Document Type : Research Paper
Authors
^{1} Department of Mathematics, Government Girls Degree College, Behat247121, Uttar Pradesh, India
^{2} Department of Mathematics, Government P.G. College, Gopeshwar246401, Uttarakhand, India
Abstract
Keywords
Main Subjects
Free Vibration and Buckling Analyses of Nanobeam Embedded in Pasternak Foundation
^{a }Department of Mathematics, Government Girls Degree College, Behat247121, Uttar Pradesh, India
^{b }Department of Mathematics, Government P.G. College, Gopeshwar246401, Uttarakhand, India
KEYWORDS 

ABSTRACT 
Free vibration; Buckling; Nonlocal; Nanobeam; EulerBernoulli beam theory; Pasternak foundation; PDQM; HDQM. 
In this paper, free transverse vibration and buckling analyses of a nanobeam are presented by coupling the EulerBernoulli beam (EBT) theory and Eringen’s nonlocal elasticity theory. The nanobeam is embedded in the Pasternak foundation. Hamilton’s energy principle is used to derive governing differential equations. The Lagrange polynomialbased differential quadrature method (PDQM) and a harmonic differential quadrature method (HDQM) are used to convert the governing differential equation and boundary conditions into a set of linear algebraic equations. The first three frequencies and the lowest critical buckling loads for clampedclamped, clampedsimply supported, and simply supportedsimply supported boundary conditions are obtained by implementing the bisection method through a computer program written in C++. The impacts of nonlocal Eringen’s parameter (scaling effect parameter), boundary conditions, axial force, and elastic foundation moduli on frequencies are examined. The effects of nonlocal Eringen’s parameter, boundary conditions, and elastic foundation moduli on critical buckling load are also studied. A convergence study of both versions of DQM is conducted to validate the present analysis. A comparison of frequencies and critical buckling loads with those available in the literature is presented. 
After the invention of carbon nanotubes (Iijima [1]), smallscale structures and devices have been developed due to advancements in nanotechnology and nanoscience. Nanobeams and carbon nanotubes are used in microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS), microactuators, transistors and microsensors, etc. Due to the wide applications of nanotechnology and nanoscience in modern science and technology, researchers have carried out extensive research on nanomaterials. Nanobeams are used in cancer treatment. To propose new designs, an analysis of the mechanical behavior of nanobeams becomes necessary. This problem can be solved by atomistic mechanics (Baughman et al. [2]), hybrid atomisticcontinuum mechanics (Wang et al. [3]), and continuum mechanics approach. The first two approaches are computationally expensive and timeconsuming. Therefore, the continuum mechanics approach has been used by researchers to model nanosystems as rods, beams, plates, and shells. In this approach, crystal material architecture is replaced by a continuous medium having homogeneous properties. This continuous medium can predict the overall response of the nanomaterial. Hence, new constitutive laws are required to study the nanoscale size effects. Up to a certain size, classical theories can be used to study the behavior of structures. Eringen’s nonlocal elasticity theory [47] has been proposed to incorporate the nanoscale effect. The work related to free vibration and buckling of a nanobeam has been reported in the literature as follows: Lu et al. [8] derived frequency equations and mode functions of a nonlocal EulerBernoulli beam. Xu [9] investigated free transverse vibration of nanotomicron scale beams using the integral equation approach. Wang et al. [10] solved the free vibration problem of micro/nano beams analytically for different combinations of classical boundary conditions. Challamel and Wang [11] presented a small length scale effect for a nonlocal cantilever beam. Murmu and Pradhan [12] used the differential quadrature method to obtain the natural frequencies of a nonuniform cantilever nanobeam. Roque et al. [13] studied the bending, buckling, and free vibration of Timoshenko nanobeams using the meshless method. Mohammadi and Ghannadpour [14] used Chebyshev polynomials in the RayleighRitz method to study the free vibration of Timoshenko nanobeams. Li et al. [15] investigated the dynamics and stability of transverse vibration of nonlocal nanobeams under variable axial load. Vibration analysis of EulerBernoulli nanobeams has been presented by Eltaher et al. [16] using the finite element method. Behra and Chakraverty [17] used orthogonal polynomials in the RayleighRitz method to obtain frequency parameters and mode shapes of Euler and Timoshenko nanobeams. Rahmani and Pedram [18] presented a closedform solution for the vibration behavior of functionally graded Timoshenko nanobeams. In another paper, Behra and Chakraverty [19] applied the differential quadrature method to study the free vibration of nanobeams based on various nonlocal theories. Ebrahimi and Salari [20] studied free flexural vibration of functionally graded nanobeams using the differential transform method. Ebrahimi et al. [21] investigated the vibrational characteristics of functionally graded nanobeams using the differential transform method. Tuna and Kirca [22] studied free vibration and buckling of nonlocal EulerBernoulli beams utilizing the Laplace transform method. HamzaCherif et al. [23] employed a differential transform method to study the free vibration of a singlewalled carbon nanotube resting on an elastic foundation under thermal effect. Aria and Friswell [24] used the finite element method to examine the free vibration and the buckling behavior of functionally graded Timoshenko nanobeams. Nikam and Sayyad [25] presented closedform solutions for bending, buckling, and