/90°) LC beams $(E_{1} / E_{2} = 40)$ .

BC	Lay-ups	Reference	Theory	L/h
				5	10	50
S-S	(0°/90°/0°)	Present	HBT	9.204	13.620	17.463
		Nguyen et al.⁴⁹	HBT	9.208	13.614	17.462
		Khdeir and Reddy⁵⁴	HBT	9.208	13.614	-
		Mantari and Canales⁶⁵	Quasi-3D	9.208	13.610	-
	(0°/90°)	Present	HBT	6.108	6.938	7.302
		Nguyen et al.⁴⁹	HBT	6.128	6.945	7.302
		Khdeir and Reddy⁵⁴	HBT	6.128	6.945	-
		Mantari and Canales⁶⁵	Quasi-3D	6.109	6.913	-
C-F	(0°/90°/0°)	Present	HBT	4.223	5.493	6.267
		Nguyen et al.⁴⁹	HBT	4.234	5.498	6.267
		Khdeir and Reddy⁵⁴	HBT	4.234	5.495	-
		Mantari and Canales⁶⁵	Quasi-3D	4.221	5.490	-
	(0°/90°)	Present	HBT	2.377	2.541	2.605
		Nguyen et al.⁴⁹	HBT	2.383	2.543	2.605
		Khdeir and Reddy⁵⁴	HBT	2.386	2.544	-
		Mantari and Canales⁶⁵	Quasi-3D	2.375	2.532	-
C-C	(0°/90°/0°)	Present	HBT	11.478	19.652	37.651
		Nguyen et al.⁴⁹	HBT	11.607	19.728	37.679
		Khdeir and Reddy⁵⁴	HBT	11.603	19.712	-
		Mantari and Canales⁶⁵	Quasi-3D	11.486	19.652	-
	(0°/90°)	Present	HBT	9.904	13.597	16.423
		Nguyen et al.⁴⁹	HBT	10.027	13.670	16.429
		Khdeir and Reddy⁵⁴	HBT	10.026	13.660	-
		Mantari and Canales⁶⁵	Quasi-3D	9.974	13.628	-

Table 5.

Nondimensional critical buckling loads of (0°/90°/0°) and (0°/90°) LC beams $(E_{1} / E_{2} = 40)$ .

BC	Lay-ups	Reference	Theory	L/h
				5	10	50
S-S	(0°/90°/0°)	Present	HBT	8.605	18.851	30.909
		Nguyen et al.⁴⁹	HBT	8.613	18.832	30.906
		Khdeir and Reddy⁵³	HBT	8.613	18.832	-
		Mantari and Canales⁶⁵	Quasi-3D	8.585	18.796	-
	(0°/90°)	Present	HBT	3.880	4.932	5.405
		Nguyen et al.⁴⁹	HBT	3.907	4.942	5.406
		Mantari and Canales⁶⁵	Quasi-3D	3.856	4.887	-
C-F	(0°/90°/0°)	Present	HBT	4.713	6.776	7.887
		Nguyen et al.⁴⁹	HBT	4.708	6.772	7.886
		Khdeir and Reddy⁵³	HBT	4.708	6.772	-
		Mantari and Canales⁶⁵	Quasi-3D	4.673	6.757	-
	(0°/90°)	Present	HBT	1.233	1.324	1.356
		Nguyen et al.⁴⁹	HBT	1.236	1.324	1.356
		Mantari and Canales⁶⁵	Quasi-3D	1.221	1.311	-
C-C	(0°/90°/0°)	Present	HBT	11.494	34.421	114.439
		Nguyen et al.⁴⁹	HBT	11.652	34.453	114.398
		Khdeir and Reddy⁵³	HBT	11.652	34.453	-
		Mantari and Canales⁶⁵	Quasi-3D	11.502	34.365	-
	(0°/90°)	Present	HBT	8.517	15.519	21.365
		Nguyen et al.⁴⁹	HBT	8.674	15.626	21.372
		Mantari and Canales⁶⁵	Quasi-3D	8.509	15.468	-

