Static Bending Behaviour of FG-CNTRC Microbeams Resting on Elastic Foundation Using Higher-Order Shear Deformation Beam Theories and Modified Couple Stress Theory

The rapid advancement of micro- and nano-scale structures has profoundly influenced modern engineering and technology. Microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) have become essential components in diverse applications such as sensors, actuators, biomedical devices, and communication systems [1], [2], [3]. These systems offer distinct advantages, including miniaturization, high sensitivity, low power consumption, and compatibility with large-scale integration. Progress in micro/nanofabrication techniques has enabled precise control at extremely small scales, leading to devices with superior mechanical performance and higher operational frequencies [4]. Understanding the mechanical behavior of MEMS and NEMS is crucial to ensure their reliability and optimal performance; at such small scales, effects such as surface forces, thermal fluctuations, and size-dependent material properties play dominant roles [5], [6], [7].

Functionally graded materials (FGMs) are advanced composites distinguished by a gradual variation in composition and microstructure across their volume, resulting in corresponding changes in mechanical, thermal, and physical properties [8], [9]. This smooth gradation enables FGMs to achieve tailored property distributions that meet specific functional requirements, effectively mitigating stress concentrations or interfacial failures typically associated with abrupt material transitions [10]. At micro- and nano-scales, FGMs provide notable advantages. Their graded characteristics can be engineered to enhance stiffness, thermal stability, and durability of microbeams and miniature components [11], [12]. Shariati et al. [13] studied how non-uniform material distribution in a viscoelastic, axially-moving Rayleigh beam affects its vibration behavior and stability, deriving exact critical velocities and mapping divergence and flutter instability regions. A refined high-order shear deformation theory was proposed by Hadji et al. [14] to analyze the static bending of functionally graded sandwich plates, showing how material gradation, lay-up, and aspect ratio influence deflections and stresses. Rizov [15] analyzed how delamination in U-shaped, multilayered viscoelastic FG structures influences the strain energy release rate, using equilibrium equations and the J-integral to explore the effect of material gradients and support conditions.

Functionally graded carbon nanotube–reinforced composites (FG-CNTRCs) have emerged as a novel class of nanomaterials engineered to optimize the outstanding mechanical, thermal, and electrical properties of carbon nanotubes (CNTs) within composite structures [16], [17]. By varying the distribution of CNTs across the thickness of the polymer matrix, FG-CNTRCs can achieve optimized mechanical, thermal, and electrical properties tailored for specific engineering applications. Typical CNT distribution patterns include uniform distribution (UD), “V-shaped” distribution (FG-V), “O-shaped” distribution (FG-O), and “X-shaped” distribution (FG-X) [17].

Extensive research has been devoted to analyzing the mechanical responses of FG-CNTRC beams under different environments [18], [19], [20], [21], [22], [23]. With the rapid advancement of MEMS/NEMS technologies, it has become evident that classical continuum mechanics fails to accurately predict the mechanical behavior of micro- and nano-scale structures [24], [25], [26], [27]. To address these scale-dependent deviations, various higher-order continuum theories were developed [28], [29], [30], [31], [32], [33]. Thai et al. [34] reviewed advanced continuum mechanics theories—including nonlocal elasticity, modified couple stress, and strain-gradient models—used to capture size effects in small-scale beams and plates, and discusses their finite-element implementations and future directions. Ebrahimi and Barati [35] developed a nonlocal, third-order shear deformation beam model to study how size effects and material gradation influence the free vibration frequencies of functionally graded nanobeams. Shariati et al. [36] investigated how axial gradation, nonlocal effects, and viscoelastic damping influence the vibration characteristics and stability (divergence and flutter) of moving AFG nanobeams, showing that proper material distribution can significantly improve stability. Kong [37] reviewed size-dependent modelling of micro-beams and micro-plates based on Modified Couple Stress Theory (MCST), emphasising how the inclusion of a material length-scale parameter alters bending, buckling, vibration and pull-in instability predictions compared to classical theories. Simsek and Reddy [38] proposed a new higher-order beam theory coupled with modified couple stress theory to analyze the static bending and free vibration of functionally graded micro-beams, revealing significant size-dependency effects via a material length-scale parameter. Ke and Wang [39] investigated how micro-scale size effects, via the modified couple stress theory, influence the dynamic stability (e.g. parametric resonance) of functionally graded microbeams. A size‑dependent Euler–Bernoulli beam model was developed by Gorji Azandariani et al. [40] using modified couple stress theory to perform a nonlinear static analysis of a bidirectional functionally graded microbeam on a nonlinear elastic foundation, revealing how gradient indices, length‑scale parameter, and foundation stiffness significantly affect its deflection. Wattanasakulpong et al. [41] investigated how size‑dependent effects (via modified couple stress theory) influence the vibration behavior of functionally graded sandwich microbeams under different boundary conditions, showing that microscale stiffness increases, raising natural frequencies. Shenas et al. [42] presented a unified higher‑order beam theory based on a stress–strain gradient framework to analyze free vibration and buckling of CNT‑reinforced functionally graded microbeams embedded in an elastic medium, highlighting how nonlocal effects, CNT distribution, and foundation stiffness influence stability and natural frequencies. The nonlinear dynamic behavior of non‑uniform microscale composite beams reinforced with CNTs, incorporating size effects through the modified couple stress theory and geometric nonlinearity was investigated by Alimoradzadeh et al. [43].

