Figure 1.
Gradient types.
Figure 1.
Gradient types.
Figure 2.
Material structure: (a) composite laminate; (b) functionally graded materials (FGM).
Figure 2.
Material structure: (a) composite laminate; (b) functionally graded materials (FGM).
Figure 3.
Mathematical model of the functionally graded plate placed on elastic foundation.
Figure 3.
Mathematical model of the functionally graded plate placed on elastic foundation.
Figure 4.
Impact of the ratio a/b and a/h on the critical buckling temperature for case of linear temperature change across plate thickness: (a) a/h = 10, k0 = 10, k1 = 10; (b) a/b = 1, k0 = 10, k1 = 10.
Figure 4.
Impact of the ratio a/b and a/h on the critical buckling temperature for case of linear temperature change across plate thickness: (a) a/h = 10, k0 = 10, k1 = 10; (b) a/b = 1, k0 = 10, k1 = 10.
Figure 5.
Impact of the ratio a/b and a/h on the critical buckling temperature for case of nonlinear temperature change across plate thickness: (a) a/h = 10, s = 3, k0 = 10, k1 = 10; (b) a/b = 1, s = 3, k0 = 10, k1 = 10.
Figure 5.
Impact of the ratio a/b and a/h on the critical buckling temperature for case of nonlinear temperature change across plate thickness: (a) a/h = 10, s = 3, k0 = 10, k1 = 10; (b) a/b = 1, s = 3, k0 = 10, k1 = 10.
Figure 6.
Impact of the power law index p and parameter s on the critical buckling temperature for case of linear and nonlinear temperature change across plate thickness: (a) a/h = 10, a/b = 1, s = 1; (b) a/h = 10, a/b = 1, s = 3.
Figure 6.
Impact of the power law index p and parameter s on the critical buckling temperature for case of linear and nonlinear temperature change across plate thickness: (a) a/h = 10, a/b = 1, s = 1; (b) a/h = 10, a/b = 1, s = 3.
Figure 7.
Impact of the Winkler coefficient k0 and Pasternak coefficient k1 on the critical buckling temperature for case of linear temperature change across plate thickness: (a) a/h = 10, a/b = 1, k1 = 10; (b) a/h = 10, a/b = 1, k0 = 10.
Figure 7.
Impact of the Winkler coefficient k0 and Pasternak coefficient k1 on the critical buckling temperature for case of linear temperature change across plate thickness: (a) a/h = 10, a/b = 1, k1 = 10; (b) a/h = 10, a/b = 1, k0 = 10.
Figure 8.
Effect of the Winkler coefficient k0 and Pasternak coefficient k1 on the critical buckling temperature for case of nonlinear temperature change across plate thickness: (a) a/h = 10, a/b = 1, s = 3, k1 = 10; (b) a/h = 10, a/b = 1, k0 = 10.
Figure 8.
Effect of the Winkler coefficient k0 and Pasternak coefficient k1 on the critical buckling temperature for case of nonlinear temperature change across plate thickness: (a) a/h = 10, a/b = 1, s = 3, k1 = 10; (b) a/h = 10, a/b = 1, k0 = 10.
Figure 9.
Impact of the Winkler coefficient k0, Pasternak coefficient k1, power law index p and ratio a/b and a/h on non-dimensional natural frequencies : (a) a/h = 5, a/b = 0.5, m = 1, n = 1; (b) a/h = 5, m = 1, n = 1, k0 = 100, k1 = 100; (c) a/b = 0.5, m = 1, n = 1, k0 = 100, k1 = 100.
Figure 9.
Impact of the Winkler coefficient k0, Pasternak coefficient k1, power law index p and ratio a/b and a/h on non-dimensional natural frequencies : (a) a/h = 5, a/b = 0.5, m = 1, n = 1; (b) a/h = 5, m = 1, n = 1, k0 = 100, k1 = 100; (c) a/b = 0.5, m = 1, n = 1, k0 = 100, k1 = 100.
Figure 10.
