# Free Vibration Analysis of Functionally Graded Shells Using an Edge-Based Smoothed Finite Element Method

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulation

#### 2.1. Functionally Grade Material

#### 2.2. The FGM Shell Model

#### 2.3. Finite Element Formulation for Shell Analysis

## 3. Formulation of ES-MITC3 Finite Element Method for FGM Shells

#### 3.1. Brief on the MITC3 Formulation

#### 3.2. The ES-MITC3 Formulation

## 4. Numerical Results

_{3}N

_{4}) and Stainless steel (SUS304), which material properties are ${E}_{c}=322.2715$ GPa, ${v}_{c}=0.24$, ${\rho}_{c}=2370\text{}{\mathrm{kg}/\mathrm{m}}^{3}$, ${E}_{m}=207.7877$ GPa, ${v}_{m}=0.31776$, and ${\rho}_{m}=8166\text{}{\mathrm{kg}/\mathrm{m}}^{3}$. The first four non-dimensional frequency of the present method list in Table 1 are compared with MITC3 [16], a higher-order theory based on radial basis functions collocation including transverse normal deformation (HSDT RBFC-1) and discarding transverse normal deformation (HSDT RBFC-2) by Neves et al. [17], a higher-order theory and finite element formalation (HSDT FEM) by Pradyumna and Bandyopadhyay [6], a higher-order theory and semi-analytical method relied on Galerkin (HSDT SAG) by Yang and Shen [18], and Quasi-3D Ritz model (ED

_{555}) by Fazzolari and Carrera [19]. From Table 1 we can see that this proposed method (ES-MITC3) is more accurate than other methods, such as MITC3, HSDT RBFC-1, HSDT RBFC-2, HSDT FEM and HSDT SAG. The errors are less than 3% in comparison with the exact solution ED

_{555}[19]. Figure 7 shows non-dimensional frequency parameter for four first modes of clamped functionally graded cylindrical shell using various methods.

_{555}[19]. Figure 8, Figure 9, Figure 10 and Figure 11 show non-dimensional frequency parameter for the first mode of cylindrical FGM shell and spherical FGM shell with different n, respectively. The first six mode shapes of simply supported spherical FGM shell are illustrated in Figure 12.

## 5. Conclusions

_{555}[19] and a higher-order theory based on radial basis functions [17]. The ES-MITC3 presented herein is promising to be a simple and effective finite element method for analysis of functionally graded shells in practice.

^{8}) due to overcoming the shear locking phenomenon.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 7.**Non-dimensional frequency parameter for four first modes. (

**a**) Mode 1; (

**b**) mode 2; (

**c**) mode 3; (

**d**) mode 4.

**Figure 8.**Non-dimensional frequency parameter for the first mode of clamped cylindrical FGM shell with different n. (

**a**) R/a = 5; (

**b**) R/a = 10; (

**c**) R/a = 50.

**Figure 9.**Non-dimensional frequency parameter for the first mode of simple supported cylindrical FGM shell with different n. (

**a**) R/a = 5; (

**b**) R/a = 10; (

**c**) R/a = 50.

**Figure 10.**Non-dimensional frequency parameter for the first mode of simply supported spherical FGM shell with different n. (

**a**) R/a = 5; (

**b**) R/a = 10; (

**c**) R/a = 50.

**Figure 11.**Non-dimensional frequency parameter for the first mode of clamped spherical FGM shell with different n. (

**a**) R/a = 5; (

**b**) R/a = 10; (

**c**) R/a = 50.

**Figure 12.**First six mode shapes of simply supported spherical FGM shell (R/a = 10, a/h = 10, n = 0.2).

**Table 1.**Non-dimensional frequency parameter for clamped cylindrical functionally graded materials (FGM) shell with $R/a=100$, and relative error between methods (ED

_{555}[19] is fixed). $\mathrm{Error}\text{}(\%)=\frac{100\times \left|\mathrm{Method}-{\mathrm{ED}}_{555}\left[19\right]\right|}{{\mathrm{ED}}_{555}\left[19\right]}$.