free vibration of simply supported nanobeams. Arian et al. [26] studied the effect of the thicknesstolength ratio on the frequency ratio of Timoshenko nanobeams and nanoplates. Ufuk [27] used the Ritz method in free transverse vibration analysis of cantilever nanobeam with intermediate support. Wang et al. [28] presented an analytical buckling analysis of micro and nano roads/tubes based on nonlocal Timoshenko beam theory. Reddy [29] reformulated EulerBernoulli, Timoshenko, Reddy, and Levinson beam theories using the nonlocal theory for bending, buckling, and vibrations of beams. Pradhan and Phadikar [30] solved buckling, bending, and vibration problems of nonhomogeneous nanotubes using the differential quadrature method. Aydogdu [31] proposed a generalized nonlocal beam theory to study the bending, buckling, and free vibration of nanobeams. Thai [32] presented analytical solutions for bending, buckling, and free vibration of nanobeams based on a nonlocal shear deformation beam theory. Eltaher et al. [33] used Galerkin finite element method to study static deflection and buckling response of functionally graded nanobeams for different combinations of boundary conditions. Ebrahimi and Salari [34] developed a differential transform method solution for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position. Buckling analysis of twodirectional functionally graded EulerBernoulli nanobeam has been presented by Nejad et al. [35] using the generalized differential quadrature method. Safarabadi et al. [36] studied the effect of surface energy on the free vibration of rotating nanobeam using the differential quadrature method. Mohammadi et al. [37] used the differential quadrature method to present the hygromechanical vibration of a rotating viscoelastic nanobeam resting on a viscoPasternak foundation subjected to nonlinear temperature variation. An asymptotic solution for critical buckling load of EulerBernoulli nanobeam using Eringen’s twophase nonlocal theory has been presented by Zhu et al. [38]. Khaniki and HosseiniHashemi [39] presented a buckling analysis of tapered nanobeam using nonlocal strain gradient theory and the generalized differential quadrature method. Finite element analysis of bending, buckling, and free vibration problems of EulerBernoulli nanobeam has been presented by Tuna and Kirca [40] using Eringen’s nonlocal integral model. Buckling analysis of nonuniform nonlocal strain gradient beams has been presented by Bakhshi et al. [41] using the generalized differential quadrature method. Stability analysis of simply supported nonlocal EulerBernoulli beam with varying crosssections and resting on the Pasternak foundation has been presented by Soltani and Mohammadi [42]. Xu and Zheng [43] solved the buckling problem of the nonlocal strain gradient Timoshenko beam in closed form. The differential quadrature method and the differential transform method have been used by Jena and Chakraverty [44] for the buckling analysis of nanobeams. Ragb et al. [45] obtained natural frequencies of a piezoelectric nanobeam resting on a nonlinear Pasternak foundation using different versions of the differential quadrature method. Nazmul and Devnath [46] derived closedform solutions for bending and buckling of functionally graded nanobeam using the Laplace transform. Karmakar and Chakravarty [47] studied the thermal vibration of an Euler nanobeam resting on the WinklerPasternak foundation using the differential quadrature method and the Adomian decomposition method. Zewei et al. [48] used the differential quadrature method in free vibration, buckling, and dynamic stability analyses of Timoshenko micro/nanobeams resting on the Pasternak foundation under axial load. Civalek et al. [49] studied the stability of restrained FGM nonlocal beams using the Fourier series. Jalaei et al. [50] presented an analytical transient response of porous viscoelastic functionally graded nanobeam subjected to dynamic load and magnetic field. Beni [51] investigated free vibration and static torsion of an electromechanical flexoelectric micro/nanotube. Beni and Beni [52] studied the dynamic stability of an isotropic viscoelastic/piezoelectric EulerBernoulli nanobeam using the Galerkin method. Numanoğlu et al. [53] presented a thermomechanical vibration analysis of Timoshenko nanobeam using the nonlocal finite element method. Karmakar and Chakraverty [54] used the Adomian decomposition method and the homotopy perturbation method to study the thermal vibration of nonhomogeneous Euler nanobeam resting on the Winkler foundation.
This paper aims at providing a numerical solution for free transverse vibration and buckling of a nanobeam under axial load and resting on the Pasternak foundation. Eringen’s nonlocal elasticity theory along with the EulerBernoulli beam theory is used to develop the mathematical model. The PDQM and the HDQM are used to obtain the first three frequencies and the lowest critical buckling loads for clampedclamped (CC), clampedsimply supported (CS), and simply supportedsimply supported (SS) beams.
The impacts of nonlocal parameters, axial force, and elastic foundation moduli on frequencies are studied. The effects of nonlocal parameters, boundary conditions, and elastic foundation moduli on critical buckling load are also studied.
Consider an isotropic uniform nanobeam of length crosssection area , density and transverse deflection as shown in Fig. 1. Here, t denotes the time. The beam is resting on the Pasternak foundation and is also subjected to uniform compressive load p.
The fundamental assumptions of the EulerBernoulli beam theory (Abbas et al. [55]) are:
This theory does not account for the effect of transverse shear strain and overpredicts natural frequencies.
Fig. 1. Geometry of a EulerBernoulli nanobeam
resting on Pasternak foundation
The strain energy of the beam is expressed as (Wang et al. [56]):