To the best of our knowledge, this study represents the first comprehensive analytical investigation that integrates the modified couple stress theory (MCST) with high‐order shear deformation beam theories (HoSDBTs) to model static bending of FG-CNTRC microbeams resting on an elastic foundation under simply-supported boundary conditions. This paper introduces four distinct CNT distribution patterns, derives governing equations and boundary conditions via Hamilton’s Principle, and presents exact analytical solutions for these configurations. The objectives of this study are to: (1) define the material properties of the functionally‐graded carbon-nanotube-reinforced composite (FG-CNTRC); (2) formulate the governing equations based on the modified couple stress theory (MCST) together with the high‐order shear deformation beam theory (HoSDBT); and (3) apply the developed formulation to perform static bending analysis of FG-CNTRC microbeams.

Despite the analytical rigor of the current model, some limitations must be acknowledged. First, only simply-supported (S–S) boundary conditions were considered but these might not reflect the full range of realistic support conditions encountered in MEMS/NEMS devices (e.g., clamped, free–clamped, and mixed). Second, the model assumes perfect alignment, dispersion, and bonding of carbon nanotubes (CNTs) within the FG-CNTRC matrix and neglects the effects of CNT agglomeration, interface debonding or micro-defects, which could degrade the mechanical performance in practice. Third, the elastic foundation was modelled with simplified assumptions (e.g., linear Winkler/Pasternak behavior) and did not account for possible nonlinear, time-dependent or substrate-microstructure interaction phenomena. Finally, size-dependent effects were captured via the selected higher-order shear deformation theory and modified couple stress theory, but other effects such as surface energy, residual stresses, roughness, and manufacturing imperfections are outside the scope of this study as these may become significant at extreme micro‐/nano-scales.

As illustrated in Figure 1, an FG-CNTRC microbeam resting on an elastic foundation and subject to a transverse distributed load has been considered in this study. The microbeam has a rectangular cross-section with length $L$, width $b$, and thickness $h$. A Cartesian coordinate system $Oxyz$ is defined such that the $x$-axis coincides with the longitudinal axis of the microbeam, the $y$-axis corresponds to the width direction, and the $z$-axis represents the thickness direction. The origin $O$ is located at the mid-point of the left end of the microbeam. The microbeam is simply supported at both ends and subject to a transverse distributed load $q(x)$ acting in the $z$-direction, which may vary spatially. To model the interaction between the microbeam and its supporting medium, a two-parameter Winkler–Pasternak foundation model was employed. The Winkler component provides a linear elastic reaction proportional to the local deflection $w(x)$ through the modulus $k_w$ while the Pasternak shear layer introduces a shear modulus $k_p$ that accounts for transverse shear interactions between adjacent points in the foundation.