To further verify the proposed theory’s accuracy, symmetric and unsymmetric LC beams with arbitrary angle-ply are examined. Tables 6 –9 present the NFF and NCBL of beams with various angle-plies, BCs, and L/h ratios. The results obtained from the proposed theory are compared to those derived from Quasi-3D of Mantari and Canales.^65,66 The outcomes indicate the deviation between the proposed theory and Mantari and Canales⁶⁵ and Canales and Mantari⁶⁶ is negligible for all angle-plies, BCs, and L/h ratios. Additionally, as the angle-ply increases, the NFF and NCBL decrease for all BCs and L/h ratios. This phenomenon can be explained that an increase in angle-ply leads to a reduction in the beam’s stiffness. Figure 4 displays NFF of $0^{0} / 60^{0} / 0^{0}$ and $0^{0} / 60^{0}$ beams $(E_{1} / E_{2} = 40)$ with respect to L/h ratio. It is seen that the NFF of beams increases as the L/h ratio increases for all BCs. Similarly, the NCBL of beams increases with an increase in L/h ratio, as shown in Figure 5.

Table 6.

Nondimensional fundamental frequency of LC beams with 1 or 2 lay-ups $(E_{1} / E_{2} = 40)$ .

L/h	Lay-ups	Reference	Theory	BC
				S-S	C-F	C-C
5	30°	Present	HBT	8.5391	3.7198	11.1157
		Mantari and Canales⁶⁵	Quasi-3D	8.5439	3.7192	11.1164
		Canales and Mantari⁶⁶	Quasi-3D	8.5439	3.7200	11.1300
	45°	Present	HBT	7.0389	2.8741	10.0015
		Mantari and Canales⁶⁵	Quasi-3D	7.0330	2.8705	9.9895
		Canales and Mantari⁶⁶	Quasi-3D	7.0330	2.8709	10.0028
	60°	Present	HBT	4.7548	1.8073	8.0771
		Mantari and Canales⁶⁵	Quasi-3D	4.7242	1.7953	8.0632
		Canales and Mantari⁶⁶	Quasi-3D	4.7242	1.7954	8.0721
	0°/30°	Present	HBT	9.0391	4.0271	11.5693
		Mantari and Canales⁶⁵	Quasi-3D	9.0505	4.0282	11.5881
		Canales and Mantari⁶⁶	Quasi-3D	9.0505	4.0291	11.6013
	0°/45°	Present	HBT	8.2738	3.5367	11.1127
		Mantari and Canales⁶⁵	Quasi-3D	8.2848	3.5383	11.1328
		Canales and Mantari⁶⁶	Quasi-3D	8.2848	3.5389	11.1458
	0°/60°	Present	HBT	7.0693	2.8531	10.4907
		Mantari and Canales⁶⁵	Quasi-3D	7.0785	2.8537	10.5358
		Canales and Mantari⁶⁶	Quasi-3D	7.0785	2.8540	10.5479
10	30°	Present	HBT	11.5789	4.4840	18.3502
		Mantari and Canales⁶⁵	Quasi-3D	11.5698	4.4807	18.3587
		Canales and Mantari⁶⁶	Quasi-3D	11.5698	4.4810	18.3778
	45°	Present	HBT	8.6101	3.2162	15.3292
		Mantari and Canales⁶⁵	Quasi-3D	8.5905	3.2089	15.3240
		Canales and Mantari⁶⁶	Quasi-3D	8.5905	3.2091	15.3382
	60°	Present	HBT	5.2208	1.8956	10.6053
		Mantari and Canales⁶⁵	Quasi-3D	5.1757	1.8796	10.5544
		Canales and Mantari⁶⁶	Quasi-3D	5.1757	1.8798	10.5611
	0°/30°	Present	HBT	12.7086	4.9999	19.3822
		Mantari and Canales⁶⁵	Quasi-3D	12.7037	4.9980	19.4026
		Canales and Mantari⁶⁶	Quasi-3D	12.7038	4.9985	19.4223
	0°/45°	Present	HBT	10.8594	4.1555	17.8855
		Mantari and Canales⁶⁵	Quasi-3D	10.8524	4.1523	17.9129
		Canales and Mantari⁶⁶	Quasi-3D	10.8523	4.1526	17.9295
	0°/60°	Present	HBT	8.4753	3.1499	15.5335
		Mantari and Canales⁶⁵	Quasi-3D	8.4608	3.1437	15.5712
		Canales and Mantari⁶⁶	Quasi-3D	8.4608	3.1438	15.5819

Table 7.

Nondimensional fundamental frequency of symmetric and unsymmetric LC beams $(E_{1} / E_{2} = 40)$ .

