**Acknowledgements**

**BC** *h/R0 p = 0 p = 0.6 p = 1 p = 2 p = 5 p = 10 p = 20*

0.1 368.31 356.53 354.28 354.66 358.44 358.13 355.04 0.3 558.81 545.07 542.41 542.83 548.35 550.42 549.32 0.5 669.03 660.42 657.10 652.86 649.27 647.64 645.98

0.1 82.564 91.860 95.778 101.94 109.61 113.78 116.41 0.3 381.14 384.20 386.62 392.04 400.91 405.96 408.78 0.5 539.57 539.80 540.83 544.12 551.17 556.07 559.18

0.1 80.874 89.444 93.013 98.592 105.46 109.13 111.41 0.3 331.30 347.67 354.18 364.51 377.62 384.51 388.49 0.5 515.44 522.81 525.80 531.29 539.88 545.19 548.45

**Table 12.** Fundamental frequencies of FGM cylindrical shells with different elasticity boundary conditions.

**4. Conclusions**

**4. Conclusions**

Fig. 5 Mode shapes of FGM cylindrical shells with CC boundary

conditions with *h/R*<sup>0</sup> = 0.5, *L/R*<sup>0</sup> = 2 and *p*=1.

Mode shapes 1–2 Mode shape 3

Mode shapes 4–5 Mode shape 6

Fig. 6 Mode shapes of FGM cylindrical shells with CF boundary

**Figure 6.** Mode shapes of FGM cylindrical shells with CF boundary conditions with *h/R*0 = 0.5, *L/R*0 = 2 and *p* = 1.

conditions with *h/R*<sup>0</sup> = 0.5, *L/R*<sup>0</sup> = 2 and *p* = 1.

A new 3‐D exact solution for free vibration analysis of thick

A new 3-D exact solution for free vibration analysis of thick functionally graded plates and cylindrical shells with arbitrary boundary conditions is presented in this chapter. The effective material properties of functionally graded structures vary continuously in the thickness direction according to the simple power-law distributions in terms of volume fraction of constituents and are estimated by Voigt's rule of mixture. By using the artificial spring

functionally graded plates and cylindrical shells with arbitrary

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116 Advances in Functionally Graded Materials and Structures

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The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51175098 and 51279035) and the Fundamental Research Funds for the Central Universities of China (No. HEUCFQ1401).