The FG-CNTRC microbeam is composed of a polymer matrix reinforced with carbon nanotubes (CNTs). Unlike conventional composites with uniformly distributed reinforcements, the CNT volume fraction in the FG-CNTRC microbeam varies continuously along the thickness direction, z-axis, according to predefined distribution patterns such as uniform (UD), V-shaped (FG-V), O-shaped (FG-O), and X-shaped (FG-X) profiles. These four CNT distribution patterns, as illustrated in Figure 2, demonstrate the functionally graded nature of the composite structure. Such gradation enables tailored mechanical performance by enhancing stiffness, strength, and resistance to buckling and vibration. The CNTs are assumed to be perfectly bonded to the polymer matrix and uniformly dispersed in the in-plane directions, with the volume fraction varying only through the thickness. The effective material properties, namely Young’s modulus, shear modulus, Poisson’s ratio, and density were evaluated using the extended rule of mixtures, which introduces CNT efficiency parameters to account for nanoscale effects, load-transfer mechanisms, and interfacial interactions. This micromechanical approach ensures accurate characterization of the composite behavior, particularly when structural dimensions reach the microscale. Based on the extended rule of mixtures, the effective Young’s moduli ($E_{11}, E_{22}$) and shear modulus ($G_{12}$) of the FG-CNTRC microbeam are expressed [16], [17], [18], [19], [20], [21], [22], [23]:

where, $E_{11}^{C N T}$ and $E_{22}^{C N T}$ denote the longitudinal and transverse Young’s moduli of CNTs, respectively, and $G_{12}^{C N T}$ represents their shear modulus. The isotropic polymer matrix possesses Young’s modulus $E^m$ and shear modulus $G^m$. In modeling FG-CNTRC materials, efficiency parameters ($\eta_1, \eta_2$, and $\eta_3$) are introduced as correction factors to bridge the gap between idealized micromechanical predictions and the actual behavior of nanocomposites. These parameters account for nanoscale imperfections and complexities such as CNT waviness, partial alignment, interfacial debonding, and non-uniform dispersion within the polymer matrix. Because conventional rule-of-mixture formulations often overestimate stiffness by assuming perfect bonding and uniform stress transfer, the efficiency parameters serve to appropriately scale the CNT contribution in the longitudinal, transverse, and shear directions. Their values are typically determined through molecular dynamic simulations, experimental calibration, or inverse modeling techniques, and they play a crucial role in achieving realistic estimations of effective material properties in FG-CNTRCs. The volume fractions of CNTs ($V_{C N T}$) and the polymer matrix ($V_m$) are related by [16], [17], [18], [19], [20], [21], [22], [23]:

which guarantees that at each point through the thickness, the composite is fully occupied by the two constituent phases.

For the UD case, the CNTs are uniformly dispersed through the thickness of the microbeam, so the local volume fraction of CNTs at coordinate $z$ is equal to the overall volume fraction of CNTs, $V_{C N T}^*$, as shown below [16], [17], [18], [19], [20], [21], [22], [23]:

For the FG-O, FG-X, and FG-V distributions, the CNT volume fraction $V_{C N T}$ along the thickness direction $z$ can be expressed as follows [16], [17], [18], [19], [20], [21], [22], [23]:

where, $f(z)$ defines the distribution profile of CNTs through the thickness coordinate $z$. For each graded distribution pattern of CNTs, the corresponding shape function $f(z)$ is specified as follows [16], [17], [18], [19], [20], [21], [22], [23]:

From Eqs. (5)–(7), the CNT volume fraction, $V_{C N T}$, for the four distribution types can be written as follows:

Figure 3 shows the variation of the effective Young's moduli $\left(E_{11}, E_{22}\right)$ and the shear modulus $\left(G_{12}\right)$ of the FG-CNTRC microbeam to the normalized thickness $(z / h)$ for the case of $V_{C N T}^*=0.12$. The total volume fraction of CNTs, $V_{C N T}^*$, is defined as [16], [17], [18], [19], [20], [21], [22], [23]:

In this expression, $w_{C N T}$ denotes the mass fraction of CNTs, to represent the overall or average volume fraction of CNTs in the composite microbeam. $\rho^{C N T}$ and $\rho^m$ are the mass densities of the CNTs and the isotropic polymer matrix, respectively. No efficiency parameters are required to adjust the Poisson's ratios ($v_{12}, v_{21}$) of the FG-CNTRC microbeam which are estimated as follows [16], [17], [18], [19], [20], [21], [22], [23]:

where, $v_{12}^{C N T}$ and $v^m$ denote the Poisson's ratios of the CNTs and the polymer matrix, respectively. The effective elastic and shear moduli are then calculated as follows [19]:

According to the MCST introduced by Yang et al. [32], the strain energy density of a linear, elastic, and isotropic material depends not only on the classical strain tensor energetically conjugates to the Cauchy stress tensor but also on the symmetric curvature tensor, which conjugates to the deviatoric part of the couple‐stress tensor. The inclusion of a material length scale parameter ($l$) enables the MCST to capture size‐dependent mechanical behavior at the microscale. Consequently, the total strain energy stored in a deformed body of volume $V$ can be expressed as [32]:

where, $\sigma_{i j}$ and $\varepsilon_{i j}$ denote the Cauchy stress tensor and the strain tensor, respectively; $m_{i j}$ is the deviatoric part of the couple stress tensor, and $\chi_{i j}$ is the symmetric curvature tensor. These tensors are mathematically defined as follows [32]:

where, $u_i$ are denoted for the components of the displacement vector, $\lambda$ and $\mu$ denote the Lamé constants of the isotropic material, $I$ is the identity tensor, and $\theta_i$ are the components of the rotation vector, which can be expressed in terms of the displacement field as [32]:

A unified higher-order shear deformation beam theory was adopted in this study, in order to precisely model the mechanical behavior of FG-CNTRC microbeams. This unified framework consolidates various shear deformation theories into a single formulation, which enables it to capture both transverse shear deformation and normal-strain effects without the need for shear-correction factors. By introducing higher-order terms in the displacement field, the formulation attains greater accuracy compared to classical and first-order theories, particularly when analyzing thick microbeams and size-dependent phenomena governed by the modified couple-stress theory. According to this theory, the displacement components are expressed as [38]:

where, $u(x)$ and $w(x)$ are denoted for the axial and transverse displacement of any point on the neutral axis of the FG-CNTRC microbeam, respectively; $\gamma(x)$ denotes the higher-order rotation related to transverse shear deformation of any point on the neutral axis of the microbeam. The higher-order rotation $\gamma(x)$ can be expressed as [38]:

where, the linear rotation term $\phi(x)$ accounts for the bending of the cross-section at any point on the neutral axis The presence of the higher-order shear deformation term $g(z) \gamma(x)$ is essential for capturing the non-linear variation of shear strains through the thickness. The through-thickness shape function $g(z)$ is specifically chosen so that the traction-free boundary conditions at the top and bottom surfaces of the microbeam are satisfied. The particular expressions of $g(z)$ for different higher-order beam theories are summarized in Table 1.

By substituting the displacement field defined in Eq. (19) to Eq. (21) into Eq. (15), the individual components of the strain tensor are derived as follows:

By applying Eq. (18) and Eqs. (19)–(21), the components of the rotation vector can be written as follows:

Substituting Eq. (26) into Eq. (17), the components of the symmetric curvature tensor can be obtained as:

In this study, the governing equations of motion are derived in a systematic way using Hamilton’s principle, which provides a variational framework for dynamic equilibrium. Mathematically, this principle is stated as:

where, $W_e, U_s$, and $U_f$ denote the work done by external forces, the strain energy, and the potential energy of the elastic foundation, respectively. From Eq. (13), the strain energy of the microbeam can be expressed as:

Consequently, the virtual variation of the strain energy can be expressed as:

where,

where,

Here, $k_s$ denotes the shear correction factor. This factor is introduced to compensate for the discrepancy between the assumed and actual shear-strain energy distributions across the microbeam’s thickness. For first-order shear deformation beam theory (FSDBT), a constant shear strain is assumed and leads to an overestimation of shear stiffness, hence a commonly accepted value of $k_s=5 / 6$ adopted for FSDBT. In contrast, for higher-order shear deformation beam theories (HoSDBTs) such as PSDBT, TSDBT, HSDBT, ESDBT, and ASDBT, the assumed displacement fields more accurately represent the parabolic variation of shear strains and satisfy the traction-free boundary conditions on the top and bottom surfaces. Therefore, the shear correction factor is typically taken as $k_s= 1$ . The virtual potential energy of the elastic foundation can be expressed as:

The virtual work done by external force, including the transverse distributed load $q(x)$, is given by:

By substituting the expressions for the virtual variation of strain energy from Eq. (32), the virtual potential energy of the elastic foundation from Eq. (43), and the virtual work done by external force from Eq. (44) into Hamilton's principle in Eq. (30), then performing integration by parts and collecting the coefficients of the generalized displacements $\delta u, \delta w$ and $\delta \gamma$, this study obtains the equilibrium equations for the FG-CNTRC microbeam based on the MCST as follows:

and the corresponding boundary conditions (BCs) at $x=0$ and $x=L$, as:

The equilibrium equations of FG-CNTRC microbeam in terms of the displacements $(u, w$, and $\gamma)$, are obtained by substituting Eqs. (33)–(39) into Eqs. (49)–(51), yielding:

This analysis is carried out under several simplifying assumptions that may limit its practical accuracy. The FG-CNTRC microbeams are modeled as homogeneous continua with smoothly varying properties, thus neglecting microstructural defects and non-uniform CNT distribution. Linear elasticity and small deformations are assumed, so geometric and material nonlinearities under large loads are not considered. The adopted HoSDBTs use simplified kinematic assumptions, which may reduce accuracy for beams with extreme aspect ratios or highly graded materials. The supports are idealized as linear Winkler–Pasternak foundations with simply supported boundaries, which may not fully represent complex or nonlinear substrate behaviors. Finally, the MCST involves a single material length-scale parameter, omitting effects such as surface energy or microstructural interactions that could become important at very small scales.

The current study employed the Navier method using Fourier sine series to derive closed-form solution for the bending analysis of simply-supported FG-CNTRC microbeams. By integrating the MCST with higher-order shear deformation effects, the solution explicitly accounted for both nanoscale length-scale effects and transverse shear behavior. To begin with, the problem’s boundary conditions were imposed. For simply-supported ends, the boundary equations are written as follows:

In formulation, the Navier method was used to derive closed-form solutions. This study started by representing the displacement fields as series of trigonometric functions multiplied by unknown coefficients. This form both satisfied the governing differential equations and inherently enforced the simply-supported boundary conditions at $x=0$ and $x=L$, as shown below:

where, $\alpha_n=n \pi / L$ corresponds to the $n$-th half-wave, $\left(U_n, W_n, \Psi_n\right)$ re the unknown modal amplitudes to be determined. The distributed load via a Fourier sine series was represented below [38]:

where, each coefficient $Q_n$ denotes the contribution of the $n$-th half-wave of the loading. Expanding the load in this way ensures perfect compatibility with the Navier series expansion of the displacement field. By matching loading $q(x)$ mode-by-mode with the assumed displacement form, this study could derive explicit expressions for deflection, internal bending moment, and the modal amplitudes ($U_n, W_n, \Psi_n$) in a straightforward and elegant manner. The coefficient $Q_n$ is determined by exploiting the orthogonality of the sine functions [38]:

Below are explicit expressions for three typical loading cases: uniform load, sinusoidal load, and point load [38]:

For uniformly distributed load: $q(x)=q_0$

where, $q_0$ is the intensity of the uniformly distributed load.