L/h	Lay-ups	Reference	Theory	BC
				S-S	C-F	C-C
5	0°/30°/0°	Present	HBT	9.4516	4.3207	11.8474
		Mantari and Canales⁶⁵	Quasi-3D	9.4651	4.3218	11.8724
		Canales and Mantari⁶⁶	Quasi-3D	9.4651	4.3229	11.8854
	0°/45°/0°	Present	HBT	9.3691	4.2853	11.7201
		Mantari and Canales⁶⁵	Quasi-3D	9.3801	4.2855	11.7378
		Canales and Mantari⁶⁶	Quasi-3D	9.3801	4.2866	11.7508
	0°/60°/0°	Present	HBT	9.2868	4.2528	11.5966
		Mantari and Canales⁶⁵	Quasi-3D	9.2946	4.2519	11.6087
		Canales and Mantari⁶⁶	Quasi-3D	9.2946	4.2530	11.6214
	0°/30°/−30°/0°	Present	HBT	9.4022	4.2801	11.7748
		Mantari and Canales⁶⁵	Quasi-3D	9.4194	4.2821	11.7949
		Canales and Mantari⁶⁶	Quasi-3D	9.4194	4.2833	11.8549
	0°/45°/−45°/0°	Present	HBT	9.2753	4.2112	11.5727
		Mantari and Canales⁶⁵	Quasi-3D	9.2928	4.2129	11.5797
		Canales and Mantari⁶⁶	Quasi-3D	9.2928	4.2141	11.5937
	0°/60°/−60°/0°	Present	HBT	9.1535	4.1537	11.3784
		Mantari and Canales⁶⁵	Quasi-3D	9.1699	4.1548	11.3749
		Canales and Mantari⁶⁶	Quasi-3D	9.1699	4.1560	11.3892
10	0°/30°/0°	Present	HBT	13.8861	5.5802	20.2009
		Mantari and Canales⁶⁵	Quasi-3D	13.8823	5.5791	20.2212
		Canales and Mantari⁶⁶	Quasi-3D	13.8823	5.5797	20.2421
	0°/45°/0°	Present	HBT	13.7848	5.5428	20.0154
		Mantari and Canales⁶⁵	Quasi-3D	13.7795	5.5412	20.0293
		Canales and Mantari⁶⁶	Quasi-3D	13.7795	5.5419	20.0504
	0°/60°/0°	Present	HBT	13.6964	5.5137	19.8324
		Mantari and Canales⁶⁵	Quasi-3D	13.6889	5.5116	19.8395
		Canales and Mantari⁶⁶	Quasi-3D	13.6889	5.5122	19.8604
	0°/30°/−30°/0°	Present	HBT	13.7302	5.4986	20.0848
		Mantari and Canales⁶⁵	Quasi-3D	13.7306	5.4982	20.1107
		Canales and Mantari⁶⁶	Quasi-3D	13.7306	5.4988	20.1325
	0°/45°/−45°/0°	Present	HBT	13.5070	5.3990	19.7912
		Mantari and Canales⁶⁵	Quasi-3D	13.5092	5.3987	19.8145
		Canales and Mantari⁶⁶	Quasi-3D	13.5092	5.3994	19.8373
	0°/60°/−60°/0°	Present	HBT	13.3349	5.3294	19.5121
		Mantari and Canales⁶⁵	Quasi-3D	13.3371	5.3289	19.5302
		Canales and Mantari⁶⁶	Quasi-3D	13.3371	5.3296	19.5538

Table 8.

Nondimensional critical buckling load of LC beams with 1 or 2 lay-ups $(E_{1} / E_{2} = 40)$ .