3D diagrams of the non-dimensional natural frequencies of the plates placed on an elastic foundation for different ratios a/h, a/b, values k0, k1 and index p: (a) p = 5, m = 1, n = 1, k0 = 100, k1 = 100; (b) a/b = 1, m = 1, n = 1, k0 = 100, k1 = 100; (c) a/b = 1, a/h = 5, m = 1, n = 1.
Figure 10.
3D diagrams of the non-dimensional natural frequencies of the plates placed on an elastic foundation for different ratios a/h, a/b, values k0, k1 and index p: (a) p = 5, m = 1, n = 1, k0 = 100, k1 = 100; (b) a/b = 1, m = 1, n = 1, k0 = 100, k1 = 100; (c) a/b = 1, a/h = 5, m = 1, n = 1.
Table 1.
Different type of shape functions.
Table 1.
Different type of shape functions.
Number of Function | Shape Function, f(z) |
---|
No. 1 [41] | |
No. 2 [42] | |
No. 3 [43] | |
No. 4 [44] | |
No. 5–6 [45] | |
No. 7 [46,47] | |
No. 8 [48] | |
No. 9 [49] | |
No. 10 [50] | |
No. 11 [51] | |
No. 12 [51] | |
No. 13 [52] | |
Table 2.
Material properties of functionally graded materials (FGM) constituents.
Table 2.
Material properties of functionally graded materials (FGM) constituents.
Material | Material Properties |
---|
Elasticity Modulus | Poisson’s Ratio | Thermal Expansion Coefficient | Density |
---|
Aluminum () | | | | |
Alumina () | | | | |
Table 3.
Critical buckling temperatures (∆tcr) of FGM plates placed on an elastic foundation for case of linear temperature change across plate thickness (a/b = 1, m = n = 1, and Tm = 5°).
Table 3.
Critical buckling temperatures (∆tcr) of FGM plates placed on an elastic foundation for case of linear temperature change across plate thickness (a/b = 1, m = n = 1, and Tm = 5°).
p | Source | ∆tcr |
---|
k0 = 0, k1 = 0 | k0 = 10, k1 = 0 | k0 = 10, k1 = 10 |
---|
a/h = 10 | a/h = 20 | a/h = 10 | a/h = 20 | a/h = 10 | a/h = 20 |
---|
0 | [55] | 3.2276 | 0.833 | 3.3154 | 0.855 | 5.0479 | 1.2881 |
[25] | 3.2273 | 0.833 | 3.3151 | 0.855 | 5.0476 | 1.2881 |
Present study | 3.2274 | 0.8331 | 3.3151 | 0.855 | 5.0476 | 1.2881 |
No. 1 | 3.2273 | 0.833 | 3.3151 | 0.855 | 5.0476 | 1.2881 |
No. 2 | 3.2273 | 0.833 | 3.3151 | 0.855 | 5.0476 | 1.2881 |
No. 3 | 3.2276 | 0.833 | 3.3154 | 0.855 | 5.0479 | 1.2881 |
No. 4 | 3.2333 | 0.8334 | 3.3211 | 0.8554 | 5.0536 | 1.2885 |
No. 5 | 3.2273 | 0.833 | 3.3151 | 0.855 | 5.0476 | 1.2881 |
