**Figure 1.**
The beam model with bending and shear deflection. (**a**) Total deflection and rotation, (**b**) pure bending and rotation, (**c**) transverse shear deflection, (**d**) shear angle rotation.

**Figure 1.**
The beam model with bending and shear deflection. (**a**) Total deflection and rotation, (**b**) pure bending and rotation, (**c**) transverse shear deflection, (**d**) shear angle rotation.

**Figure 2.**
Modeling of variable thickness FGM beam.

**Figure 2.**
Modeling of variable thickness FGM beam.

**Figure 3.**
The buckled form of the beam.

**Figure 3.**
The buckled form of the beam.

**Figure 4.**
The non-dimensional transverse deflection of S-S and C-C supported FGM beam with different values of index $p$ when $L/{h}_{0}=10$ and $n=1$.

**Figure 4.**
The non-dimensional transverse deflection of S-S and C-C supported FGM beam with different values of index $p$ when $L/{h}_{0}=10$ and $n=1$.

**Figure 5.**
The non-dimensional transverse deflection of S-S and C-C supported FGM beam with different values of index $n$ with $L/{h}_{0}=10$ and $p=0.5$.

**Figure 5.**
The non-dimensional transverse deflection of S-S and C-C supported FGM beam with different values of index $n$ with $L/{h}_{0}=10$ and $p=0.5$.

**Figure 6.**
The maximum non-dimensional transverse deflection, ${\overline{w}}_{\mathrm{max}}$, of S-S and C-C supported FGM beam depend on index $p$ with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $n=1$.

**Figure 6.**
The maximum non-dimensional transverse deflection, ${\overline{w}}_{\mathrm{max}}$, of S-S and C-C supported FGM beam depend on index $p$ with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $n=1$.

**Figure 7.**
The maximum non-dimensional transverse deflection, ${\overline{w}}_{\mathrm{max}}$, of S-S and C-C supported FGM beam depending on index $n$ with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $p=0.5$.

**Figure 7.**
The maximum non-dimensional transverse deflection, ${\overline{w}}_{\mathrm{max}}$, of S-S and C-C supported FGM beam depending on index $n$ with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $p=0.5$.

**Figure 8.**
Distribution of axial normal stress, ${\sigma}_{x}$, and shear stress, ${\tau}_{xz}$, at the mid-span of the beam across the height of S-S supported FGM beam under uniform load with $L/{h}_{0}=\text{\hspace{0.17em}}100$.

**Figure 8.**
Distribution of axial normal stress, ${\sigma}_{x}$, and shear stress, ${\tau}_{xz}$, at the mid-span of the beam across the height of S-S supported FGM beam under uniform load with $L/{h}_{0}=\text{\hspace{0.17em}}100$.

**Figure 9.**
The influence of index $p$ on the fundamental non-dimensional natural frequencies, ${\overline{\omega}}_{1}$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $n=1$.

**Figure 9.**
The influence of index $p$ on the fundamental non-dimensional natural frequencies, ${\overline{\omega}}_{1}$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $n=1$.

**Figure 10.**
The influence of $n$ on the fundamental non-dimensional natural frequencies, ${\overline{\omega}}_{1}$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $p=0.5$.

**Figure 10.**
The influence of $n$ on the fundamental non-dimensional natural frequencies, ${\overline{\omega}}_{1}$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $p=0.5$.

**Figure 11.**
The first four mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=2,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Figure 11.**
The first four mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=2,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Figure 12.**
The first mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=2,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Figure 12.**
The first mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=2,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Figure 13.**
The influence of index $p$ on the non-dimensional critical load, ${\overline{Q}}_{cr}$, of an S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $n=1$.

**Figure 13.**
The influence of index $p$ on the non-dimensional critical load, ${\overline{Q}}_{cr}$, of an S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $n=1$.

**Figure 14.**
The influence of index $n$ on the non-dimensional critical load, ${\overline{Q}}_{cr}$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $p=0.5$.

**Figure 14.**
The influence of index $n$ on the non-dimensional critical load, ${\overline{Q}}_{cr}$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10,\text{\hspace{0.17em}}20,\text{\hspace{0.17em}}50,\text{\hspace{0.17em}}100$ and $p=0.5$.

**Figure 15.**
The first buckling mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=0.5,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Figure 15.**
The first buckling mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=0.5,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Figure 16.**
The first buckling mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=2,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Figure 16.**
The first buckling mode shapes of the FGM beam with $p=0.5,\text{\hspace{0.17em}}n=2,\text{\hspace{0.17em}}L/{h}_{0}=10$.