L/h	Lay-ups	Reference	Theory	BC
				S-S	C-F	C-C
5	30°	Present	HBT	7.4262	3.4107	10.9060
		Mantari and Canales⁶⁵	Quasi-3D	7.4054	3.3891	10.9007
	45°	Present	HBT	5.0728	1.8888	8.9090
		Mantari and Canales⁶⁵	Quasi-3D	5.0384	1.8743	8.8541
	60°	Present	HBT	2.3377	0.6955	5.7292
		Mantari and Canales⁶⁵	Quasi-3D	2.2921	0.6830	5.6307
	0°/30°	Present	HBT	8.3122	4.1072	11.7212
		Mantari and Canales⁶⁵	Quasi-3D	8.3028	4.0810	11.7465
	0°/45°	Present	HBT	6.9896	3.0046	10.8439
		Mantari and Canales⁶⁵	Quasi-3D	6.9773	2.9870	10.8545
	0°/60°	Present	HBT	5.1502	1.8361	9.6209
		Mantari and Canales⁶⁵	Quasi-3D	5.1342	1.8233	9.6241
10	30°	Present	HBT	13.6417	4.3297	29.7048
		Mantari and Canales⁶⁵	Quasi-3D	13.5993	4.3163	29.6490
	45°	Present	HBT	7.5545	2.1542	20.2912
		Mantari and Canales⁶⁵	Quasi-3D	7.5067	2.1417	20.1852
	60°	Present	HBT	2.7816	0.7302	9.3508
		Mantari and Canales⁶⁵	Quasi-3D	2.7283	0.7176	9.2078
	0°/30°	Present	HBT	16.4276	5.4671	33.2488
		Mantari and Canales⁶⁵	Quasi-3D	16.3914	5.4531	33.2384
	0°/45°	Present	HBT	12.0177	3.6758	27.9583
		Mantari and Canales⁶⁵	Quasi-3D	11.9825	3.6634	27.9399
	0°/60°	Present	HBT	7.3438	2.0578	20.6006
		Mantari and Canales⁶⁵	Quasi-3D	7.3051	2.0459	20.5747

Table 9.

Nondimensional critical buckling load of symmetric and unsymmetric LC beams $(E_{1} / E_{2} = 40)$ .

L/h	Lay-ups	Reference	Theory	BC
				S-S	C-F	C-C
5	0°/30°/0°	Present	HBT	9.0759	4.8993	12.2330
		Mantari and Canales⁶⁵	Quasi-3D	9.0718	4.8633	12.2267
	0°/45°/0°	Present	HBT	8.9177	4.8279	11.9842
		Mantari and Canales⁶⁵	Quasi-3D	7.6533	4.7909	12.0091
	0°/60°/0°	Present	HBT	8.7610	4.7660	11.7375
		Mantari and Canales⁶⁵	Quasi-3D	8.7473	4.7275	11.7534
	0°/30°/−30°/0°	Present	HBT	8.9819	4.7903	12.1280
		Mantari and Canales⁶⁵	Quasi-3D	8.9843	4.7569	12.1462
	0°/45°/−45°/0°	Present	HBT	8.7408	4.6359	11.7716
		Mantari and Canales⁶⁵	Quasi-3D	8.7439	4.6034	11.7924
	0°/60°/−60°/0°	Present	HBT	8.5123	4.5184	11.4174
		Mantari and Canales⁶⁵	Quasi-3D	8.5136	4.4857	11.4270
10	0°/30°/0°	Present	HBT	19.5958	6.9635	36.3035
		Mantari and Canales⁶⁵	Quasi-3D	19.5591	6.9473	36.3133
	0°/45°/0°	Present	HBT	19.3104	6.8763	35.6706
		Mantari and Canales⁶⁵	Quasi-3D	19.2700	6.8596	35.6626
	0°/60°/0°	Present	HBT	19.0629	6.8138	35.0439
		Mantari and Canales⁶⁵	Quasi-3D	19.0166	6.7569	35.0138
	0°/30°/−30°/0°	Present	HBT	19.1599	6.7392	35.9276
		Mantari and Canales⁶⁵	Quasi-3D	19.1350	6.7246	35.9634
	0°/45°/−45°/0°	Present	HBT	18.5425	6.4884	34.9633
		Mantari and Canales⁶⁵	Quasi-3D	18.5228	6.4746	35.0019
	0°/60°/−60°/0°	Present	HBT	18.0725	6.3243	34.0491
		Mantari and Canales⁶⁵	Quasi-3D	18.0533	6.3106	34.0797

Figure 4.

Nondimensional fundamental frequency of symmetric and unsymmetric LC beam $(E_{1} / E_{2} = 40)$ with respect to L/h ratio: (a) $0^{0} / 60^{0} / 0^{0}$ LC beam and (b) $0^{0} / 60^{0}$ LC beam.

Figure 5.

Nondimensional critical buckling load of symmetric and unsymmetric LC beam $(E_{1} / E_{2} = 40)$ with respect to L/h ratio: (a) $0^{0} / 60^{0} / 0^{0}$ LC beam and (b) $0^{0} / 60^{0}$ LC beam.