No. 6 | 3.2282 | 0.8331 | 3.316 | 0.855 | 5.0485 | 1.2882 |
No. 7 | 3.2284 | 0.8331 | 3.3162 | 0.855 | 5.0487 | 1.2882 |
No. 8 | 3.2285 | 0.8331 | 3.3163 | 0.855 | 5.0488 | 1.2882 |
No. 9 | 3.2285 | 0.8331 | 3.3163 | 0.855 | 5.0488 | 1.2882 |
No. 10 | 3.2273 | 0.833 | 3.3151 | 0.855 | 5.0476 | 1.2881 |
No. 11 | 3.2288 | 0.8331 | 3.3166 | 0.8551 | 5.0491 | 1.2882 |
No. 12 | 3.2275 | 0.833 | 3.3152 | 0.855 | 5.0477 | 1.2881 |
No. 13 | 3.2273 | 0.833 | 3.3151 | 0.855 | 5.0476 | 1.2881 |
1 | [55] | 1.413 | 0.3587 | 1.4897 | 0.3778 | 3.004 | 0.7564 |
[25] | 1.4129 | 0.3587 | 1.4896 | 0.3778 | 3.0039 | 0.7564 |
Present study | 1.413 | 0.3587 | 1.4897 | 0.3779 | 3.0039 | 0.7564 |
No. 1 | 1.4129 | 0.3587 | 1.4896 | 0.3778 | 3.0039 | 0.7564 |
No. 2 | 1.4129 | 0.3587 | 1.4896 | 0.3778 | 3.0039 | 0.7564 |
No. 3 | 1.413 | 0.3587 | 1.4897 | 0.3778 | 3.004 | 0.7564 |
No. 4 | 1.4152 | 0.3588 | 1.4919 | 0.378 | 3.0061 | 0.7565 |
No. 5 | 1.4129 | 0.3587 | 1.4896 | 0.3778 | 3.0039 | 0.7564 |
No. 6 | 1.4133 | 0.3587 | 1.49 | 0.3779 | 3.0042 | 0.7564 |
No. 7 | 1.4133 | 0.3587 | 1.49 | 0.3779 | 3.0043 | 0.7564 |
No. 8 | 1.4134 | 0.3587 | 1.4901 | 0.3779 | 3.0043 | 0.7564 |
No. 9 | 1.4134 | 0.3587 | 1.4901 | 0.3779 | 3.0043 | 0.7564 |
No. 10 | 1.4129 | 0.3587 | 1.4896 | 0.3778 | 3.0039 | 0.7564 |
No. 11 | 1.4135 | 0.3587 | 1.4902 | 0.3779 | 3.0044 | 0.7564 |
No. 12 | 1.413 | 0.3587 | 1.4897 | 0.3778 | 3.0039 | 0.7564 |
No. 13 | 1.4129 | 0.3587 | 1.4896 | 0.3778 | 3.0039 | 0.7564 |
5 | [55] | 1.16 | 0.2986 | 1.2576 | 0.323 | 3.1839 | 0.8046 |
[25] | 1.1606 | 0.2987 | 1.2582 | 0.3231 | 3.1845 | 0.8046 |
Present study | 1.1608 | 0.2987 | 1.2584 | 0.3231 | 3.1846 | 0.8047 |
No. 1 | 1.1606 | 0.2987 | 1.2582 | 0.3231 | 3.1845 | 0.8046 |
No. 2 | 1.1606 | 0.2987 | 1.2582 | 0.3231 | 3.1845 | 0.8046 |
No. 3 | 1.16 | 0.2986 | 1.2576 | 0.323 | 3.1839 | 0.8046 |
No. 4 | 1.1604 | 0.2986 | 1.2579 | 0.323 | 3.1842 | 0.8046 |
No. 5 | 1.1607 | 0.2987 | 1.2582 | 0.3231 | 3.1845 | 0.8046 |
No. 6 | 1.1626 | 0.2988 | 1.2602 | 0.3232 | 3.1865 | 0.8048 |
No. 7 | 1.1597 | 0.2986 | 1.2573 | 0.323 | 3.1835 | 0.8045 |
No. 8 | 1.1597 | 0.2986 | 1.2572 | 0.323 | 3.1835 | 0.8045 |
No. 9 | 1.1597 | 0.2986 | 1.2572 | 0.323 | 3.1835 | 0.8045 |
No. 10 | 1.1607 | 0.2987 | 1.2583 | 0.3231 | 3.1846 | 0.8046 |