Mode | Method | n | ||||
---|---|---|---|---|---|---|

0 | 0.2 | 2 | 10 | $\mathbf{\infty}$ | ||

1 | ES-MITC3 | 75.4587 | 61.3587 | 40.9880 | 35.3951 | 33.0594 |

% | 0.2776 | 0.0298 | 0.3963 | 0.7275 | 0.5532 | |

MITC3 [16] | 72.7508 | 59.0689 | 39.4771 | 34.0744 | 31.8220 | |

% | 3.3209 | 3.7031 | 4.0679 | 4.4317 | 4.2754 | |

HSDT FEM [6] | 72.9613 | 60.0269 | 39.1457 | 33.3666 | 32.0274 | |

% | 3.0412 | 2.1413 | 4.8733 | 6.4169 | 3.6576 | |

HSDT RBFC-1 [17] | 74.2634 | 60.0061 | 40.5259 | 35.1663 | 32.6108 | |

% | 1.3108 | 2.1752 | 1.5193 | 1.3693 | 1.9026 | |

HSDT RBFC-2 [17] | 74.5821 | 60.3431 | 40.8262 | 35.4229 | 32.8593 | |

% | 0.8873 | 1.6258 | 0.7895 | 0.6496 | 1.1551 | |

HSDT SAG [18] | 74.5180 | 57.4790 | 40.7500 | 35.8520 | 32.7610 | |

% | 0.9725 | 6.2950 | 0.9747 | 0.5539 | 1.4508 | |

ED_{555} [19] | 75.2498 | 61.3404 | 41.1511 | 35.6545 | 33.2433 | |

2 | ES-MITC3 | 144.4760 | 117.6462 | 78.5402 | 67.7320 | 63.3473 |

% | 0.6724 | 0.6147 | 0.5174 | 0.3138 | 0.4088 | |

MITC3 [16] | 140.8063 | 114.5113 | 76.4785 | 65.9309 | 61.6559 | |

% | 1.8847 | 2.0664 | 2.1212 | 2.3537 | 2.2722 | |

HSDT FEM [6] | 138.5552 | 113.8806 | 74.2915 | 63.2869 | 60.5546 | |

% | 3.4533 | 2.6058 | 4.9201 | 6.2695 | 4.0178 | |

HSDT RBFC-1 [17] | 141.6779 | 114.3788 | 76.9725 | 66.6482 | 61.9329 | |

% | 1.2773 | 2.1797 | 1.4889 | 1.2913 | 1.8331 | |

HSDT RBFC-2 [17] | 142.4281 | 115.2134 | 77.6639 | 67.1883 | 62.4886 | |

% | 0.7546 | 1.4660 | 0.6041 | 0.4914 | 0.9523 | |

HSDT SAG [18] | 144.6630 | 111.7170 | 78.8170 | 69.0750 | 63.3140 | |

% | 0.8027 | 4.4562 | 0.8717 | 2.3029 | 0.3560 | |

ED_{555} [19] | 143.5110 | 116.9275 | 78.1359 | 67.5201 | 63.0894 | |

3 | ES-MITC3 | 145.1510 | 118.1985 | 78.9069 | 68.0474 | 63.6440 |

% | 1.0284 | 0.9602 | 0.8727 | 2.2434 | 0.7658 | |

MITC3 [16] | 141.7861 | 115.3112 | 77.0122 | 66.3906 | 62.0864 | |

% | 1.3137 | 1.5061 | 1.5494 | 4.6235 | 1.7003 | |

HSDT FEM [6] | 138.5552 | 114.0266 | 74.3868 | 63.3668 | 60.6302 | |

% | 3.5625 | 2.6033 | 4.9056 | 8.9675 | 4.0058 | |

HSDT RBFC-1 [17] | 141.8485 | 114.5495 | 77.0818 | 66.7332 | 62.0082 | |

% | 1.2702 | 2.1567 | 1.4604 | 4.1314 | 1.8241 | |

HSDT RBFC-2 [6] | 142.6024 | 115.3665 | 77.7541 | 67.2689 | 62.5668 | |

% | 0.7455 | 1.4588 | 0.6010 | 3.3618 | 0.9397 | |

HSDT SAG [18] | 145.7400 | 112.5310 | 79.4070 | 67.5946 | 63.8060 | |