FGP beams

To assess the proposed theory’s efficiency and accuracy, symmetric (SPD) and unsymmetric (ASP) beams $(r_{0} = 0.5)$ with various BCs and L/h ratios are considered. Tables 10 and 11 present the NFF and NCBL of beams, respectively. The results obtained from the proposed theory are compared to those obtained from FBT by Chen et al.,^37,39 Jamshidi and Arghavani,¹³ HBT by Nguyen et al.,⁴⁷ Karamanli et al.,⁶⁷ and Quasi-3D by Fang et al.⁶⁴ and Karamanli et al.,⁶⁷ The outcomes demonstrate that the proposed results agree with those of previous studies, evidencing the efficiency of the proposed theory. Figures 6 and 7 display the NFF and NCBL of SPD and ASP beams with respect to L/h ratio. As L/h ratio increases, the NFF and NCBL decrease for all BCs and distribution porosity types.

Table 10.

Nondimensional fundamental frequency of FGP beams ( $r_{0} = 0.5$ ).

BC	Distribution type	Reference	Theory	L/h
				10	20	50
S-S	SPD	Present	HBT	0.2792	0.1421	0.0571
		Nguyen et al.⁴⁷	HBT	0.2791	0.1421	-
		Chen et al.³⁷	FBT	0.2798	0.1422	0.0571
		Karamanli et al.⁶⁷	Quasi-3D	0.2639	-	-
	APD	Present	HBT	0.2549	0.1293	0.0519
		Nguyen et al.⁴⁷	HBT	0.2549	0.1293	-
		Chen et al.³⁷	FBT	0.2599	0.1318	0.0529
		Karamanli et al.⁶⁷	Quasi-3D	0.2414	-	-
C-F	SPD	Present	HBT	0.1007	0.0508	0.0204
		Nguyen et al.⁴⁷	HBT	0.1007	0.0508	-
		Chen et al.³⁷	FBT	0.1008	0.0508	0.0204
		Karamanli et al.⁶⁷	Quasi-3D	0.0953	-	-
		Fang et al.⁶⁴	Quasi-3D	0.0875	-	-
	APD	Present	HBT	0.0917	0.0462	0.0185
		Nguyen et al.⁴⁷	HBT	0.0917	0.0462	-
		Chen et al.³⁷	FBT	0.0917	0.0462	0.0185
		Karamanli et al.⁶⁷	Quasi-3D	0.0870	-	-
		Fang et al.⁶⁴	Quasi-3D	0.0870	-	-
C-C	SPD	Present	HBT	0.5902	0.3159	0.1291
		Nguyen et al.⁴⁷	HBT	0.5900	0.3159	-
		Chen et al.³⁷	FBT	0.5944	0.3166	0.1291
		Karamanli et al.⁶⁷	Quasi-3D	0.5654	-	-
		Fang et al.⁶⁴	Quasi-3D	0.5653	-	-
	APD	Present	HBT	0.5476	0.2888	0.1174
		Nguyen et al.⁴⁷	HBT	0.5476	0.2888
		Chen et al.³⁷	FBT	0.5475	0.2888	0.1274
		Karamanli et al.⁶⁷	Quasi-3D	0.5248	-
		Fang et al.⁶⁴	Quasi-3D	0.5248	-

Table 11.

Nondimensional critical buckling load of FGP beams ( $r_{0} = 0.5$ ).