(1) 
where is normal stress.
Normal strain in terms of deflection is represented as

(2) 
Using (2), strain energy given by (1) becomes

(3) 
where M is the bending moment and is defined as

(4) 
The kinetic energy of the beam is expressed as (Wang et al. [56])

(5) 
The potential energy due to the Pasternak foundation is represented as

(6) 
where and are the Winkler and the Pasternak (shear) foundation stiffnesses, respectively.
Work done by the compressive load p is expressed as

(7) 
Using Hamilton’s principle

(8) 
The equation of motion of the beam is obtained as follows:

(9) 
Based on Eringen’s nonlocal elasticity theory, the nonlocal stress tensor at a point is given as (Murmu and Adhikari [57])

(10) 
where and are classical stress and nonlocal modulus, respectively.
The differential form of the above integral constitutive relation (Murmu and Adhikari [57]) is

(11) 
where is a material constant, is the Laplace operator and a is the internal characteristic length.
For EulerBernoulli nanobeam, the relationship between local and nonlocal stresses (Eringen [5]) given by Eq. (11) can be rewritten as follows:

(12) 
Multiplying the equation (12) by z dA and integrating over the area A, we obtain

(13) 
where E is Young’s modulus and

(14) 
is the second moment of the area.
By using Eqs. (9) and (13), the equation of motion is obtained as follows:

(15) 
where
is the nonlocal parameter and is the frequency parameter.
The buckling problem is obtained by putting in Eq. (15) and is given as follows:

(16) 
where is the critical buckling load.
Boundary conditions at the ends and are given as
For CC beam

(17a17d) 
For CS beam



For SS beam

(19a19d) 

Using the PDQM and the HDQM, free vibration and buckling problems are reduced to eigenvalue problems which are solved for frequencies and critical buckling loads, respectively.
The displacement and its derivatives are approximated by a weighted linear sum of functional values at grid points as follows:

(20) 

(21) 
and 
(22) 
and 
(23) 
and

(24) 
In this method, the weighting coefficients of the first, second, third, and fourth order derivatives are obtained as follows:

(25) 

(26) 
and 
(27) 

(28) 

(29) 

(30) 
The GaussChebyshevLobatto grid points (Chang [60]) in the range [0, 1] are given below as:

(31) 
We obtain a set of equations in unknowns by discretizing the equation of motion (15) at grid points
as follows:

(32) 
Similarly, the buckling problem (16) can be discretized as follows:

(33) 
By discretizing boundary conditions
(17a17d) for the CC beam, we obtain a set of four equations in unknowns

(34a34d) 

Equations (32) and (34a34d) form a set of algebraic linear equations in unknowns. This set represents a generalized eigenvalue problem that is solved to obtain frequencies for the CC beam.
The boundary conditions for CS and SS beams are discretized as

(35a35d) 
and 


(36a36d) 
Similarly, critical buckling loads for CC, CS and SS beams can be obtained by using equation (33) and discretizing boundary conditions represented by equations
(17a17d), (18a18d), and (19a19d), respectively.
The first three frequencies and the lowest critical buckling loads are calculated by solving the corresponding eigenvalue problem and the buckling problem, respectively. The Convergence of results for both versions of DQM with an increasing number of grid points is shown in Tables 1 and 2 for
. The value of N has been fixed as 17 as we get results correct to four decimal places. Results are shown in the Tables (36).
Tables (35) present the first three frequencies while Table 6 shows critical buckling loads for CC, CS, and SS nanobeams for different combinations of parameters. A comparison of frequencies in special cases is presented in Table (7) while a comparison of critical buckling loads in special cases is shown in Table (8). It is evident from the comparison tables that the results are in good agreement with those available in the literature.
Table 1. Convergence of frequency parameter of EulerBernoulli nanobeam using PDQM






10 
13 
14 
15 
16 
17 
Mode 
CC 

I 
5.0562 
5.0562 
5.0562 
5.0562 
5.0562 
5.0562 
II 
7.4587 
7.4640 
7.4640 
7.4640 
7.4640 
7.4640 
III 
9.6622 
9.5491 
9.5481 
9.5475 
9.5475 
9.5475 

CS 

I 
4.4727 
4.4728 
4.4728 
4.4728 
4.4728 
4.4728 
II 
6.8302 
6.8326 
6.8325 
6.8325 
6.8325 
6.8325 
III 
9.2086 
8.9567 
8.9573 
8.9560 
8.9560 
8.9560 

SS 

I 
4.0068 
4.0068 
4.0068 
4.0068 
4.0068 
4.0068 
II 
6.2130 
6.2169 
6.2169 
6.2169 
6.2169 
6.2169 
III 
8.5510 
8.3685 
8.3675 
8.3667 
8.3667 
8.3667 
Table 2. Convergence of frequency parameter of EulerBernoulli nanobeam using HDQM






10 
13 
14 
15 
16 
17 
Mode 
CC 

I 
5.0650 
5.0564 
5.0566 
5.0563 
5.0562 
5.0562 
II 
7.4664 
7.4651 
7.4640 
7.4641 
7.4640 
7.4640 
III 
9.5614 
9.5477 
9.548 
9.5475 
9.5475 
9.5475 

CS 

I 
4.4859 
4.4702 
4.4737 
4.4722 
4.4730 
4.4728 
II 
6.8247 
6.8355 
6.8316 
6.8332 
6.8323 
6.8325 
III 
8.9794 
8.9547 
8.9572 
8.9556 
8.9563 
8.9560 

SS 

I 
3.9863 
4.0068 
4.0056 
4.0068 
4.0065 
4.0068 
II 
6.2266 
6.2169 
6.2173 
6.2169 
6.2169 
6.2169 
III 
8.3869 
8.3667 
8.3675 
8.3667 
8.3669 
8.3667 
Table 3. First three values of frequency parameter of CC EulerBernoulli nanobeam








0 
0.1 
0.2 
0.3 
0.4 








Mode 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
1 
0 
I 
4.8170 
4.6427 
4.7119 
4.4725 
4.4740 
4.0547 
4.2233 
3.5302 
4.0158 
2.9427 


