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Free vibration of functionally graded truncated conical shells under internal pressure

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Abstract

The influence of internal pressure on the free vibration behavior of functionally graded (FG) truncated conical shells are investigated based on the first-order shear deformation theory (FSDT) of shells. The initial mechanical stresses are obtained by solving the static equilibrium equations. Using Hamilton’s principle and by including the influences of initial stresses, the free vibration equations of motion around this equilibrium state together with the related boundary conditions are derived. The material properties are assumed to be graded in the thickness direction. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the governing equations and the related boundary conditions. The convergence behavior of the method is numerically investigated and its accuracy is demonstrated by comparing the results in the limit cases with existing solutions in literature. Finally, the effects of internal pressure together with the material property graded index, the semi-vertex angle and the other geometrical parameters on the frequency parameters of the FG truncated conical shells subjected to different boundary conditions are studied.

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Abbreviations

R 1 and R 2 :

Small and large end mean radius

h :

Thickness of the truncated conical shell

L :

Length of the truncated conical shell

P i :

Internal pressure

p :

Power law index

P M :

Material property of the metal

P C :

Material property of the ceramic

N s :

Total number of nodes along the s-direction

G :

Shear rigidity

E :

Young’s modulus

k s :

Shear correction factor

t :

Time

m :

Circumferential wave number

s,θ,z :

Cylindrical coordinate variable

u 0,v 0,w 0 :

Initial displacement components

u,v,w :

Displacement components

r :

Mean radius of the truncated conical shell

K and U :

Kinetic and the potential energy of the shell

ω m :

Natural frequency

δK and δU :

Kinetic and the potential energy

λ mi :

Non-dimensional natural frequency parameters

β :

Semi-vertex angle

ν :

Poisson’s ratio

ρ :

Density

\(\varphi_{0}^{s}\), \(\varphi_{0}^{\theta}\) :

Initial bending rotation of the unit normal to the mid-surface of the shell about the θ- and s-axis, respectively

φ s and φ θ :

Bending rotation of the unit normal to the mid-surface of the shell about the θ- and s-axis, respectively

ε 0ss , ε 0θθ , γ 0sz :

Initial normal and the shear components of the strain tensor

σ 0ss , σ 0θθ , σ 0sz :

Initial normal and the shear components of the stress tensor

ε ss , ε θθ , γ sz , γ sθ , γ θz :

Normal and the shear components of the strain tensor

σ ss , σ θθ , σ sz , σ sθ , σ θz :

Normal and the shear components of the stress tensor

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., Tehran, 15875-4413, Iran

    Y. Heydarpour & M. M. Aghdam

  2. Department of Mechanical Engineering, School of Engineering, Persian Gulf University, Bushehr, 75168, Iran

    P. Malekzadeh

Authors
  1. Y. Heydarpour
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  2. P. Malekzadeh
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  3. M. M. Aghdam
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Corresponding author

Correspondence to Y. Heydarpour.

Appendix

Appendix

Based on the three-dimensional elasticity theory, the nonlinear strain displacement relations in the conical coordinate system can be presented as,