For sinusoidal load: $q(x)=q_0 \sin \left(\frac{\pi x}{L}\right)$

Only the coefficient with $n=1$ is non-zero:

For point load: $q(x)=P \delta\left(x-x_p\right)$

where, $P$ denotes the magnitude of the point load, $x_p$ is the location at which the point load is applied, and $\delta(\cdot)$ is the Dirac delta function. Substituting Eqs. (59)–(61) into Eqs. (52)–(54) yields the following system of algebraic equations:

where, the coefficients $K_{i j} (i, j=1,2,3)$ are given by:

Solving Eq. (67), the modal amplitudes ($U_n, W_n, \Psi_n$) can be obtained as:

The analytical solution for static bending of FG-CNTRC microbeam can be obtained by substituting the expressions of the modal amplitudes $\left(U_n, W_n, \Psi_n\right)$ obtained in Eqs. (69)–(71) into Eqs. (59)–(61).

Figure 4 presents a schematic flow diagram that outlines the overall methodological framework adopted in this study. It depicts the sequence of key stages beginning with the definition of the FG-CNTRC properties, followed by the formulation under the MCST and HoSDBTs, and culminating in the static bending analysis of FG-CNTRC microbeams.

This study conducted a comprehensive numerical investigation into the static bending, buckling, and free vibration behavior of simply supported FG-CNTRC microbeams with the previously obtained closed-form solutions. The microbeams were reinforced with CNTs embedded in a polymer matrix, with material properties characterized by the extended rule of mixtures. In this study, poly methyl methacrylate (PMMA) was used as the polymer matrix. The material properties of PMMA are given as follows [16], [17]: $E^m=2.5 \mathrm{GPa}$, $\rho^m=1190 \mathrm{~kg} / \mathrm{m}^3, v^m=0.3$. The $(10,10)$ SWCNTs are considered as the reinforcement material, which have the material properties as [16], [17]: $E_{11}^{C N T}=5646.6 \mathrm{GPa}, E_{22}^{C N T}=7080 \mathrm{GPa}, G_{12}^{C N T}=1944.5 \mathrm{GPa}, v_{12}^{C N T}=0.175$, $\rho^{C N T}=2100 \mathrm{~kg} / \mathrm{m}^3$. The CNT efficiency parameters $\left(\eta_1, \eta_2, \eta_3\right)$ can be determined by matching the results from molecular dynamics [16], [17], as presented in Table 2.

In the context of the MCST, the material length-scale parameter $l$ is not a universal constant but depends on the microstructure, geometry, and material composition of the system [44]. Experimental and theoretical studies have shown that $l$ varies with structural dimensions (especially thickness or diameter) as well as inherent material features such as grain size or crystalline texture [44]. To enable a quantitative investigation of size effects in functionally graded microbeams, Simsek and Reddy [38] assumed a value for the material length-scale parameter, $l=15 \mu \mathrm{~m}$. The present work likewise treated $l=15 \mu \mathrm{~m}$ as a parameter (i.e., variable) to explore its influence on the size-dependent behavior of FG-CNTRC microbeams. Moreover, to clearly reflect the relative importance of size effects, this study normalized $l$ by the microbeam thickness $h$, and presented results in terms of the nondimensional ratio $h / l$.

For convenience in analyzing the static bending behavior of FG-CNTRC microbeams, the following dimensionless parameters were defined:

Dimensionless transverse deflection:

Dimensionless transverse deflection:

where, $I=b h^3 / 12$ is the area moment of inertia of the microbeam’s cross-section. Also, the dimensionless elastic constants of the elastic foundation were defined as:

To validate the analytical solutions derived from using the MCST and HoSDBTs for simply-supported FG-CNTRC microbeams, a set of comparative studies was performed. In comparison, the static bending results in this study were benchmarked against those reported by Wattanasakulpong and Ungbhakorn [23]. The comparison is summarized in Table 3. The nondimensional deflection is defined as $\hat{w}=100 \frac{E^m h^3}{q_0 L^4} w\left(\frac{L}{2}\right) \cdot$ The comparative results showed very close agreement, thus demonstrating high accuracy and consistency of the present analysis.