**Table 1.**
Comparison of non-dimensional mid-span deflection of an S-S supported isotropic beam for different values of the thickness parameter, $\lambda ,\text{\hspace{0.17em}}n$ with $L/{h}_{0}=10$.

**Table 1.**
Comparison of non-dimensional mid-span deflection of an S-S supported isotropic beam for different values of the thickness parameter, $\lambda ,\text{\hspace{0.17em}}n$ with $L/{h}_{0}=10$.

$\mathit{\lambda}$ | n = 1 | n = 2 | n = 3 |
---|

[35] | Abaqus | Present | [35] | Abaqus | Present | [35] | Abaqus | Present |
---|

0.05 | 1.3370 | 1.3500 | 1.3405 | 1.3111 | 1.3237 | 1.3144 | 1.3349 | 1.3356 | 1.3385 |

0.10 | 1.3441 | 1.3584 | 1.3493 | 1.2889 | 1.3021 | 1.2920 | 1.3359 | 1.3368 | 1.3395 |

0.20 | 1.3727 | 1.3967 | 1.3849 | 1.2479 | 1.2626 | 1.2508 | 1.3400 | 1.3416 | 1.3437 |

0.30 | 1.4204 | 1.4625 | 1.4497 | 1.2103 | 1.2279 | 1.2138 | 1.3467 | 1.3500 | 1.3510 |

0.40 | 1.4872 | 1.5534 | 1.5502 | 1.1747 | 1.1956 | 1.1801 | 1.3561 | 1.3428 | 1.3632 |

**Table 2.**
Comparison of non-dimensional deflection at the central point of the S-S supported FGM beam with constant thickness and different values of the ratio of $L/h=5,$, $L/h=20$.

**Table 2.**
Comparison of non-dimensional deflection at the central point of the S-S supported FGM beam with constant thickness and different values of the ratio of $L/h=5,$, $L/h=20$.

p | L/h = 5 | L/h = 20 |
---|

[20] | SBT [35] | HBT [35] | Present | [20] | SBT [35] | HBT [35] | Present |
---|

0 | 3.1657 | 3.1649 | 3.1654 | 3.1657 | 2.8962 | 2.8962 | 2.8962 | 2.8962 |

0.5 | 4.8292 | 4.8278 | 4.8285 | 4.8348 | 4.4645 | 4.4644 | 4.4644 | 4.4648 |

1 | 6.2599 | 6.2586 | 6.2594 | 6.2599 | 5.8049 | 5.8049 | 5.8049 | 5.8049 |

2 | 8.0602 | 8.0683 | 8.0675 | 8.0303 | 7.4415 | 7.4421 | 7.4420 | 7.4397 |

5 | 9.7802 | 9.8367 | 9.8271 | 9.6483 | 8.8151 | 8.8188 | 8.8181 | 8.8069 |

10 | 10.8979 | 10.9420 | 10.9375 | 10.7194 | 9.6879 | 9.6908 | 9.6905 | 9.6767 |

**Table 3.**
Maximum non-dimensional transverse deflection, ${\overline{w}}_{\mathrm{max}}$, of a variable thickness beam.

**Table 3.**
Maximum non-dimensional transverse deflection, ${\overline{w}}_{\mathrm{max}}$, of a variable thickness beam.

$\mathit{L}/{\mathit{h}}_{0}$ | $\mathit{p}$ | S-S | C-C |
---|

n = 0 | n = 0.5 | n = 1 | n = 2 | n = 0 | n = 0.5 | n = 1 | n = 2 |
---|

10 | 0 | 2.9501 | 5.0428 | 7.6450 | 12.4967 | 0.6475 | 1.2046 | 1.5965 | 2.1175 |

0.5 | 4.5388 | 7.7639 | 11.7761 | 19.2590 | 0.9867 | 1.8432 | 2.4462 | 3.2483 |

1 | 5.8959 | 10.0885 | 15.3053 | 25.0363 | 1.2763 | 2.3884 | 3.1717 | 4.2139 |

2 | 7.5578 | 12.9313 | 19.6172 | 32.0881 | 1.6376 | 3.0634 | 4.0674 | 5.4034 |

5 | 8.9752 | 15.3402 | 23.2520 | 38.0049 | 1.9745 | 3.6699 | 4.8622 | 6.4472 |

10 | 9.8853 | 16.8831 | 25.5737 | 41.7764 | 2.1995 | 4.0688 | 5.3818 | 7.1264 |

20 | 0 | 2.8962 | 4.9748 | 7.5688 | 12.4132 | 0.5936 | 1.1383 | 1.5228 | 2.0363 |