No. 11 | 1.1631 | 0.2988 | 1.2607 | 0.3232 | 3.187 | 0.8048 |
No. 12 | 1.1602 | 0.2986 | 1.2578 | 0.323 | 3.184 | 0.8046 |
No. 13 | 1.1606 | 0.2987 | 1.2582 | 0.323 | 3.1844 | 0.8046 |
10 | [55] | 1.2183 | 0.3156 | 1.3317 | 0.344 | 3.5699 | 0.9035 |
[25] | 1.2186 | 0.3156 | 1.332 | 0.344 | 3.5701 | 0.9035 |
Present study | 1.2187 | 0.3157 | 1.3321 | 0.344 | 3.5703 | 0.9036 |
No. 1 | 1.2186 | 0.3156 | 1.332 | 0.344 | 3.5701 | 0.9035 |
No. 2 | 1.2186 | 0.3156 | 1.332 | 0.344 | 3.5701 | 0.9035 |
No. 3 | 1.2183 | 0.3156 | 1.3317 | 0.344 | 3.5699 | 0.9035 |
No. 4 | 1.2206 | 0.3158 | 1.3339 | 0.3441 | 3.5721 | 0.9037 |
No. 5 | 1.2186 | 0.3156 | 1.332 | 0.344 | 3.5702 | 0.9035 |
No. 6 | 1.2201 | 0.3157 | 1.3335 | 0.3441 | 3.5717 | 0.9036 |
No. 7 | 1.2184 | 0.3156 | 1.3318 | 0.344 | 3.57 | 0.9035 |
No. 8 | 1.2185 | 0.3156 | 1.3318 | 0.344 | 3.57 | 0.9035 |
No. 9 | 1.2185 | 0.3156 | 1.3318 | 0.344 | 3.57 | 0.9035 |
No. 10 | 1.2186 | 0.3156 | 1.332 | 0.344 | 3.5702 | 0.9035 |
No. 11 | 1.2203 | 0.3158 | 1.3337 | 0.3441 | 3.5719 | 0.9036 |
No. 12 | 1.2184 | 0.3156 | 1.3318 | 0.344 | 3.5699 | 0.9035 |
No. 13 | 1.2186 | 0.3156 | 1.3319 | 0.344 | 3.5701 | 0.9035 |
Table 4.
Critical buckling temperatures (∆tcr) of FGM plates placed on an elastic foundation for case of nonlinear temperature change across plate thickness (a/b = 1, m = n = 1, s = 3, and Tm = 5°).
Table 4.
Critical buckling temperatures (∆tcr) of FGM plates placed on an elastic foundation for case of nonlinear temperature change across plate thickness (a/b = 1, m = n = 1, s = 3, and Tm = 5°).
p | Source | ∆tcr |
---|
k0 = 0, k1 = 0 | k0 = 10, k1 = 0 | k0 = 10, k1 = 10 |
---|
a/h = 10 | a/h = 20 | a/h = 10 | a/h = 20 | a/h = 10 | a/h = 20 |
---|
0 | [55] | 6.4552 | 1.6661 | 6.6308 | 1.71 | 10.0958 | 2.5763 |
Present study | 6.4547 | 1.6661 | 6.6303 | 1.71 | 10.0953 | 2.5763 |
No. 1 | 6.4547 | 1.6661 | 6.6302 | 1.71 | 10.0953 | 2.5762 |
No. 2 | 6.4547 | 1.6661 | 6.6302 | 1.71 | 10.0953 | 2.5762 |
No. 3 | 6.4552 | 1.6661 | 6.6308 | 1.71 | 10.0958 | 2.5763 |
No. 4 | 6.4667 | 1.6669 | 6.6422 | 1.7108 | 10.1073 | 2.577 |
No. 5 | 6.4547 | 1.6661 | 6.6302 | 1.7182 | 10.0953 | 2.5762 |
No. 6 | 6.4564 | 1.6662 | 6.632 | 1.71 | 10.097 | 2.5764 |
No. 7 | 6.4569 | 1.6662 | 6.6324 | 1.7101 | 10.0974 | 2.5764 |