II 
7.9240 
7.7814 
7.2571 
7.0188 
6.2603 
5.7842 
5.5576 
4.7433 
5.1047 
3.8372 


III 
11.0512 
10.9396 
9.3728 
9.1401 
7.6302 
7.1119 
6.6414 
5.7228 
6.0581 
4.6072 

10 
I 
5.0692 
4.9229 
5.0431 
4.8528 
4.9852 
4.6988 
4.9273 
4.5449 
4.8830 
4.4251 


II 
8.1452 
8.0148 
7.6084 
7.4037 
6.8721 
6.5259 
6.4215 
5.9490 
6.1701 
5.6052 


III 
11.2299 
11.1237 
9.7262 
9.5189 
8.3129 
7.9246 
7.6364 
7.0905 
7.3007 
6.6405 

30 
I 
5.4780 
5.3657 
5.5558 
5.4172 
5.6941 
5.5092 
5.8005 
5.5797 
5.8676 
5.6238 


II 
8.5381 
8.4262 
8.1924 
8.0304 
7.7561 
7.5225 
7.5171 
7.2384 
7.3947 
7.0906 


III 
11.5632 
11.4663 
10.3357 
10.1642 
9.3206 
9.0525 
8.9129 
8.5873 
8.7377 
8.3804 

50 
I 
5.8067 
5.7145 
5.9546 
5.8436 
6.2073 
6.0667 
6.3970 
6.2356 
6.5156 
6.3415 


II 
8.8809 
8.7824 
8.6724 
8.5368 
8.4120 
8.2313 
8.2736 
8.0684 
8.2043 
7.9865 


III 
11.8693 
11.7800 
10.8533 
10.7058 
10.0788 
9.8693 
9.7993 
9.5586 
9.6901 
9.4338 
10 
0 
I 
4.8370 
4.6650 
4.7333 
4.4975 
4.4989 
4.0881 
4.2529 
3.5802 
4.0501 
3.0273 


II 
7.9285 
7.7862 
7.2630 
7.0253 
6.2695 
5.7958 
5.5707 
4.7642 
5.1215 
3.8764 


III 
11.0528 
10.9413 
9.3756 
9.1430 
7.6353 
7.1181 
6.6491 
5.7348 
6.0682 
4.6301 

10 
I 
5.0864 
4.9416 
5.0606 
4.8724 
5.0033 
4.7204 
4.9461 
4.5687 
4.9022 
4.4509 


II 
8.1494 
8.0192 
7.6135 
7.4093 
6.8790 
6.5340 
6.4299 
5.9597 
6.1797 
5.6180 


III 
11.2315 
11.1253 
9.7286 
9.5215 
8.3168 
7.9292 
7.6415 
7.0968 
7.3065 
6.6482 

30 
I 
5.4917 
5.3802 
5.5689 
5.4313 
5.7062 
5.5226 
5.8120 
5.5926 
5.8787 
5.6365 


II 
8.5417 
8.4299 
8.1965 
8.0347 
7.7609 
7.5278 
7.5224 
7.2443 
7.4002 
7.0970 


III 
11.5646 
11.4678 
10.3378 
10.1663 
9.3234 
9.0556 
8.9161 
8.5908 
8.7410 
8.3842 

50 
I 
5.8182 
5.7265 
5.9653 
5.8548 
6.2167 
6.0768 
6.4056 
6.2448 
6.5237 
6.3503 


II 
8.8841 
8.7858 
8.6758 
8.5404 
8.4158 
8.2354 
8.2776 
8.0727 
8.2084 
7.9909 


III 
11.8707 
11.7814 
10.8551 
10.7077 
10.0810 
9.8716 
9.8017 
9.5612 
9.6926 
9.4365 
1000 
0 
I 
6.2618 
6.1852 
6.2150 
6.1160 
6.1165 
5.9689 
6.0243 
5.8288 
5.9568 
5.7247 


II 
8.3843 
8.2646 
7.8372 
7.6506 
7.0957 
6.7842 
6.6478 
6.2287 
6.4003 
5.9049 


III 
11.2317 
11.1255 
9.6624 
9.4509 
8.1392 
7.7229 
7.3664 
6.7465 
6.9595 
6.1704 

10 
I 
6.3824 
6.3110 
6.3694 
6.2782 
6.3409 
6.2093 
6.3131 
6.1448 
6.2922 
6.0976 


II 
8.5725 
8.4612 
8.1213 
7.9545 
7.5384 
7.2825 
7.2080 
6.8884 
7.0342 
6.6758 


III 
11.4022 
11.3009 
9.9869 
9.7961 
8.7172 
8.3848 
8.1443 
7.7062 
7.8719 
7.3657 

30 
I 
6.6018 
6.5387 
6.6467 
6.5673 
6.7290 
6.6197 
6.7943 
6.6607 
6.8365 
6.6868 


II 
8.9138 
8.8158 
8.6131 
8.4744 
8.2434 
8.0509 
8.0465 
7.8224 
7.9472 
7.7063 


III 
11.7214 
11.6285 
10.5549 
10.3941 
9.6148 
9.3719 
9.2465 
8.9571 
9.0901 
8.7759 

50 
I 
6.7982 
6.7414 
6.8920 
6.8213 
7.0595 
6.9652 
7.1907 
7.0787 
7.2751 
7.1518 


II 
9.2178 
9.1300 
9.0323 
8.9127 
8.8034 
8.6466 
8.6831 
8.5068 
8.6233 
8.4372 


III 
12.0160 
11.9299 
11.0436 
10.9038 
10.3143 
10.1194 
10.0545 
9.8326 
9.9536 
9.7182 
Table 4. First three values of frequency parameter of CS EulerBernoulli nanobeam