0.5 | 4.4648 | 7.6707 | 11.6719 | 19.1450 | 0.9127 | 1.7523 | 2.3451 | 3.1368 |

1 | 5.8049 | 9.9740 | 15.1774 | 24.8963 | 1.1853 | 2.2770 | 3.0475 | 4.0769 |

2 | 7.4397 | 12.7829 | 19.4513 | 31.9066 | 1.5194 | 2.9186 | 3.9062 | 5.2256 |

5 | 8.8069 | 15.1288 | 23.0158 | 37.7459 | 1.8063 | 3.4628 | 4.6323 | 6.1939 |

10 | 9.6767 | 16.6219 | 25.2822 | 41.4567 | 1.9910 | 3.8118 | 5.0969 | 6.8128 |

50 | 0 | 2.8812 | 4.9578 | 7.5456 | 12.4021 | 0.5785 | 1.1233 | 1.5022 | 2.0150 |

0.5 | 4.4441 | 7.6483 | 11.6397 | 19.1344 | 0.8920 | 1.7328 | 2.3168 | 3.1082 |

1 | 5.7794 | 9.9470 | 15.1372 | 24.8863 | 1.1598 | 2.2536 | 3.0127 | 4.0421 |

2 | 7.4066 | 12.7482 | 19.3989 | 31.8952 | 1.4864 | 2.8883 | 3.8610 | 5.1806 |

5 | 8.7597 | 15.0791 | 22.9408 | 37.7255 | 1.7591 | 3.4164 | 4.5675 | 6.1296 |

10 | 9.6183 | 16.5627 | 25.1875 | 41.4396 | 1.9326 | 3.7526 | 5.0161 | 6.7344 |

100 | 0 | 2.8790 | 4.9590 | 7.5518 | 12.4096 | 0.5764 | 1.1247 | 1.5013 | 2.0129 |

0.5 | 4.4413 | 7.6516 | 11.6519 | 19.1490 | 0.8890 | 1.7356 | 2.3162 | 3.1058 |

1 | 5.7758 | 9.9518 | 15.1541 | 24.9056 | 1.1561 | 2.2575 | 3.0123 | 4.0393 |

2 | 7.4018 | 12.7551 | 19.4218 | 31.9211 | 1.4816 | 2.8933 | 3.8607 | 5.1771 |

5 | 8.7529 | 15.0879 | 22.9710 | 37.7591 | 1.7524 | 3.4219 | 4.5666 | 6.1242 |

10 | 9.6098 | 16.5766 | 25.2311 | 41.4862 | 1.9242 | 3.7590 | 5.0162 | 6.7286 |

**Table 4.**
Comparison of the first three non-dimensional natural frequencies for a cantilever isotropic bean with different values of the taper ratio.

**Table 4.**
Comparison of the first three non-dimensional natural frequencies for a cantilever isotropic bean with different values of the taper ratio.

c | ${\overline{\mathit{\omega}}}_{1}$ | ${\overline{\mathit{\omega}}}_{2}$ | ${\overline{\mathit{\omega}}}_{3}$ |
---|

[34] | Abaqus | Present | [34] | Abaqus | Present | [34] | Abaqus | Present |
---|

0.1 | 3.559 | 3.562 | 3.553 | 21.338 | 21.140 | 21.132 | 58.980 | 57.510 | 57.663 |

0.2 | 3.608 | 3.612 | 3.603 | 20.621 | 20.453 | 20.439 | 56.192 | 54.939 | 55.045 |

0.3 | 3.667 | 3.669 | 3.662 | 19.881 | 19.739 | 19.720 | 53.322 | 52.269 | 52.331 |

0.4 | 3.737 | 3.739 | 3.732 | 19.114 | 18.996 | 18.975 | 50.354 | 49.487 | 49.513 |

0.5 | 3.824 | 3.826 | 3.819 | 18.317 | 18.222 | 18.198 | 47.265 | 46.568 | 46.565 |

0.6 | 3.934 | 3.936 | 3.930 | 17.488 | 17.413 | 17.391 | 44.025 | 43.482 | 43.464 |

0.7 | 4.082 | 4.083 | 4.078 | 16.625 | 16.568 | 16.548 | 40.588 | 40.186 | 40.155 |

0.8 | 4.292 | 4.293 | 4.290 | 15.743 | 15.701 | 15.691 | 36.885 | 36.608 | 36.583 |

0.9 | 4.631 | 4.631 | 4.630 | 14.931 | 14.902 | 14.911 | 32.833 | 32.671 | 32.688 |

**Table 5.**
Comparison of the first three non-dimensional natural frequencies of $Al/A{l}_{2}{O}_{3}$ beam with $L/h=5$ and $L/h=20$.