No. 8 | 6.4571 | 1.6663 | 6.6326 | 1.7101 | 10.0977 | 2.5764 |
No. 9 | 6.4571 | 1.6663 | 6.6326 | 1.7101 | 10.0977 | 2.5764 |
No. 10 | 6.4547 | 1.6661 | 6.6302 | 1.7101 | 10.0953 | 2.5762 |
No. 11 | 6.4577 | 1.6663 | 6.6332 | 1.7139 | 10.0983 | 2.5764 |
No. 12 | 6.455 | 1.6661 | 6.6305 | 1.7102 | 10.0956 | 2.5762 |
No. 13 | 6.4547 | 1.6661 | 6.6302 | 1.71 | 10.0953 | 2.5762 |
1 | [55] | 2.8269 | 0.7176 | 2.9804 | 0.756 | 6.0097 | 1.5133 |
Present study | 2.8268 | 0.7176 | 2.9802 | 0.756 | 6.0096 | 1.5133 |
No. 1 | 2.8267 | 0.7176 | 2.9802 | 0.7559 | 6.0095 | 1.5133 |
No. 2 | 2.8267 | 0.7176 | 2.9802 | 0.7559 | 6.0095 | 1.5133 |
No. 3 | 2.8269 | 0.7176 | 2.9804 | 0.756 | 6.0097 | 1.5133 |
No. 4 | 2.8312 | 0.7179 | 2.9847 | 0.7562 | 6.014 | 1.5136 |
No. 5 | 2.8267 | 0.7176 | 2.9802 | 0.7559 | 6.0095 | 1.5133 |
No. 6 | 2.8274 | 0.7176 | 2.9808 | 0.756 | 6.0102 | 1.5133 |
No. 7 | 2.8275 | 0.7176 | 2.981 | 0.756 | 6.0103 | 1.5133 |
No. 8 | 2.8276 | 0.7176 | 2.9811 | 0.756 | 6.0104 | 1.5133 |
No. 9 | 2.8276 | 0.7176 | 2.9811 | 0.756 | 6.0104 | 1.5133 |
No. 10 | 2.8267 | 0.7176 | 2.9802 | 0.7559 | 6.0095 | 1.5133 |
No. 11 | 2.8278 | 0.7177 | 2.9813 | 0.756 | 6.0106 | 1.5134 |
No. 12 | 2.8268 | 0.7176 | 2.9803 | 0.756 | 6.0096 | 1.5133 |
No. 13 | 2.8267 | 0.7176 | 2.9802 | 0.7559 | 6.0095 | 1.5133 |
5 | [55] | 2.0152 | 0.5188 | 2.1847 | 0.5612 | 5.5309 | 1.3977 |
Present study | 2.0165 | 0.5189 | 2.186 | 0.5613 | 5.5322 | 1.3978 |
No. 1 | 2.0162 | 0.5188 | 2.1858 | 0.5612 | 5.5319 | 1.3978 |
No. 2 | 2.0162 | 0.5188 | 2.1858 | 0.5612 | 5.5319 | 1.3978 |
No. 3 | 2.0152 | 0.5188 | 2.1847 | 0.5611 | 5.5309 | 1.3977 |
No. 4 | 2.0157 | 0.5188 | 2.1853 | 0.5612 | 5.5315 | 1.3977 |
No. 5 | 2.0163 | 0.5188 | 2.1858 | 0.5612 | 5.532 | 1.3978 |
No. 6 | 2.0197 | 0.5191 | 2.1892 | 0.5615 | 5.5354 | 1.398 |
No. 7 | 2.0146 | 0.5187 | 2.1841 | 0.5611 | 5.5303 | 1.3977 |
No. 8 | 2.0145 | 0.5187 | 2.1841 | 0.5611 | 5.5302 | 1.3977 |
No. 9 | 2.0145 | 0.5187 | 2.1841 | 0.5611 | 5.5302 | 1.3977 |
No. 10 | 2.0164 | 0.5189 | 2.1859 | 0.5612 | 5.5321 | 1.3978 |
No. 11 | 2.0206 | 0.5191 | 2.1901 | 0.5615 | 5.5363 | 1.3981 |
No. 12 | 2.0154 | 0.5188 | 2.1849 | 0.5612 | 5.5311 | 1.3977 |
No. 13 | 2.0161 | 0.5188 | 2.1856 | 0.5612 | 5.5318 | 1.3978 |
10 | [55] | 2.0971 | 0.5433 | 2.2923 | 0.5921 | 6.1448 | 1.5552 |