0 
0.1 
0.2 
0.3 
0.4 








Mode 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 

1 
0 
I 
4.0655 
3.7797 
3.9825 
3.6454 
3.7920 
3.3102 
3.5875 
2.8782 
3.4149 
2.3753 



II 
7.1586 
6.9764 
6.5937 
6.3298 
5.7313 
5.2570 
5.1121 
4.3304 
4.7087 
3.5081 



III 
10.2760 
10.1435 
8.7705 
8.5286 
7.1604 
6.6543 
6.2284 
5.3495 
5.6740 
4.2981 


10 
I 
4.4351 
4.2257 
4.4020 
4.1661 
4.3280 
4.0326 
4.2531 
3.8955 
4.1945 
3.7858 



II 
7.4343 
7.2727 
6.9738 
6.7535 
6.3289 
5.9920 
5.9283 
5.4830 
5.7039 
5.1771 



III 
10.4861 
10.3616 
9.1342 
8.9213 
7.8214 
7.4461 
7.1736 
6.6556 
6.8449 
6.2233 


30 
I 
4.9708 
4.8292 
4.9924 
4.8377 
5.0270 
4.8478 
5.0497 
4.8500 
5.0621 
4.8483 



II 
7.9073 
7.7745 
7.5880 
7.4194 
7.1786 
6.9553 
6.9549 
6.6943 
6.8427 
6.5603 



III 
10.8717 
10.7604 
9.7535 
9.5801 
8.7899 
8.5329 
8.3815 
8.0736 
8.1961 
7.8604 


50 
I 
5.3684 
5.2589 
5.4239 
5.3054 
5.5165 
5.3833 
5.5854 
5.4409 
5.6287 
5.4767 



II 
8.3066 
8.1930 
8.0812 
7.9428 
7.8028 
7.6313 
7.6607 
7.4694 
7.5943 
7.3922 



III 
11.2198 
11.1188 
10.2733 
10.1256 
9.5151 
9.3149 
9.2184 
8.9913 
9.0909 
8.8502 

10 
0 
I 
4.0985 
3.8207 
4.0176 
3.6910 
3.8326 
3.3706 
3.6352 
2.9682 
3.4701 
2.5278 



II 
7.1648 
6.9830 
6.6016 
6.3387 
5.7432 
5.2724 
5.1288 
4.3578 
4.7301 
3.5591 



III 
10.2781 
10.1457 
8.7738 
8.5322 
7.1665 
6.6619 
6.2377 
5.3641 
5.6863 
4.3261 


10 
I 
4.4606 
4.2552 
4.4281 
4.1969 
4.3555 
4.0665 
4.2820 
3.9330 
4.2247 
3.8266 



II 
7.4398 
7.2786 
6.9805 
6.7608 
6.3378 
6.0025 
5.9391 
5.4966 
5.7160 
5.1932 



III 
10.4880 
10.3636 
9.1371 
8.9245 
7.8261 
7.4516 
7.1797 
6.6632 
6.8519 
6.2326 


30 
I 
4.9890 
4.8490 
5.0104 
4.8574 
5.0446 
4.8674 
5.0671 
4.8696 
5.0794 
4.8679 



II 
7.9118 
7.7792 
7.5932 
7.4249 
7.1847 
6.9620 
6.9615 
6.7018 
6.8497 
6.5682 



III 
10.8735 
10.7622 
9.7559 
9.5826 
8.7932 
8.5365 
8.3853 
8.0779 
8.2002 
7.8650 


50 
I 
5.3829 
5.2743 
5.4380 
5.3205 
5.5298 
5.3977 
5.5983 
5.4548 
5.6413 
5.4903 



II 
8.3105 
8.1971 
8.0855 
7.9472 
7.8076 
7.6363 
7.6657 
7.4748 
7.5994 
7.3977 



III 
11.2214 
11.1205 
10.2754 
10.1278 
9.5177 
9.3177 
9.2213 
8.9943 
9.0939 
8.8534 

1000 
0 
I 
5.9722 
5.8895 
5.9467 
5.8555 
5.8927 
5.7838 
5.8418 
5.7162 
5.8043 
5.6663 



II 
7.7595 
7.6179 
7.3316 
7.1437 
6.7516 
6.4796 
6.4040 
6.0623 
6.2136 
5.8240 



III 
10.4988 
10.3748 
9.1193 
8.9055 
7.7608 
7.3758 
7.0738 
6.5297 
6.7169 
6.0506 


10 
I 
6.1014 
6.0251 
6.0888 
6.0049 
6.0614 
5.9620 
6.0347 
5.9212 
6.0145 
5.8911 



II 
7.9792 
7.8496 
7.6159 
7.4493 
7.1431 
6.9162 
6.8751 
6.6047 
6.7350 
6.4375 



III 
10.6963 
10.5792 
9.4456 
9.2540 
8.2980 
7.9888 
7.7712 
7.3768 
7.5178 
7.0703 


30 
I 
6.3339 
6.2673 
6.3444 
6.2712 
6.3614 
6.2759 
6.3727 
6.2769 
6.3789 
6.2761 



II 
8.3702 
8.2588 
8.1045 
7.9672 
7.7752 
7.6017 
7.6014 
7.4053 
7.5161 
7.3073 



III 
11.0611 
10.9555 
10.0122 
9.8523 
9.1366 
8.9092 
8.7768 
8.5113 
8.6163 
8.3307 


50 
I 
6.5402 
6.4806 
6.5711 
6.5057 
6.6239 
6.5484 
6.6641 
6.5807 
6.6897 
6.6010 



II 
8.7117 
8.6136 
8.5178 
8.4001 
8.2825 
8.1401 
8.1644 
8.0077 
8.1096 
7.9452 



III 
11.3926 
11.2962 
10.4963 
10.3581 
9.7926

9.6096 
9.5219 
9.3167 
9.4065 
9.1903 

Table 5. First three values of frequency parameter of SS EulerBernoulli nanobeam








0 
0.1 
0.2 
0.3 
0.4 








Mode 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
1 
0 
I 
3.3637 
2.8800 
3.3047 
2.7837 
3.1658 
2.5340 
3.0110 
2.1894 
2.8756 
1.7397 