**Table 5.**
Comparison of the first three non-dimensional natural frequencies of $Al/A{l}_{2}{O}_{3}$ beam with $L/h=5$ and $L/h=20$.

L/h | p | ${\overline{\mathit{\omega}}}_{1}$ | ${\overline{\mathit{\omega}}}_{2}$ | ${\overline{\mathit{\omega}}}_{3}$ |
---|

SBT [14] | HBT [14] | Present | SBT [14] | HBT [14] | Present | SBT [14] | HBT [14] | Present |
---|

5 | 0 | 5.1531 | 5.1527 | 5.2220 | 17.8868 | 17.8810 | 18.4730 | 34.2344 | 34.2085 | 35.6198 |

0.5 | 4.4110 | 4.4107 | 4.4693 | 15.4631 | 15.4587 | 15.9861 | 29.8569 | 29.8373 | 31.1588 |

1 | 3.9907 | 3.9904 | 4.0497 | 14.0138 | 14.0098 | 14.5588 | 27.1152 | 27.0971 | 28.5214 |

2 | 3.6263 | 3.6265 | 3.6936 | 12.6411 | 12.6407 | 13.2636 | 24.3237 | 24.3151 | 25.9539 |

5 | 3.3998 | 3.4014 | 3.4882 | 11.5324 | 11.5444 | 12.3067 | 21.6943 | 21.7187 | 23.6695 |

10 | 3.2811 | 3.2817 | 3.3644 | 11.0216 | 11.0246 | 11.7210 | 20.5581 | 20.5569 | 22.2828 |

20 | 0 | 5.4603 | 5.4603 | 5.4658 | 21.5736 | 21.5732 | 21.6578 | 47.5950 | 47.5930 | 47.9905 |

0.5 | 4.6511 | 4.6511 | 4.6556 | 18.3965 | 18.3962 | 18.4665 | 40.6542 | 40.6526 | 40.9852 |

1 | 4.2051 | 4.2051 | 4.2096 | 16.6347 | 16.6344 | 16.7048 | 36.7692 | 36.7679 | 37.1020 |

2 | 3.8361 | 3.8361 | 3.8413 | 15.1617 | 15.1619 | 15.2418 | 33.4681 | 33.4691 | 33.8471 |

5 | 3.6484 | 3.6485 | 3.6554 | 14.3728 | 14.3748 | 14.4806 | 31.5699 | 31.5789 | 32.0740 |

10 | 3.5389 | 3.5390 | 3.5457 | 13.9255 | 13.9264 | 14.0289 | 30.5337 | 30.5373 | 31.0136 |

**Table 6.**
First three non-dimensional natural frequencies, ${\overline{\omega}}_{i},\text{\hspace{0.17em}}i=1,2,3$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10$ depending on index $p$ and index $n$.

**Table 6.**
First three non-dimensional natural frequencies, ${\overline{\omega}}_{i},\text{\hspace{0.17em}}i=1,2,3$, of S-S and C-C supported FGM beam with $L/{h}_{0}=10$ depending on index $p$ and index $n$.

Mode | p | S-S | C-C |
---|

n = 0 | n = 0.5 | n = 1 | n = 2 | n = 0 | n = 0.5 | n = 1 | n = 2 |
---|

1 | 0 | 5.4144 | 4.5186 | 3.9296 | 3.3404 | 11.6989 | 9.3900 | 8.7793 | 8.3438 |

0.5 | 4.6165 | 3.8514 | 3.3485 | 2.8457 | 10.0244 | 8.0292 | 7.5020 | 7.1257 |

1 | 4.1761 | 3.4834 | 3.0282 | 2.5733 | 9.0882 | 7.2725 | 6.7930 | 6.4505 |

2 | 3.8104 | 3.1785 | 2.7632 | 2.3481 | 8.2881 | 6.6336 | 6.1967 | 5.8846 |

5 | 3.6201 | 3.0214 | 2.6277 | 2.2339 | 7.8125 | 6.2736 | 5.8667 | 5.5765 |

10 | 3.5072 | 2.9282 | 2.5476 | 2.1664 | 7.5246 | 6.0571 | 5.6688 | 5.3921 |

2 | 0 | 20.8896 | 17.5473 | 15.6625 | 13.8003 | 30.2375 | 25.1511 | 23.2809 | 21.5200 |