Present study | 2.0978 | 0.5434 | 2.2929 | 0.5922 | 6.1455 | 1.5553 |
No. 1 | 2.0976 | 0.5433 | 2.2928 | 0.5921 | 6.1453 | 1.5553 |
No. 2 | 2.0976 | 0.5433 | 2.2928 | 0.5921 | 6.1453 | 1.5553 |
No. 3 | 2.0971 | 0.5433 | 2.2923 | 0.5921 | 6.1448 | 1.5552 |
No. 4 | 2.101 | 0.5436 | 2.2961 | 0.5924 | 6.1487 | 1.5555 |
No. 5 | 2.0976 | 0.5433 | 2.2928 | 0.5921 | 6.1453 | 1.5553 |
No. 6 | 2.1002 | 0.5435 | 2.2954 | 0.5923 | 6.1479 | 1.5554 |
No. 7 | 2.0973 | 0.5433 | 2.2925 | 0.5921 | 6.145 | 1.5552 |
No. 8 | 2.0973 | 0.5433 | 2.2925 | 0.5921 | 6.145 | 1.5552 |
No. 9 | 2.0973 | 0.5433 | 2.2925 | 0.5921 | 6.145 | 1.5552 |
No. 10 | 2.0977 | 0.5433 | 2.2928 | 0.5921 | 6.1454 | 1.5553 |
No. 11 | 2.1005 | 0.5435 | 2.2957 | 0.5923 | 6.1482 | 1.5555 |
No. 12 | 2.0972 | 0.5433 | 2.2924 | 0.5921 | 6.1449 | 1.5552 |
No. 13 | 2.0975 | 0.5433 | 2.2927 | 0.5921 | 6.1452 | 1.5553 |
Table 5.
Non-dimensional natural frequencies of rectangular FGM plates placed on an elastic foundation for different values of the Winkler coefficient k0, Pasternak coefficient k1, and power law index p (a/b = 0.5, a/h = 5, m = 1, n = 1).
Table 5.
Non-dimensional natural frequencies of rectangular FGM plates placed on an elastic foundation for different values of the Winkler coefficient k0, Pasternak coefficient k1, and power law index p (a/b = 0.5, a/h = 5, m = 1, n = 1).
a/b | k0 | k1 | Theory | |
---|
a/h = 5 |
---|
p = 0 | p = 1 | p = 5 | p = 10 |
---|
0.5 | 0 | 0 | Present study | 6.761 | 5.2016 | 4.3761 | 4.206 |
No. 1 | 6.7609 | 5.2015 | 4.3757 | 4.2058 |
No. 2 | 6.7609 | 5.2015 | 4.3757 | 4.2058 |
No. 3 | 6.7616 | 5.202 | 4.3733 | 4.205 |
No. 4 | 6.775 | 5.2108 | 4.3753 | 4.2136 |
No. 5 | 6.7609 | 5.2015 | 4.3757 | 4.2058 |
No. 6 | 6.7628 | 5.2027 | 4.3832 | 4.211 |
No. 7 | 6.7636 | 5.2033 | 4.3722 | 4.2055 |
No. 8 | 6.7638 | 5.2034 | 4.3721 | 4.2056 |
No. 9 | 6.7638 | 5.2034 | 4.3721 | 4.2056 |
No. 10 | 6.7609 | 5.2015 | 4.3759 | 4.2059 |
No. 11 | 6.7642 | 5.2036 | 4.3852 | 4.2117 |
No. 12 | 6.8031 | 5.2291 | 4.4434 | 4.2771 |
No. 13 | 6.7609 | 5.2015 | 4.3754 | 4.2056 |
100 | 0 | Present study | 7.2125 | 5.8654 | 5.2358 | 5.1214 |
No. 1 | 7.2125 | 5.8653 | 5.2354 | 5.1211 |
No. 2 | 7.2125 | 5.8653 | 5.2354 | 5.1211 |
No. 3 | 7.2132 | 5.8657 | 5.2336 | 5.1205 |