II 
6.4002 
6.1613 
5.9303 
5.6234 
5.1867 
4.6973 
4.6355 
3.8724 
4.2700 
3.1268 


III 
9.5037 
9.3445 
8.1657 
7.9091 
6.6878 
6.1893 
5.8167 
4.9733 
5.2951 
3.9890 

10 
I 
3.8803 
3.5975 
3.8423 
3.5496 
3.7566 
3.4393 
3.6674 
3.3212 
3.5952 
3.2226 


II 
6.7474 
6.5458 
6.3556 
6.1112 
5.7831 
5.4491 
5.4099 
4.9893 
5.1930 
4.7058 


III 
9.7524 
9.6055 
8.5461 
8.3242 
7.3312 
6.9668 
6.7148 
6.2233 
6.3968 
5.8125 

30 
I 
4.5380 
4.3706 
4.5145 
4.3442 
4.4624 
4.2856 
4.4102 
4.2264 
4.3693 
4.1799 


II 
7.3144 
7.1582 
7.0147 
6.8364 
6.6092 
6.3938 
6.3696 
6.1270 
6.2407 
5.9813 


III 
10.1995 
10.0716 
9.1829 
9.0058 
8.2645 
8.0176 
7.8563 
7.5658 
7.6645 
7.3497 

50 
I 
4.9930 
4.8695 
4.9753 
4.8505 
4.9367 
4.8087 
4.8983 
4.7672 
4.8686 
4.7349 


II 
7.7737 
7.6445 
7.5275 
7.3847 
7.2069 
7.0432 
7.0251 
6.8477 
6.9300 
6.7448 


III 
10.5946 
10.4807 
9.7094 
9.5605 
8.9588 
8.7675 
8.6452 
8.4312 
8.5030 
8.2775 
10 
0 
I 
3.4213 
2.9699 
3.3653 
2.8827 
3.2345 
2.6622 
3.0903 
2.3780 
2.9659 
2.0643 


II 
6.4088 
6.1709 
5.9411 
5.6360 
5.2028 
4.7189 
4.6580 
3.9106 
4.2987 
3.1979 


III 
9.5063 
9.3472 
8.1698 
7.9137 
6.6953 
6.1988 
5.8281 
4.9915 
5.3102 
4.0240 

10 
I 
3.9183 
3.6449 
3.8814 
3.5989 
3.7983 
3.4934 
3.7122 
3.3810 
3.6427 
3.2879 


II 
6.7547 
6.5538 
6.3643 
6.1210 
5.7947 
5.4629 
5.4240 
5.0073 
5.2090 
4.7272 


III 
9.7549 
9.6080 
8.5497 
8.3280 
7.3369 
6.9734 
6.7222 
6.2327 
6.4054 
5.8240 

30 
I 
4.5619 
4.3973 
4.5387 
4.3713 
4.4875 
4.3139 
4.4362 
4.2559 
4.3961 
4.2104 


II 
7.3202 
7.1643 
7.0212 
6.8435 
6.6170 
6.4024 
6.3783 
6.1367 
6.2499 
5.9918 


III 
10.2017 
10.0738 
9.1858 
9.0088 
8.2685 
8.0220 
7.8609 
7.5710 
7.6695 
7.3554 

50 
I 
5.0110 
4.8889 
4.9935 
4.8701 
4.9553 
4.8288 
4.9173 
4.7878 
4.8880 
4.7560 


II 
7.7785 
7.6495 
7.5328 
7.3903 
7.2129 
7.0496 
7.0316 
6.8547 
6.9367 
6.7521 


III 
10.5965 
10.4827 
9.7119 
9.5630 
8.9619 
8.7708 
8.6487 
8.4349 
8.5067 
8.2814 
1000 
0 
I 
5.7941 
5.7164 
5.7828 
5.7047 
5.7583 
5.6791 
5.7342 
5.6541 
5.7158 
5.6349 


II 
7.1930 
7.0283 
6.8764 
6.6866 
6.4425 
6.2086 
6.1822 
5.9147 
6.0406 
5.7519 


III 
9.7821 
9.6366 
8.5901 
8.3717 
7.4005 
7.0472 
6.8044 
6.3352 
6.5001 
5.9486 

10 
I 
5.9169 
5.8441 
5.9064 
5.8332 
5.8834 
5.8093 
5.8609 
5.7859 
5.8436 
5.7680 


II 
7.4447 
7.2968 
7.1617 
6.9947 
6.7835 
6.5853 
6.5632 
6.3429 
6.4458 
6.2123 


III 
10.0112 
9.8757 
8.9209 
8.7270 
7.8963 
7.6105 
7.4205 
7.0704 
7.1906 
6.8018 

30 
I 
6.1420 
6.0771 
6.1325 
6.0673 
6.1120 
6.0461 
6.0919 
6.0254 
6.0766 
6.0095 


II 
7.8829 
7.7591 
7.6474 
7.5114 
7.3428 
7.1885 
7.1715 
7.0053 
7.0822 
6.9093 


III 
10.4272 
10.3076 
9.4897 
9.3298 
8.6753 
8.4636 
8.3273 
8.0863 
8.1675 
7.9111 

50 
I 
6.3447 
6.2859 
6.3361 
6.2771 
6.3176 
6.2580 
6.2994 
6.2393 
6.2855 
6.2250 


II 
8.2582 
8.1510 
8.0550 
7.9392 
7.7974 
7.6694 
7.6555 
7.5199 
7.5824 
7.4427 


III 
10.7986 
10.6912 
9.9715 
9.8343 
9.2876 
9.1166 
9.0082 
8.8202 
8.8830 
8.6865 
Table 6. Critical buckling load parameter of EulerBernoulli nanobeam