0.5 | 17.8784 | 14.9960 | 13.3746 | 11.7763 | 26.0655 | 21.6021 | 19.9692 | 18.4359 |

1 | 16.1999 | 13.5793 | 12.1067 | 10.6567 | 23.6957 | 19.6054 | 18.1126 | 16.7125 |

2 | 14.7756 | 12.3871 | 11.0448 | 9.7226 | 21.5964 | 17.8751 | 16.5163 | 15.2415 |

5 | 13.9541 | 11.7253 | 10.4682 | 9.2250 | 20.1629 | 16.7856 | 15.5430 | 14.3714 |

10 | 13.4588 | 11.3281 | 10.1235 | 8.9283 | 19.2868 | 16.1225 | 14.9524 | 13.8450 |

3 | 0 | 44.4998 | 37.8226 | 34.0651 | 30.3772 | 55.1307 | 47.1283 | 43.6107 | 40.0379 |

0.5 | 38.2849 | 32.4491 | 29.1820 | 25.9891 | 47.8038 | 40.6706 | 37.5628 | 34.4232 |

1 | 34.7730 | 29.4349 | 26.4536 | 23.5455 | 43.5761 | 36.9917 | 34.1349 | 31.2560 |

2 | 31.6987 | 26.8401 | 24.1253 | 21.4760 | 39.6909 | 33.7104 | 31.1133 | 28.4944 |

5 | 29.6875 | 25.2494 | 22.7497 | 20.2930 | 36.7108 | 31.4169 | 29.0861 | 26.7146 |

10 | 28.4609 | 24.2835 | 21.9179 | 19.5804 | 34.8885 | 30.0141 | 27.8481 | 25.6299 |

**Table 7.**
Fundamental non-dimensional natural frequencies, ${\overline{\omega}}_{1}$ of (S-S), and (C-C) FGM beams depend on ratio, $L/{h}_{0}$, and index $n$.

**Table 7.**
Fundamental non-dimensional natural frequencies, ${\overline{\omega}}_{1}$ of (S-S), and (C-C) FGM beams depend on ratio, $L/{h}_{0}$, and index $n$.

$\mathit{L}/{\mathit{h}}_{0}$ | p | S-S | C-C |
---|

n = 0 | n = 0.5 | n = 1 | n = 2 | n = 0 | n = 0.5 | n = 1 | n = 2 |
---|

10 | 0 | 5.4144 | 4.5186 | 3.9296 | 3.3404 | 11.6989 | 9.3900 | 8.7793 | 8.3438 |