% | 1.4383 | 3.8808 | 1.5121 | 2.8939 | 1.0223 | |

ED_{555} [19] | 143.6735 | 117.0744 | 78.2242 | 69.6090 | 63.1603 | |

4 | ES-MITC3 | 204.0647 | 166.3177 | 111.0461 | 95.6539 | 89.5229 |

% | 1.1780 | 1.2302 | 1.3863 | 1.2447 | 1.2996 | |

MITC3 [16] | 195.3261 | 158.8135 | 106.1329 | 91.3802 | 85.4901 | |

% | 3.1547 | 3.3373 | 3.0995 | 3.2788 | 3.2637 | |

HSDT FEM [6] | 195.5366 | 160.6235 | 104.7687 | 89.1970 | 85.1788 | |

% | 3.0503 | 2.2357 | 4.3450 | 5.5896 | 3.6160 | |

HSDT RBFC-1 [17] | 199.1566 | 160.7355 | 107.9484 | 93.3350 | 86.8160 | |

% | 1.2555 | 2.1675 | 1.4419 | 1.2097 | 1.7634 | |

HSDT RBFC-2 [17] | 200.3158 | 162.0337 | 108.9677 | 94.0923 | 87.6341 | |

% | 0.6808 | 1.3773 | 0.5113 | 0.4081 | 0.8377 | |

HSDT SAG [18] | 206.9920 | 159.8550 | 112.4570 | 98.3860 | 90.3700 | |

% | 2.6294 | 2.7034 | 2.6745 | 4.1365 | 2.2581 | |

ED_{555} [19] | 201.6888 | 164.2966 | 109.5277 | 94.4779 | 88.3744 |

**Table 2.**Non-dimensional frequency parameter for clamped cylindrical FGM shell with $a/h=10$ and different $R/a$ ratios.

$\frac{\mathit{R}}{\mathit{a}}$ | Method | n | ||||||
---|---|---|---|---|---|---|---|---|

0 | 0.2 | 0.5 | 1 | 2 | 10 | $\mathbf{\infty}$ | ||

5 | ES-MITC3 | 73.4741 | 67.2928 | 60.6591 | 53.9842 | 48.1650 | 41.4718 | 33.4122 |

HSDT FEM [6] | 71.8861 | 68.1152 | 63.1896 | 56.5546 | 36.2487 | 33.6611 | 32.4802 | |

HSDT RBFC-1 [17] | 73.1640 | 66.6620 | 60.2477 | 53.5430 | 47.5205 | 40.8099 | 33.0576 | |

HSDT RBFC-2 [17] | 73.6436 | 67.1004 | 60.6568 | 53.9340 | 47.9060 | 41.0985 | 33.2743 | |

10 | ES-MITC3 | 72.6253 | 65.5578 | 60.0417 | 53.4874 | 47.7863 | 41.1837 | 33.0311 |

HSDT FEM [6] | 71.0394 | 67.3320 | 62.4687 | 55.8911 | 35.6633 | 31.1474 | 32.0976 | |

HSDT RBFC-1 [17] | 72.3304 | 65.8808 | 59.5215 | 52.8800 | 46.9447 | 40.4145 | 32.6810 | |

HSDT RBFC-2 [17] | 72.8141 | 66.3235 | 59.9353 | 53.2759 | 47.3343 | 40.7046 | 32.8995 | |

50 | ES-MITC3 | 72.3439 | 66.3519 | 59.9114 | 53.4282 | 47.7802 | 41.1529 | 32.9058 |

HSDT FEM [6] | 70.7660 | 67.0801 | 62.2380 | 55.6799 | 35.4745 | 32.9812 | 31.9741 | |

HSDT RBFC-1 [17] | 72.0614 | 65.6371 | 59.3022 | 52.6864 | 46.7820 | 40.3028 | 32.5594 | |

HSDT RBFC-2 [17] | 72.5465 | 66.0814 | 59.7178 | 53.0841 | 47.1726 | 40.5923 | 32.7786 |

**Table 3.**Non-dimensional frequency parameter for simply supported cylindrical FGM shell with $a/h=10$ and different $R/a$ ratios.