No. 4 | 7.2256 | 5.8734 | 5.2353 | 5.1274 |
No. 5 | 7.2125 | 5.8653 | 5.2355 | 5.1212 |
No. 6 | 7.2142 | 5.8664 | 5.2415 | 5.1252 |
No. 7 | 7.215 | 5.8668 | 5.2327 | 5.1209 |
No. 8 | 7.2152 | 5.867 | 5.2326 | 5.121 |
No. 9 | 7.2152 | 5.867 | 5.2326 | 5.121 |
No. 10 | 7.2125 | 5.8653 | 5.2357 | 5.1212 |
No. 11 | 7.2155 | 5.8672 | 5.2431 | 5.1258 |
No. 12 | 7.2517 | 5.8893 | 5.2902 | 5.1777 |
No. 13 | 7.2125 | 5.8653 | 5.2352 | 5.121 |
0 | 100 | Present study | 11.115 | 10.845 | 10.992 | 11.079 |
No. 1 | 11.115 | 10.845 | 10.9919 | 11.0793 |
No. 2 | 11.115 | 10.845 | 10.9919 | 11.0793 |
No. 3 | 11.1154 | 10.8452 | 10.9914 | 11.0791 |
No. 4 | 11.1226 | 10.8484 | 10.9922 | 11.0814 |
No. 5 | 11.115 | 10.845 | 10.9919 | 11.0793 |
No. 6 | 11.116 | 10.8455 | 10.9936 | 11.0803 |
No. 7 | 11.1164 | 10.8457 | 10.9912 | 11.0793 |
No. 8 | 11.1166 | 10.8457 | 10.9912 | 11.0794 |
No. 9 | 11.1166 | 10.8457 | 10.9912 | 11.0794 |
No. 10 | 11.115 | 10.845 | 10.992 | 11.0793 |
No. 11 | 11.1168 | 10.8458 | 10.994 | 11.0804 |
No. 12 | 11.138 | 10.8552 | 11.0077 | 11.0949 |
No. 13 | 11.115 | 10.845 | 10.9918 | 11.0792 |
100 | 100 | Present study | 11.395 | 11.178 | 11.3593 | 11.4558 |
No. 1 | 11.3952 | 11.178 | 11.3593 | 11.4558 |
No. 2 | 11.3952 | 11.178 | 11.3593 | 11.4558 |
No. 3 | 11.3956 | 11.1782 | 11.3588 | 11.4557 |
No. 4 | 11.4026 | 11.1812 | 11.3596 | 11.4578 |
No. 5 | 11.3952 | 11.178 | 11.3593 | 11.4558 |
No. 6 | 11.3962 | 11.1784 | 11.3608 | 11.4567 |
No. 7 | 11.3966 | 11.1786 | 11.3587 | 11.4559 |
No. 8 | 11.3967 | 11.1787 | 11.3587 | 11.4559 |
No. 9 | 11.3967 | 11.1787 | 11.3587 | 11.4559 |
No. 10 | 11.3952 | 11.178 | 11.3593 | 11.4558 |
No. 11 | 11.3969 | 11.1787 | 11.3612 | 11.4568 |
No. 12 | 11.4174 | 11.1876 | 11.3737 | 11.47 |
No. 13 | 11.3952 | 11.178 | 11.3592 | 11.4557 |
Table 6.
Non-dimensional natural frequencies of square FGM plates placed on an elastic foundation for different values of the Winkler coefficient k0, Pasternak coefficient k1, and power law index p (a/b = 1, a/h = 5, m = 1, n = 1).
Table 6.
Non-dimensional natural frequencies of square FGM plates placed on an elastic foundation for different values of the Winkler coefficient k0, Pasternak coefficient k1, and power law index p (a/b = 1, a/h = 5, m = 1, n = 1).