0.5 | 4.6165 | 3.8514 | 3.3485 | 2.8457 | 10.0244 | 8.0292 | 7.5020 | 7.1257 |

1 | 4.1761 | 3.4834 | 3.0282 | 2.5733 | 9.0882 | 7.2725 | 6.7930 | 6.4505 |

2 | 3.8104 | 3.1785 | 2.7632 | 2.3481 | 8.2881 | 6.6336 | 6.1967 | 5.8846 |

5 | 3.6201 | 3.0214 | 2.6277 | 2.2339 | 7.8125 | 6.2736 | 5.8667 | 5.5765 |

10 | 3.5072 | 2.9282 | 2.5476 | 2.1664 | 7.5246 | 6.0571 | 5.6688 | 5.3921 |

20 | 0 | 2.7329 | 2.2752 | 1.9750 | 1.6760 | 6.1175 | 4.8350 | 4.4997 | 4.2589 |

0.5 | 2.3278 | 1.9377 | 1.6820 | 1.4273 | 5.2178 | 4.1212 | 3.8349 | 3.6291 |

1 | 2.1048 | 1.7520 | 1.5207 | 1.2904 | 4.7208 | 3.7276 | 3.4683 | 3.2820 |

2 | 1.9207 | 1.5987 | 1.3877 | 1.1775 | 4.3072 | 3.4012 | 3.1647 | 2.9947 |

5 | 1.8277 | 1.5215 | 1.3208 | 1.1209 | 4.0899 | 3.2328 | 3.0088 | 2.8478 |

10 | 1.7729 | 1.4759 | 1.2814 | 1.0875 | 3.9606 | 3.1328 | 2.9163 | 2.7608 |

50 | 0 | 1.0961 | 0.9118 | 0.7912 | 0.6708 | 2.4797 | 1.9477 | 1.8127 | 1.7133 |

0.5 | 0.9334 | 0.7763 | 0.6737 | 0.5712 | 2.1120 | 1.6585 | 1.5437 | 1.4589 |

1 | 0.8438 | 0.7019 | 0.6091 | 0.5164 | 1.9096 | 1.4993 | 1.3957 | 1.3190 |

2 | 0.7700 | 0.6405 | 0.5558 | 0.4712 | 1.7425 | 1.3682 | 1.2736 | 1.2036 |

5 | 0.7331 | 0.6097 | 0.5292 | 0.4486 | 1.6584 | 1.3025 | 1.2124 | 1.1457 |

10 | 0.7114 | 0.5915 | 0.5135 | 0.4352 | 1.6087 | 1.2636 | 1.1763 | 1.1114 |

100 | 0 | 0.5483 | 0.4559 | 0.3955 | 0.3353 | 1.2422 | 0.9734 | 0.9069 | 0.8572 |

0.5 | 0.4668 | 0.3881 | 0.3368 | 0.2855 | 1.0578 | 0.8287 | 0.7721 | 0.7299 |

1 | 0.4221 | 0.3509 | 0.3044 | 0.2581 | 0.9563 | 0.7492 | 0.6981 | 0.6598 |

2 | 0.3851 | 0.3202 | 0.2778 | 0.2355 | 0.8727 | 0.6837 | 0.6370 | 0.6021 |

5 | 0.3667 | 0.3048 | 0.2645 | 0.2242 | 0.8308 | 0.6509 | 0.6064 | 0.5732 |

10 | 0.3558 | 0.2957 | 0.2566 | 0.2175 | 0.8062 | 0.6314 | 0.5883 | 0.5560 |

**Table 8.**
Comparison of the buckling load of the constant thickness beam with different values of the ratio of $L/h$ for S-S and C-C support conditions.

**Table 8.**
Comparison of the buckling load of the constant thickness beam with different values of the ratio of $L/h$ for S-S and C-C support conditions.

L/h | S-S | C-C |
---|

Analytical Solution [8] | [52] | [8] | Abaqus | Present | Analytical Solution [8] | [52] | [8] | Abaqus | Present |
---|

10 | 8013.8 | 8021.8 | 8013.86 | 8020.9 | 8013.83 | 29766 | 29877 | 29770 | 29864 | 29767.2 |

100 | 8.223 | 8.231 | 8.2225 | 8.2258 | 8.2225 | 32.864 | 32.999 | 32.864 | 32.917 | 32.864 |

1000 | 0.0082 | 0.0082 | 0.00822 | 0.00823 | 0.00822 | 0.0329 | 0.0330 | 0.0329 | 0.03295 | 0.0329 |

**Table 9.**
Comparison of the nondimensional buckling load of the constant thickness FGM beam with the ratio of $L/h=5$ and $L/h=10$ for the S-S and C-C support condition.

**Table 9.**
Comparison of the nondimensional buckling load of the constant thickness FGM beam with the ratio of $L/h=5$ and $L/h=10$ for the S-S and C-C support condition.

p | L/h = 5, C-C | L/h = 5, S-S | L/h = 10, C-C | L/h = 10, S-S |
---|

[9] | Present | [9] | Present | [9] | Present | [9] | Present |
---|

0 | 154.35 | 154.37 | 48.835 | 48.836 | 195.34 | 195.35 | 52.309 | 52.308 |

0.5 | 103.22 | 103.23 | 31.967 | 31.968 | 127.87 | 127.87 | 33.996 | 33.997 |

1 | 80.498 | 80.505 | 24.687 | 24.687 | 98.749 | 98.752 | 26.171 | 26.171 |

2 | 62.614 | 62.620 | 19.245 | 19.245 | 76.980 | 76.983 | 20.416 | 20.416 |

5 | 50.384 | 50.389 | 16.024 | 16.024 | 64.096 | 64.099 | 17.192 | 17.194 |

10 | 44.267 | 44.272 | 14.427 | 14.427 | 57.708 | 57.711 | 15.612 | 15.612 |

**Table 10.**
Non-dimensional critical load, ${\overline{Q}}_{cr}$, of an S-S and C-C variable thickness FGM beam depending on the ratio of $L/{h}_{0}$ and index $n$.