$\frac{\mathit{R}}{\mathit{a}}$ | Method | n | ||||||
---|---|---|---|---|---|---|---|---|

0 | 0.2 | 0.5 | 1 | 2 | 10 | $\mathbf{\infty}$ | ||

5 | ES-MITC3 | 42.9913 | 39.3028 | 35.4690 | 31.7485 | 28.6106 | 24.7564 | 19.5592 |

HSDT FEM [6] | 42.2543 | 40.1621 | 37.2870 | 33.2268 | 27.4449 | 19.3892 | 19.0917 | |

HSDT RBFC-1 [17] | 42.6701 | 38.7168 | 34.8768 | 30.9306 | 27.5362 | 24.2472 | 19.2796 | |

HSDT RBFC-2 [17] | 42.7172 | 38.7646 | 34.9273 | 30.9865 | 27.5977 | 24.2839 | 19.3008 | |

ED_{555} [19] | 42.7160 | 39.0642 | 35.0811 | 31.0414 | 27.5634 | 24.1245 | 19.3003 | |

10 | ES-MITC3 | 42.5231 | 38.9004 | 35.1357 | 31.4868 | 28.4168 | 24.6061 | 19.3492 |

HSDT FEM [6] | 41.9080 | 39.8472 | 36.9995 | 32.9585 | 27.1879 | 19.1562 | 18.9352 | |

HSDT RBFC-1 [17] | 42.3153 | 38.3840 | 34.5672 | 30.6485 | 27.2979 | 24.1063 | 19.1193 | |

HSDT RBFC-2 [17] | 42.3684 | 38.4368 | 34.6219 | 30.7077 | 27.3616 | 24.1444 | 19.1433 | |

ED_{555} [19] | 42.3677 | 38.7377 | 34.7661 | 30.7621 | 27.3258 | 23.9848 | 19.1429 | |

50 | ES-MITC3 | 42.3669 | 38.7889 | 35.0696 | 31.4631 | 28.4233 | 24.5937 | 19.2798 |

HSDT FEM [6] | 41.7963 | 39.7465 | 36.9088 | 32.8750 | 27.0961 | 19.0809 | 18.8848 | |

HSDT RBFC-1 [17] | 42.2008 | 38.2842 | 34.4809 | 30.5759 | 27.2423 | 24.0762 | 19.0675 | |

HSDT RBFC-2 [17] | 42.2560 | 38.3384 | 34.5365 | 30.6355 | 27.3055 | 24.1125 | 19.0924 | |

ED_{555} [19] | 42.2553 | 38.6391 | 34.6904 | 30.6890 | 27.2682 | 23.9515 | 19.0922 |

**Table 4.**Non-dimensional frequency parameter for simply supported spherical FGM shell with $a/h=10$ and different $R/a$ ratios.

$\frac{\mathit{R}}{\mathit{a}}$ | Method | n | ||||||
---|---|---|---|---|---|---|---|---|