$$\begin{aligned} &{\hat{\varepsilon}_{ss} = \frac{\partial \hat{u}}{\partial s} + \frac{1}{2} \biggl[ \biggl( \frac{\partial \hat{u}}{\partial s} \biggr)^{2} + \biggl( \frac{\partial \hat{v}}{\partial s} \biggr)^{2} + \biggl( \frac{\partial \hat{w}}{\partial s} \biggr)^{2} \biggr],} \end{aligned}$$
(33a)
$$\begin{aligned} &{ \begin{aligned}[b] \hat{\varepsilon}_{\theta \theta}& = \frac{1}{r} \biggl( \frac{\partial \hat{v}}{\partial \theta} + \hat{u}\sin \beta + \hat{w}\cos \beta \biggr) \\ &\quad{} + \frac{1}{2r^{2}} \biggl[ \biggl( \frac{\partial \hat{w}}{\partial \theta} - \hat{v}\cos \beta \biggr)^{2}\end{aligned} } \\ &{ \begin{aligned}[b]&\quad{} + \biggl( \frac{\partial \hat{v}}{\partial \theta} + \hat{u}\sin \beta + \hat{w}\cos \beta \biggr)^{2} \\ &\quad{} + \biggl( \frac{\partial \hat{u}}{\partial \theta} - \hat{v}\sin \beta \biggr)^{2} \biggr],\end{aligned}} \end{aligned}$$
(33b)
$$\begin{aligned} &{\hat{\varepsilon}_{zz} = \frac{\partial \hat{w}}{\partial z} + \frac{1}{2} \biggl[ \biggl( \frac{\partial \hat{u}}{\partial z} \biggr)^{2} + \biggl( \frac{\partial \hat{v}}{\partial z} \biggr)^{2} + \biggl( \frac{\partial \hat{w}}{\partial z} \biggr)^{2} \biggr],} \end{aligned}$$
(33c)
$$\begin{aligned} &{\hat{\gamma}_{z\theta} = \frac{\partial \hat{v}}{\partial z} + \frac{1}{r} \biggl( \frac{\partial \hat{w}}{\partial \theta} - \hat{v} \cos \beta \biggr) } \\ &{\phantom{\hat{\gamma}_{z\theta} =}{} + \frac{1}{r} \biggl[ \frac{\partial \hat{w}}{\partial z} \biggl( \frac{\partial \hat{w}}{\partial \theta} - \hat{v}\cos \beta \biggr) + \frac{\partial \hat{v}}{\partial z} \biggl( \frac{\partial \hat{v}}{\partial \theta} } \\ &{\phantom{\hat{\gamma}_{z\theta} =}{} + \hat{u}\sin \beta + \hat{w}\cos \beta \biggr) }\\ &{\phantom{\hat{\gamma}_{z\theta} =}{} + \frac{\partial \hat{u}}{\partial z} \biggl( \frac{\partial \hat{u}}{\partial \theta} - \hat{v}\sin \beta \biggr) \biggr]} \end{aligned}$$
(33d)
$$\begin{aligned} &{ \begin{aligned}[b]\hat{\gamma}_{\theta s} &= \frac{\partial \hat{v}}{\partial s} + \frac{1}{r} \biggl( \frac{\partial \hat{u}}{\partial \theta} - \hat{v} \sin \beta \biggr) \\ &\quad{} + \frac{1}{r} \biggl[ \frac{\partial \hat{w}}{\partial s} \biggl( \frac{\partial \hat{w}}{\partial \theta} - \hat{v}\cos \beta \biggr) \\ &\quad {} + \frac{\partial \hat{v}}{\partial s} \biggl( \frac{\partial \hat{v}}{\partial \theta} + \hat{u}\sin \beta + \hat{w}\cos \beta \biggr) \\ &\quad + \frac{\partial \hat{u}}{\partial s} \biggl( \frac{\partial \hat{u}}{\partial \theta} - \hat{v}\sin \beta \biggr) \biggr]\end{aligned}} \end{aligned}$$
(33e)
$$\begin{aligned} &{\hat{\gamma}_{zs} = \frac{\partial \hat{w}}{\partial s} + \frac{\partial \hat{u}}{\partial z} + \frac{\partial \hat{u}}{\partial s}\frac{\partial \hat{u}}{\partial z} + \frac{\partial \hat{v}}{\partial s} \frac{\partial \hat{v}}{\partial z} + \frac{\partial \hat{w}}{\partial s}\frac{\partial \hat{w}}{\partial z} } \end{aligned}$$
(33f)

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Heydarpour, Y., Malekzadeh, P. & Aghdam, M.M. Free vibration of functionally graded truncated conical shells under internal pressure. Meccanica 49, 267–282 (2014). https://doi.org/10.1007/s11012-013-9791-y

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