**Table 10.**
Non-dimensional critical load, ${\overline{Q}}_{cr}$, of an S-S and C-C variable thickness FGM beam depending on the ratio of $L/{h}_{0}$ and index $n$.

$\mathit{L}/{\mathit{h}}_{0}$ | p | S-S | C-C |
---|

n = 0 | n = 0.5 | n = 1 | n = 2 | n = 0 | n = 0.5 | n = 1 | n = 2 |
---|

10 | 0 | 52.2381 | 29.9212 | 19.4184 | 11.9650 | 194.3954 | 103.3309 | 73.8638 | 52.8577 |

0.5 | 33.9560 | 19.4365 | 12.6075 | 7.7642 | 127.3125 | 67.4450 | 48.1566 | 34.4249 |

1 | 26.1410 | 14.9587 | 9.7007 | 5.9727 | 98.3404 | 52.0179 | 37.1226 | 26.5247 |

2 | 20.3927 | 11.6700 | 7.5685 | 4.6601 | 76.6582 | 40.5622 | 28.9508 | 20.6879 |

5 | 17.1703 | 9.8358 | 6.3844 | 3.9343 | 63.7783 | 33.9271 | 24.2601 | 17.3648 |

10 | 15.5882 | 8.9358 | 5.8042 | 3.5789 | 57.3891 | 30.6474 | 21.9459 | 15.7276 |

20 | 0 | 6.6546 | 3.7940 | 2.4532 | 1.5062 | 26.1201 | 13.5473 | 9.6064 | 6.8244 |

0.5 | 4.3168 | 2.4607 | 1.5908 | 0.9766 | 16.9785 | 8.7970 | 6.2363 | 4.4291 |

1 | 3.3203 | 1.8925 | 1.2234 | 0.7510 | 13.0709 | 6.7692 | 4.7982 | 3.4074 |

2 | 2.5907 | 1.4766 | 0.9546 | 0.5860 | 10.1967 | 5.2811 | 3.7435 | 2.6585 |

5 | 2.1884 | 1.2476 | 0.8067 | 0.4953 | 8.5855 | 4.4534 | 3.1582 | 2.2438 |

10 | 1.9917 | 1.1355 | 0.7344 | 0.4510 | 7.7944 | 4.0473 | 2.8713 | 2.0406 |

50 | 0 | 0.4282 | 0.2437 | 0.1575 | 0.0965 | 1.7075 | 0.8770 | 0.6219 | 0.4404 |

0.5 | 0.2776 | 0.1580 | 0.1021 | 0.0625 | 1.1074 | 0.5685 | 0.4032 | 0.2855 |

1 | 0.2135 | 0.1215 | 0.0785 | 0.0481 | 0.8516 | 0.4371 | 0.3101 | 0.2195 |

2 | 0.1666 | 0.0948 | 0.0613 | 0.0375 | 0.6645 | 0.3411 | 0.2419 | 0.1713 |

5 | 0.1408 | 0.0801 | 0.0518 | 0.0317 | 0.5616 | 0.2883 | 0.2045 | 0.1448 |

10 | 0.1283 | 0.0730 | 0.0472 | 0.0289 | 0.5112 | 0.2625 | 0.1863 | 0.1318 |

100 | 0 | 0.0536 | 0.0305 | 0.0197 | 0.0121 | 0.2141 | 0.1095 | 0.0778 | 0.0551 |

0.5 | 0.0347 | 0.0197 | 0.0128 | 0.0078 | 0.1388 | 0.0710 | 0.0504 | 0.0357 |

1 | 0.0267 | 0.0152 | 0.0098 | 0.0060 | 0.1067 | 0.0546 | 0.0387 | 0.0274 |

2 | 0.0208 | 0.0118 | 0.0076 | 0.0047 | 0.0833 | 0.0426 | 0.0302 | 0.0214 |

5 | 0.0176 | 0.0100 | 0.0065 | 0.0040 | 0.0704 | 0.0360 | 0.0256 | 0.0181 |

10 | 0.0160 | 0.0091 | 0.0059 | 0.0036 | 0.0641 | 0.0328 | 0.0233 | 0.0165 |