In this subsection, a numerical study was carried out to show how key design parameters affected the static bending behaviour of FG-CNTRC microbeams. The parameters analyzed included: material length scale parameter $(h / l)$ from the MCST; CNT volume fraction $\left(V_{C N T}^*\right)$; aspect ratio $(L / h)$; elastic support via Winkler and Pasternak foundations ($K_w$ and $K_p$), different beam models, and CNT gradation profiles of UD, FG-O, FG-X, and FG-V.

The maximum dimensionless deflections ($\bar{w}$) of FG-CNTRC microbeams were investigated across various beam models, the dimensionless material length scale parameter ($h/l$), aspect ratio ($L/h$), CNT volume fraction ($V_{C N T}^*$), and CNT distribution patterns of UD, FG-O, FG-X, and FG-V, as detailed in Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. The dimensionless maximum deflection ($\bar{w}$) for FG-CNTRC microbeams under a uniformly distributed load, first without an elastic foundation, then with one, is presented in Table 4 and Table 5, respectively. Table 6 and Table 7 show the dimensionless maximum deflection ($\bar{w}$) under a point load applied at the mid-span, for cases without and with an elastic foundation, respectively. Meanwhile, the dimensionless maximum deflection ($\bar{w}$) under a sinusoidal distributed load is reported for models without an elastic foundation in Table 8, and with an elastic foundation in Table 9. The results accord well with the expected trends: EBBT always yields the smallest deflection (underestimating bending) since it neglects shear deformation; FSDBT predicts the largest deflection, thus reflecting its first-order shear consideration and approximate shear correction. The higher-order theories (PSDBT, TSDBT, HSDBT, ESDBT, and ASDBT) cluster closely together, hence demonstrating their abilities to model shear deformation accurately with minimal inter-model discrepancy. This consistency confirms their capacity to represent shear profiles effectively without heavy dependence on correction factors. From Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, one can observe that significant differences among the beam theories (EBBT, FSDBT, and HoSDBTs) emerge when the slenderness ratio ($L/h$) is small (e.g., $L/h$ = 5, i.e., for thick microbeams). As $L/h$ increases (e.g., $L/h$ = 30 or 50, corresponding to slender microbeams), the predictions from all theories converge. In the thick-beam regime, reliance on EBBT or FSDBT may lead to substantial under- or over-estimation of deflection, so the use of HoSDBTs becomes essential for accuracy. Yet for slender microbeams, the simpler theories (EBBT or FSDBT) suffice and are computationally more efficient. The results clearly illustrated this convergence behavior, highlighting both the limitations of simpler models and the advantages of HoSDBTs; they provided strong methodological justification for selecting the appropriate theory based on the slenderness ratio ($L/h$).

The influence of the dimensionless material length scale parameter $(h / l)$ on the static bending behavior of FG-CNTRC microbeams is evident in Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. In the MCST framework, the ratio $h / l$ serves as the key indicator of size-dependent effects. When $h / l$ is small (i.e., the internal length scale $l$ is on the same order as the microbeam thickness $h$), couple-stress effects impart significant stiffening, thereby reducing the microbeam's deflection. In that regime, the intrinsic couple-stress contributions in the MCST markedly increase bending rigidity compared to the classical elasticity theory (CET). As $h / l$ grows larger, the microstructural influence wanes and the MCST predictions smoothly converge to those of CET, to signify diminishing size effects. This behavior aligns with prior analytical and experimental investigations using the MCST-based beam models, which show that size effects are substantial at lower $h / l$ values but become negligible once $h / l$ exceeds roughly Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. Thus, the characteristic response, stiffer behavior at small $h / l$ and a gradual return to classical behavior at large $h / l$, is a signature feature in micro- and nano-scale bending analyses under the MCST.