0 
0.1 
0.2 
0.3 
0.4 


CC 

1 
0 
39.5544 
28.3659 
15.3518 
10.3878 
5.4280 

30 
69.5544 
58.3659 
45.3518 
40.3878 
35.4280 

50 
89.5544 
78.3659 
65.3518 
60.3878 
55.4280 
10 
0 
40.2376 
28.9201 
15.7558 
10.4530 
6.0810 

30 
70.2376 
58.9201 
45.7558 
40.4530 
36.0810 

50 
90.2376 
78.9201 
65.7558 
60.4530 
56.0810 
1000 
0 
101.1910 
61.2950 
28.6333 
13.6863 
7.8113 

30 
131.1910 
91.2950 
58.6333 
43.6863 
37.8113 

50 
151.1910 
111.2950 
78.6333 
63.6863 
57.8113 


CS 

1 
0 
20.2733 
16.8759 
11.2375 
7.2283 
4.8300 

30 
50.2733 
46.8759 
41.2375 
37.2283 
34.8300 

50 
70.2733 
66.8759 
61.2375 
57.2283 
54.8300 
10 
0 
21.0149 
17.5676 
11.8463 
7.7780 
6.1985 

30 
51.0149 
47.5676 
41.8463 
37.7780 
36.1985 

50 
71.0149 
67.5676 
61.8463 
57.7780 
56.1985 
1000 
0 
74.4955 
55.8308 
25.0525 
12.7116 
7.3664 

30 
104.4950 
85.8308 
55.0525 
42.7116 
37.3664 

50 
124.4950 
105.8310 
75.0525 
62.7116 
57.3664 


SS 

1 
0 
9.9709 
9.0843 
7.1774 
5.3282 
3.9280 

30 
39.9709 
39.0843 
37.1774 
35.3282 
33.9280 

50 
59.9709 
59.0843 
57.1774 
55.3282 
53.9280 
10 
0 
10.8828 
9.9963 
8.0893 
6.2400 
4.8399 

30 
40.8828 
39.9962 
38.0893 
36.2400 
34.8399 

50 
60.8828 
59.9962 
58.0893 
56.2400 
54.8399 
1000 
0 
64.8087 
53.6346 
25.8330 
12.3344 
6.9380 

30 
94.8087 
83.6346 
55.8330 
42.3344 
36.9380 

50 
114.8087 
103.6346 
75.8330 
62.3344 
56.9380 
Table 7. Comparison of frequency parameter of EulerBernoulli nanobeam
Boundary condition 
Reference 



Mode I 
Mode II 
Mode III 
CC 
Demir [61] 
0 
0 
0 
4.73004 
7.8532 
11.0856 

Ebrahimi and Salari [34] 



4.7299 
7.8525 
10.9934 

Present 



4.7300 
7.8532 
10.9956 

Demir [61] 
0 
0 
0.05 
4.69433 
7.64178 
10.4625 

Present 



4.6943 
7.6418 
10.4042 

Demir [61] 
0 
0 
0.2 
4.27661 
6.03520 
7.28636 

Present 



4.2766 
6.0352 
7.3840 

Demir [61] 
10000 
0 
0.2 
10.08260 
10.31634 
10.64047 

Present 



10.0826 
10.3163 
10.6723 

RahbarRanji and Shahbaztabar [62] 
2.5 
2.5* 
0 
5.3200 
8.3815 
11.4280 

Present 



5.3224 
8.3821 
11.4282 

RahbarRanji and Shahbaztabar [62] 
10000 
2.5* 
0 
10.1943 
11.0546 
12.8252 

Present 



10.1943 
11.0546 
12.8251 
CS 
Wang, Zhang, and He [10] 
0 
0 
0 
3.9266 
7.0686 
10.2102 

Ebrahimi and Salari [34] 



3.9265 
7.0679 
10.2081 

Present 



3.9266 
7.0686 
10.2102 

Wang, Zhang, and He [10] 
0 
0 
0.3 
3.2828 
4.7668 
5.8371 

Present 



3.28284 
4.7668 
5.8371 

Wang, Zhang, and He [10] 
0 
0 
0.5 
2.7899 
3.8325 
4.6105 

Present 



2.7899 
3.8325 
4.6105 
SS 
Demir [61] 
0 
0 
0 
3.14159 
6.28319 
9.42394 

Present 



3.1416 
6.2832 
9.4248 

Demir [61] 
0 
0 
0.05 
3.12251 
6.13706 
8.96310 

Present 



3.1225 
6.1371 
8.9639 

Demir [61] 
0 
0 
0.2 
2.89083 
4.95805 
6.45140 

Present 



2.8908 
4.9581 
6.4520 

Demir [61] 
10000 
0 
0.2 
10.01741 
10.14776 
10.40748 

Present 



10.0174 
10.1478 
10.4076 

RahbarRanji and Shahbaztabar [62] 
10000 
2.5* 
0 
10.0842 
10.5806 
11.9042 

Present 



10.0842 
10.5806 
11.9042 
Table 8. Comparison of the lowest nondimensional critical buckling load
parameter for EulerBernoulli nanobeam without foundation
Boundary condition 
Reference 