a/b | k0 | k1 | Theory | |
---|
a/h = 5 |
---|
p = 0 | p = 1 | p = 5 | p = 0 |
---|
1 | 0 | 0 | Present study | 10.3761 | 8.0121 | 6.6687 | 6.3883 |
No. 1 | 10.3761 | 8.0121 | 6.6677 | 6.3879 |
No. 2 | 10.3761 | 8.0121 | 6.6677 | 6.3879 |
No. 3 | 10.3779 | 8.0133 | 6.663 | 6.3864 |
No. 4 | 10.4086 | 8.0336 | 6.6684 | 6.4062 |
No. 5 | 10.3761 | 8.0121 | 6.6679 | 6.3879 |
No. 6 | 10.38 | 8.0147 | 6.6838 | 6.3987 |
No. 7 | 10.3824 | 8.0163 | 6.6607 | 6.3877 |
No. 8 | 10.3831 | 8.0167 | 6.6606 | 6.388 |
No. 9 | 10.3831 | 8.0167 | 6.6606 | 6.388 |
No. 10 | 10.3761 | 8.0121 | 6.6683 | 6.3881 |
No. 11 | 10.383 | 8.0166 | 6.6881 | 6.4001 |
No. 12 | 10.4698 | 8.0739 | 6.8155 | 6.5423 |
No. 13 | 10.3762 | 8.0122 | 6.6672 | 6.3876 |
100 | 0 | Present study | 10.6722 | 8.4517 | 7.2542 | 7.0179 |
No. 1 | 10.6723 | 8.4517 | 7.2534 | 7.0175 |
No. 2 | 10.6723 | 8.4517 | 7.2534 | 7.0175 |
No. 3 | 10.674 | 8.4528 | 7.2491 | 7.0162 |
No. 4 | 10.7037 | 8.4718 | 7.2541 | 7.0339 |
No. 5 | 10.6722 | 8.4517 | 7.2535 | 7.0176 |
No. 6 | 10.676 | 8.4541 | 7.2678 | 7.0271 |
No. 7 | 10.6783 | 8.4556 | 7.2471 | 7.0174 |
No. 8 | 10.679 | 8.456 | 7.247 | 7.0177 |
No. 9 | 10.679 | 8.456 | 7.247 | 7.0177 |
No. 10 | 10.6722 | 8.4517 | 7.2539 | 7.0177 |
No. 11 | 10.6789 | 8.4559 | 7.2717 | 7.0284 |
No. 12 | 10.7629 | 8.5096 | 7.3865 | 7.1553 |
No. 13 | 10.6723 | 8.4517 | 7.2529 | 7.0173 |
0 | 100 | Present study | 15.1867 | 14.3818 | 14.3054 | 14.376 |
No. 1 | 15.1867 | 14.3818 | 14.3052 | 14.3759 |
No. 2 | 15.1867 | 14.3818 | 14.3052 | 14.3759 |
No. 3 | 15.1878 | 14.3823 | 14.304 | 14.3757 |
No. 4 | 15.2066 | 14.391 | 14.3064 | 14.3818 |
No. 5 | 15.1867 | 14.3818 | 14.3052 | 14.376 |
No. 6 | 15.1891 | 14.3829 | 14.3094 | 14.3785 |
No. 7 | 15.1906 | 14.3836 | 14.3036 | 14.3763 |
No. 8 | 15.191 | 14.3838 | 14.3036 | 14.3764 |
No. 9 | 15.191 | 14.3838 | 14.3036 | 14.3764 |
No. 10 | 15.1867 | 14.3818 | 14.3053 | 14.376 |
No. 11 | 15.191 | 14.3838 | 14.3105 | 14.3788 |
No. 12 | 15.2444 | 14.4086 | 14.3463 | 14.4167 |
No. 13 | 15.1868 | 14.3818 | 14.305 | 14.3759 |
100 | 100 | Present study | 15.3904 | 14.6304 | 14.5846 | 14.6636 |
No. 1 | 15.3904 | 14.6305 | 14.5843 | 14.6636 |
No. 2 | 15.3904 | 14.6305 | 14.5843 | 14.6636 |
No. 3 | 15.3914 | 14.6309 | 14.5833 | 14.6634 |
No. 4 | 15.4099 | 14.6394 | 14.5856 | 14.6692 |
No. 5 | 15.3904 | 14.6305 | 14.5844 | 14.6636 |
No. 6 | 15.3927 | 14.6315 | 14.5883 | 14.666 |
No. 7 | 15.3941 | 14.6322 | 14.5829 | 14.6639 |
No. 8 | 15.3945 | 14.6323 | 14.5829 | 14.664 |
No. 9 | 15.3945 | 14.6323 | 14.5829 | 14.664 |
No. 10 | 15.3904 | 14.6305 | 14.5845 | 14.6636 |
No. 11 | 15.3945 | 14.6324 | 14.5894 | 14.6663 |
No. 12 | 15.447 | 14.6564 | 14.6233 | 14.7021 |
No. 13 | 15.3904 | 14.6305 | 14.5842 | 14.6635 |