0 | 0.2 | 0.5 | 1 | 2 | 10 | $\mathbf{\infty}$ | ||

5 | ES-MITC3 | 44.4405 | 40.6238 | 36.6449 | 32.7529 | 29.4124 | 25.2893 | 20.2096 |

HSDT FEM [6] | 44.0073 | 41.7782 | 38.7731 | 34.6004 | 28.7459 | 20.4691 | 19.8838 | |

HSDT RBFC-1 [17] | 44.4555 | 40.3936 | 36.4453 | 32.3691 | 28.7833 | 25.0772 | 20.0818 | |

HSDT RBFC-2 [17] | 44.4697 | 40.4211 | 36.6004 | 32.4101 | 28.8329 | 25.1038 | 20.0927 | |

ED_{555} [19] | 44.4671 | 40.7166 | 36.6297 | 32.4645 | 28.7996 | 24.9403 | 20.0915 | |

10 | ES-MITC3 | 42.9198 | 39.2373 | 35.4098 | 31.6957 | 28.5633 | 24.7153 | 19.5267 |

HSDT FEM [6] | 42.3579 | 40.2608 | 37.3785 | 33.3080 | 27.5110 | 19.4357 | 19.1385 | |

HSDT RBFC-1 [17] | 42.7709 | 38.8074 | 34.9574 | 31.0012 | 27.5984 | 24.3034 | 19.3251 | |

HSDT RBFC-2 [17] | 42.8180 | 38.8551 | 35.0080 | 31.0572 | 27.6602 | 24.3401 | 19.3464 | |

ED_{555} [19] | 42.8169 | 39.1556 | 35.1622 | 31.1122 | 27.6258 | 24.1803 | 19.3459 | |

50 | ES-MITC3 | 42.4046 | 38.8147 | 35.0835 | 31.4662 | 28.4197 | 24.5977 | 19.2966 |

HSDT FEM [6] | 41.8145 | 39.7629 | 36.9234 | 32.8881 | 27.1085 | 19.0922 | 18.8930 | |

HSDT RBFC-1 [17] | 42.2192 | 38.2988 | 34.4922 | 30.5840 | 27.2474 | 24.0791 | 19.0759 | |

HSDT RBFC-2 [17] | 42.2741 | 38.3528 | 34.5478 | 30.6437 | 27.3109 | 24.1168 | 19.1006 | |

ED_{555} [19] | 42.2735 | 38.6538 | 34.7018 | 30.6975 | 27.2741 | 23.9567 | 19.1004 |

**Table 5.**Non-dimensional frequency parameter for clamped spherical FGM shell with different $R/a$ ratios.

$\mathit{R}/\mathit{a}$ | Method | n | ||||||
---|---|---|---|---|---|---|---|---|

0 | 0.2 | 0.5 | 1 | 2 | 10 | $\mathbf{\infty}$ | ||

5 | ES-MITC3 | 74.3416 | 68.0034 | 61.2122 | 54.3761 | 48.3922 | 41.5911 | 33.7972 |

HSDT FEM [6] | 73.5550 | 69.6597 | 64.6114 | 57.8619 | 37.3914 | 34.6658 | 33.2343 | |

HSDT RBFC-1 [17] | 74.8207 | 68.2142 | 61.6902 | 54.8597 | 48.6656 | 41.6016 | 33.8061 | |

HSDT RBFC-2 [17] | 75.2810 | 68.6329 | 62.0789 | 55.2302 | 49.0328 | 41.8796 | 34.0141 | |

10 | ES-MITC3 | 72.8831 | 66.7331 | 60.1352 | 53.5040 | 47.7428 | 41.1652 | 33.1447 |

HSDT FEM [6] | 71.4659 | 67.7257 | 62.8299 | 56.2222 | 35.9568 | 33.4057 | 32.2904 | |

HSDT RBFC-1 [17] | 72.7536 | 66.2686 | 59.8745 | 53.1956 | 47.2135 | 40.5990 | 32.8722 | |

HSDT RBFC-2 [17] | 73.2322 | 66.7063 | 60.2831 | 53.5864 | 47.5990 | 40.8883 | 33.0884 | |

50 | ES-MITC3 | 72.3889 | 66.3780 | 59.9190 | 53.4192 | 47.7612 | 41.1495 | 32.9258 |

HSDT FEM [6] | 70.7832 | 67.0956 | 62.2519 | 55.6923 | 35.4861 | 32.9916 | 31.9819 | |

HSDT RBFC-1 [17] | 72.0784 | 65.6498 | 59.3112 | 52.6921 | 46.7849 | 40.3049 | 32.5671 | |

HSDT RBFC-2 [17] | 72.5633 | 66.0938 | 59.7265 | 53.0895 | 47.1574 | 40.5946 | 32.7862 |

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**MDPI and ACS Style**

Pham, T.D.; Pham, Q.H.; Phan, V.D.; Nguyen, H.N.; Do, V.T.
Free Vibration Analysis of Functionally Graded Shells Using an Edge-Based Smoothed Finite Element Method. *Symmetry* **2019**, *11*, 684.
https://doi.org/10.3390/sym11050684

**AMA Style**

Pham TD, Pham QH, Phan VD, Nguyen HN, Do VT.
Free Vibration Analysis of Functionally Graded Shells Using an Edge-Based Smoothed Finite Element Method. *Symmetry*. 2019; 11(5):684.
https://doi.org/10.3390/sym11050684

**Chicago/Turabian Style**

Pham, Tien Dat, Quoc Hoa Pham, Van Duc Phan, Hoang Nam Nguyen, and Van Thom Do.
2019. "Free Vibration Analysis of Functionally Graded Shells Using an Edge-Based Smoothed Finite Element Method" *Symmetry* 11, no. 5: 684.
https://doi.org/10.3390/sym11050684