BC	Distribution type	Reference	Theory	L/h
				5	10	20
S-S	SPD	Present	HBT	0.046487	0.012957	0.003335
		Nguyen et al.⁴⁷	HBT	-	0.012953	0.003335
		Karamanli et al.⁶⁷	HBT	0.04643	0.01295	0.0033349
		Chen et al.³⁹	FBT	-	-	0.003338
		Jamshidi and Arghavani¹³	FBT	-	-	0.003339
		Karamanli et al.⁶⁷	Quasi-3D	0.04686	0.01299	0.0033378
	APD	Present	HBT	0.039743	0.010800	0.002760
		Nguyen et al.⁴⁷	HBT	-	0.010800	0.002760
		Karamanli et al.⁶⁷	HBT	0.03973	0.01080	0.0027601
		Chen et al.³⁹	FBT	-	-	0.002869
		Karamanli et al.⁶⁷	Quasi-3D	0.04058	0.01096	0.0027951
C-F	SPD	Present	HBT	0.012958	0.003336	0.000840
		Nguyen et al.⁴⁷	HBT	-	0.003335	0.000840
		Karamanli et al.⁶⁷	HBT	0.01295	0.00333	0.0008400
		Chen et al.³⁹	FBT	-	0.00334	0.000840
		Jamshidi and Arghavani¹³	FBT	-	-	0.000840
		Karamanli et al.⁶⁷	Quasi-3D	0.01317	0.00336	0.0008440
	APD	Present	HBT	0.010801	0.002760	0.000694
		Nguyen et al.⁴⁷	HBT	-	0.002760	0.000694
		Karamanli et al.⁶⁷	HBT	0.01080	0.00276	0.0006934
		Chen et al.³⁹	FBT	-	0.00276	0.000694
		Karamanli et al.⁶⁷	Quasi-3D	0.01111	0.00281	0.0007053
C-C	SPD	Present	HBT	0.131880	0.046487	0.012957
		Nguyen et al.⁴⁷	HBT	-	0.046433	0.012953
		Karamanli et al.⁶⁷	HBT	0.13151	0.04643	0.0129526
		Chen et al.³⁹	FBT	-	-	0.013010
		Jamshidi and Arghavani¹³	FBT	-	-	0.013010
		Karamanli et al.⁶⁷	Quasi-3D	0.13508	0.04747	0.0131092
	APD	Present	HBT	0.120545	0.039743	0.010800
		Nguyen et al.⁴⁷	HBT	-	0.039735	0.010800
		Karamanli et al.⁶⁷	HBT	0.12050	0.03973	0.0107997
		Chen et al.³⁹	FBT	-	-	0.010806
		Karamanli et al.⁶⁷	Quasi-3D	0.12599	0.04112	0.0110561

Figure 6.

Nondimensional fundamental frequency of symmetric and unsymmetric FGP beam $(r_{0} = 0.5)$ with respect to L/h ratio: (a) SPD beam and (b) ASP beam.

Figure 7.

Nondimensional critical buckling load of symmetric and unsymmetric FGP beam $(r_{0} = 0.5)$ with respect to L/h ratio: (a) SPD beam and (b) ASP beam.

Figure 8 illustrates the NFF of beams as a function of the porosity ratio. As observed, when the porosity ratio increases, NFF increases for SPD beams but decreases for APD beams. This phenomenon can be explained by the decrease in both mass inertia and rigidity of FGP beams with increased porosity. However, the reduction rate in rigidity is more significant than that in mass inertia for APD beams, whereas the opposite is true for SPD beams. In addition, Figure 9 demonstrates that the NCBL decreases as the porosity ratio increases for both SPD and APD beams, as expected.

Figure 8.

Nondimensional fundamental frequency of symmetric and unsymmetric FGP beam (L/h = 10) with respect to $r_{0}$ ratio.

Figure 9.

Nondimensional critical buckling load of symmetric and unsymmetric FGP beam (L/h = 10) with respect to $r_{0}$ ratio.

Conclusions

This study introduces a novel higher-order shear deformation theory to analyse the free vibration and buckling responses of LC and FGP beams. The proposed theory has three variables and satisfies the traction-free conditions at the top and bottom surfaces of beams. The governor equations are established by using the Lagrange principle. The Legendre-Ritz functions are utilised to solve the problem. Numerical examples are conducted to verify the proposed theory and investigate the impact of porosity distribution, boundary conditions, and span-to-height ratio on the critical buckling loads and frequency of beams. The following conclusions can be drawn based on the findings of this study:

- The proposed higher-order shear deformation theory demonstrates efficiency and accuracy comparable to those obtained from the Quasi-3D theory.

- An increase in the angle-ply angle of laminated composite beams leads to a decrease in the frequency and critical buckling load.

- An increase in porosity ratio has a detrimental effect on the critical buckling load of both SPD and APD beams. Additionally, there is a divergent impact on the natural frequency of the two types of beams, with an increase in porosity ratio resulting in an increase in frequency for SPD beams and a decrease in frequency for APD beams.

- As the same porosity ratio and slenderness, the frequency and critical buckling load of SPD beams are greater than those of APD beams.

Footnotes

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ngoc-Duong Nguyen

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A new higher-order beam theory for buckling and free vibration responses of laminated composite and functionally graded porous beams

Abstract

Keywords

Introduction

Theoretical formulation

Constitutive relations

Laminated composite beam

Functionally graded porous beam

Kinematics

Variational formulation

Strain energy

Work done by axial compression

Kinetic energy

Total energy

Ritz solution

Numerical results

Convergence study

LC beams

FGP beams

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References