CC 
Ghannadpour, Mohammadi, and Fazilati [63] 
0 
39.4784 

Pradhan and Phadikar [30] 

39.4784 

Nejad, Hadi and Rastgoo [35] 

39.4784 

Zhu, Wang and Dai [38] 

39.47842 

Present 

39.4784 

Ghannadpour, Mohammadi, and Fazilati [63] 
0.2 
15.3068 

Nejad, Hadi and Rastgoo [35] 

15.3068 

Present 

15.3069 
CS 
Ghannadpour, Mohammadi, and Fazilati [63] 
0 
20.1907 

Pradhan and Phadikar [30] 

20.1907 

Nejad, Hadi and Rastgoo [35] 

20.1907 

Zhu, Wang and Dai [38] 

20.19073 

Present 

20.1907 

Ghannadpour, Mohammadi, and Fazilati [63] 
0.2 
11.1697 

Nejad, Hadi and Rastgoo [35] 

11.1697 

Present 

11.1697 
SS 
Sari, AlKouz and Atieh [64] 
0 
9.8696 

Reddy [29] 

9.8696 

Ebrahimi and Salari [34] 

9.8696044 

Eltaher, Emam and Mahmoud [33] 

9.86973 

Wang and Cai [3] 

9.8696 

Nejad, Hadi and Rastgoo [35] 

9.8696 

Zhu, Wang and Dai [38] 

9.86960 

Present 

9.8696 

Sari, AlKouz and Atieh [64] 
0.1 
8.9830 

Reddy [29] 

8.9830 

Eltaher, Emam and Mahmoud [33] 

8.98312 

Present 

8.9830 

Sari, AlKouz and Atieh [64] 

8.2426 

Reddy [29] 

8.2426 

Eltaher, Emam and Mahmoud [33] 

8.24267 

Present 

8.2426 

Sari, AlKouz and Atieh [64] 
0.2 
7.0762 

Reddy [29] 

7.0761 

Eltaher, Emam and Mahmoud [33] 

7.07614 

Nejad, Hadi and Rastgoo [35] 

7.076 

Present 

7.0761 

Sari, AlKouz and Atieh [64] 

6.6085 

Reddy [29] 

6.6085 

Present 

6.6084 
The first three frequencies and critical buckling loads are also presented through graphs via Figs. (29). The effect of nonlocal parameter
along with the shear foundation parameter on frequency is shown in Figs. (23) for
for CC and CS beams, respectively.

Fig. 2. (a)First mode (b) second mode (c) third mode of CC nanobeam for 

Fig. 3. (a)First mode (b) second mode (c) third mode of CS nanobeam for 
Figure 4 shows the variation of frequency with respect to nonlocal parameters for different combinations of Winkler and shear foundation parameters.

Fig. 4. (a)First mode (b) second mode (c) third mode of SS nanobeam for 
The effect of the nonlocal parameter along with the Winkler foundation parameter on frequency is shown in Figs. (56) for CC and CS beams for fixed values of load parameter and shear foundation parameter.

Fig. 5. First three modes of CC nanobeam for 

Fig. 6. First three modes of CS nanobeam for 
Figure 7 shows the effect of the nonlocal parameter along with the shear foundation parameter on critical buckling loads for two different values of the Winkler foundation parameter for CC, CS, and SS beams.

Fig. 7. Critical buckling load (a) CC (b) CS (c) SS 
Figure 8 depicts a threedimensional variation of critical buckling load for different values of the Winkler foundation parameter and nonlocal parameter keeping the shear foundation parameter constant for CC, CS, and SS beams.

Fig. 8. Critical buckling load (a) CC (b) CS (c) SS 
Figure 9 presents a threedimensional variation of critical buckling load for different values of shear foundation parameter and nonlocal parameter keeping the Winkler foundation parameter constant for CC, CS, and SS beams.

Fig. 9. Critical buckling load (a) CC (b) CS (c) SS 
Free transverse vibration and buckling of an EulerBernoulli nanobeam resting on the Pasternak foundation have been studied on the basis of Eringen’s nonlocal elasticity theory. The PDQM and the HDQM are used to obtain the first three values of the frequency parameter and the lowest critical buckling load. A computer program in C++ is developed to calculate the results. In this analysis, CC, CS, and SS boundary conditions have been considered. The study shows that nonlocal parameters, boundary conditions, axial force parameters, and elastic foundation moduli have considerable impacts on the results. So, it is concluded from the present study that
Nomenclature

Crosssection area of the beam 

Clamped edge 

Weighting coefficients of m^{th} order 

Flexural rigidity 

Young’s modulus of the plate material 

Second moment of area 

Nonlocal modulus 

Foundation parameters 

Winkler and shear foundation stiffnesses 

Length of the beam 

Bending moment 

Number of grid points 
p 
Compressive load 

Critical buckling load 

Simply supported edge 

Time 

Kinetic energy of the beam 

Strain energy of the beam 

Transverse deflection 

Nondimensional transverse deflection 

Potential energy due to Pasternak foundation 

Work done by the compressive load 

Nonlocal parameter 

Circular frequency 

Density of beam material 

Normal stress 

Normal strain 

Frequency parameter 

The Laplace operator 
Algorithm of C++
Step 1: Generation of grid points
Step 2: Generation of weighting coefficients
Step 3: Discretization of governing equation at grid points
Step 4: Implementation of boundary conditions
Step 5: Implementation of bisection method to obtain frequencies and critical buckling loads.
Conflicts of Interest
The authors declare that there is no conflict of interest between the authors.
Acknowledgment
The authors are thankful to the learned reviewers for their critical comments to improve the quality of the paper.
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