The numerical findings revealed a strong interplay between CNT volume fraction $V_{C N T}^*$ and the distribution scheme. Raising the CNT volume fraction invariably stiffened the microbeam and lowered the maximum dimensionless deflection ($\bar{w}$), in agreement with prior studies showing reductions up to about 36% under comparable loading conditions. Moreover, the layout of reinforcement profoundly affected bending behavior: FG-X distributions (with CNTs concentrated near the surfaces) yielded the smallest deflection, whereas FG-O layouts produced the largest, because the outer fiber reinforcement is the most effective in resisting bending stresses. The UD and FG-V patterns fell in between. Interestingly, as the CNT volume fraction $V_{C N T}^*$ increased, the performance gap between different distribution strategies widened, indicating that optimizing stiffness demands careful selection of both volume fraction and gradient profile. These results underscored the vital influence of both CNT volume fraction and spatial gradation on the bending stiffness of FG-CNTRC microbeams. In particular, surface-rich configurations (e.g., FG-X) delivered superior stiffness and reduced deflection, in consistence with the literature [16], [17], [18], [19], [20], [21], [22], [23], whereas mid-plane-rich configurations (e.g., FG-O) led to higher deflections. As CNT loading increased, differences among distribution strategies became more pronounced. Therefore, for stiffness-targeted designs, a synergistic optimization of both CNT fraction and its distribution profile is strongly recommended.

Incorporating a Winkler–Pasternak elastic foundation markedly reduced the deflection of the microbeam across all CNT distribution schemes, though the degree of reduction depended on the material gradation. Microbeams with FG-X distributions (CNTs concentrated at the surfaces) gained the most from additional stiffening, resulting in the lowest deflection under foundation support. Meanwhile, FG-O microbeams also experienced significant deflection reduction on the order of 20 to 30%, yet remained the most flexible of the group, owing to the reinforcement concentrated near the mid-plane. The UD and FG-V distributions exhibited intermediate responses. These trends conformed to established mechanics of FG-CNT systems under elastic foundations, thus underscoring the compensatory effect that a foundation provided, especially for softer gradation profiles.

Changes in maximum dimensionless deflections ($\bar{w}$) of FG-CNTRC microbeams on an elastic foundation under uniform, point, and sinusoidal loads are depicted in Figure 5, Figure 6, and Figure 7, respectively. In these figures, the MCST-based higher-order shear deformation theories (FSDBT-MCST, and HoSDBTs-MCST) showed markedly lower deflections at $h / l \approx 1$, in some cases 40 to 50% below classical theory predictions. This underscored the capacity of the theory to capture microstructural stiffening effects that are significant at small scales. As $h/l$ increased (i.e., the microbeam became “thicker” relative to the material length scale), the influence of the intrinsic length scale parameter waned, and the MCST results gradually converged toward classical (shear deformation) curves. This behavior suggests that for thicker beams, classical shear deformation theories become sufficiently accurate.

Figure 8 illustrates how the maximum dimensionless deflection ($\bar{w}$) of FG-CNTRC microbeams on an elastic foundation varies with the slenderness ratio ($L/h$). Four CNT distribution schemes, UD, FG-O, FG-X, and FG-V, are included in this depiction. It is evident that deflections differ markedly at low $L/h$ values but tend to converge toward stable levels as $L/h$ increases. Among the four distribution types, FG-O exhibited the greatest deflection, followed by FG-V, UD, and FG-X.

The impact of the CNT volume fraction $V_{C N T}^*$ on the static bending behavior of FG-CNTRC microbeams is illustrated in Figure 9. The uniform, sinusoidal, and point loads were considered in this investigation. The input data was considered as $h=2 l$, $K_W=10$ and $K_P=5$ . It can be observed that an increase in the CNT volume fraction $V_{C N T}^*$ in the FG-CNTRC microbeams led to higher composite stiffness; under a given static bending load, the maximum deflection decreased. However, the extent of deflection reduction might diminish at larger values of $V_{C N T}^*$ as the gain in stiffness became marginal.

Figure 10 depicts how the maximum dimensionless deflection $(\bar{w})$ of FG-CNTRC microbeams on an elastic foundation varies along the normalized beam axis $(x / L)$ under uniform, point, and sinusoidal loads. The deflection profiles are symmetric about the mid-axis $(x / L=0.5)$ for all CNT distribution schemes. Once again, the influence of CNT distribution is clear the ranking of maximum deflections is FG-O $>$ FG-V $>$ UD $>$ FG-X.