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Closed-Form Solutions for Free Vibration Frequencies of Functionally Graded Euler-Bernoulli Beams

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Mechanics of Composite Materials Aims and scope

The bending vibration of a functionally graded Euler–Bernoulli beam is investigated by the transformed-section method. The material properties of the functionally graded beam (FGB) are assumed to vary across its thickness according to a simple power law. Closed-form solutions for free vibration frequencies of FGBs with classical boundary conditions are derived. Some analytical results are compared with numerical results found in the published literature to verify the accuracy of the model presented, and a good agreement between them is observed.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Chinese Culture University, Taipei, Taiwan, R.O.C.

    W. R. Chen & H. Chang

Authors
  1. W. R. Chen
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  2. H. Chang
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Correspondence to W. R. Chen.

Additional information

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 53, No. 1, pp. 119-142 , January-February, 2017.

Appendix A

Appendix A

Based on the transformed-section method, the effective second moment of area of the transformed cross section of beams about their neutral axis is given as

$$ {I}_{\mathrm{eff}}={I}_{y_1}={\displaystyle {\int}_{A_t}{z}_1^2dA.} $$
(A1)

Introducing z 1 = z − h o into Eq. (A1) yields

$$ {I}_{\mathrm{eff}}={\displaystyle {\int}_{A_t}{z}_1^2dA-{h}_o^2}{\displaystyle {\int}_{A_t}dA={I}_y-{h}_o^2{A}_t.} $$
(A2)

The quantities I y and A t are calculated as follows:

$$ {I}_y={\displaystyle {\int}_{\frac{-h}{2}}^{\frac{h}{2}}{\displaystyle {\int}_{\frac{-bn(z)}{2}}^{\frac{bn(z)}{2}}{z}^2 dydz}=}{\displaystyle {\int}_{\frac{-h}{2}}^{\frac{h}{2}}bn(z){z}^2dz}, $$
(A3)
$$ {A}_t={\displaystyle {\int}_{\frac{-h}{2}}^{\frac{h}{2}}{\displaystyle {\int}_{\frac{-bn(z)}{2}}^{\frac{bn(z)}{2}} dydz}=}{\displaystyle {\int}_{\frac{-h}{2}}^{\frac{h}{2}}bn(z){z}^2dz}. $$
(A4)

Inserting Eq. (5) into Eqs. (A3) and (A4) and performing integration, it is found that

$$ {I}_y=\frac{b{h}^3}{12}\left[\frac{E_b}{E_t}+12\left(1-\frac{E_b}{E_t}\right)\left(\frac{1}{k+3}-\frac{1}{k+2}+\frac{1}{4\left(k+1\right)}\right)\right], $$
(A5)
$$ {A}_t=bh\left[\frac{E_b}{E_t}+\left(1-\frac{E_b}{E_t}\right)\frac{1}{k+1}\right]. $$
(A6)

Using Eqs. (A5), (A6) and (6) in Eq. (A2), the effective second area moment about the neutral axis of the transformed cross section of a functionally graded beam is obtained in the form

$$ {I}_y=\frac{b{h}^3}{12}\left[\frac{E_b}{E_t}+12\left(1-\frac{E_b}{E_t}\right)\left(\frac{1}{k+3}-\frac{1}{k+2}+\frac{1}{4\left(k+1\right)}\right)-\frac{3{k}^2{\left(1-{E}_b/{E}_t\right)}^2}{\left(k+1\right){\left(k+2\right)}^2{\left(1+k\;{E}_b/{E}_t\right)}^2}\right]. $$
(A7)

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Chen, W.R., Chang, H. Closed-Form Solutions for Free Vibration Frequencies of Functionally Graded Euler-Bernoulli Beams. Mech Compos Mater 53, 79–98 (2017). https://doi.org/10.1007/s11029-017-9642-3

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  • DOI: https://doi.org/10.1007/s11029-017-9642-3

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