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Numerical thermoelastic eigenfrequency prediction of damaged layered shell panel with concentric/eccentric cutout and corrugated (TD/TID) properties

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Abstract

The vibrational responses are predicted numerically for the layered shell panel structure with and without cutout under the variable temperature loading and corrugated composite properties. The presence of variable cutout shapes (circular/elliptical/square and rectangular) and sizes are modelled via a generic mathematical macro-mechanical model in the framework of the cubic-order kinematic model. Also, the present model includes the variation of composite properties due to the change in environmental conditions, i.e. the temperature-dependent (TD) and -independent (TID) cases. The computational responses are obtained by taking advantages of the isoparametric finite element technique and the Hamilton principle to derive the final governing equation. The total Lagrangian approach is adopted to compute the responses using the specialized computer code prepared in the MATLAB platform. The frequency responses are predicted considering the effect of a cutout, including the environmental variation and compared with previously published eigenvalues. The model versatility is tested over a variety of examples considering the shell configurations (plate, cylindrical, spherical, hyperboloid, and elliptical), the influential cutout parameter (shape, size, and position) and temperature loading including the corrugated composite properties.

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Abbreviations

\(n\) :

Number of layers in the laminated shell panel

\(\theta\) :

Angle of fibre orientation

\(\left[ H \right]\) :

Thickness of coordinate matrix

\(\left\{ \sigma \right\}\) :

Stress

\(\left[ {\overline{Q}} \right]\) :

Reduced transformed elastic constant matrix

\(\varepsilon\) :

Strain

\(\alpha\) :

Thermal expansion coefficient

\(\Delta T\) :

Temperature difference

\(S\) :

Strain energy

\(T\) :

Kinetic energy

\(\left[ M \right]\) :

Mass matrix

\(\rho\) :

Mass density

\(N\) :

Shape function

\(\left[ K \right]\) :

Stiffness matrix

\(\left\{ \Phi \right\}\) :

Eigenvector

\(\omega\) :

Natural frequency

\(\omega_{{{\text{nd}}}}\) :

Normalized natural frequency

\(E_{1} ,\,E_{2} \,{\text{and}}\,E_{3}\) :

Young’s modulus

\(G_{12} ,\,G_{13} \,{\text{and}}\,G_{23}\) :

Shear modulus

\(\mu_{12} ,\,\mu_{13} \,{\text{and}}\,\mu_{23}\) :

Poisson’s ratio

\(\left[ {K_{{\text{G}}} } \right]\) :

Geometrical stiffness matrix

\(u_{{0\zeta_{x} }} ,\,\,u_{{0\zeta_{y} }}\) and \(u_{{0\zeta_{z} }} \,\) :

Displacement of a point at mid-plane

\(u_{{1\zeta_{y} }}\) and \(u_{{1\zeta_{x} }}\) :

Rotation along \(\zeta_{y}\) and \(\zeta_{y}\)

\(R_{{\zeta_{x} }}\), \(R_{{\zeta_{y} }}\) and \(R_{{\zeta_{xy} }}\) :

Radius of curvature of the shell panel

\(\zeta_{x}\), \(\zeta_{y}\) and \(\zeta_{z}\) :

Global reference axis of the laminate’s shell panel

\(\left[ D \right]\) and \(\left[ {D_{{\text{G}}} } \right]\) :

Material property matrix

\(\left[ {B_{{\text{l}}} } \right]\) and \(\left[ {B_{{\text{G}}} } \right]\) :

Strain displacement matrix

\(L,\,W\)and \(h\) :

Length, width and thickness of the shell panel

\(U_{{\zeta_{x} }} ,\,U_{{\zeta_{y} }}\) and \(U_{{\zeta_{z} }}\) :

Global displacement

\(u_{{2\zeta_{x} }} ,\,\,u_{{2\zeta_{y} }} ,\,\,u_{{3\zeta_{x} }} \,\,\) and \(u_{{3\zeta_{y} }}\) :

Higher-order deformation parameters

\(\left\{ \lambda \right\},\left\{ {\dot{\lambda }} \right\}\) and \(\left\{ {\ddot{\lambda }} \right\}\) :

Displacement, velocity and acceleration

\(\left\{ {\lambda_{0} } \right\}\) and \(\left\{ {\lambda_{0i} } \right\}\) :

Elemental and nodal displacement

\(A^{\prime}\) and \(A\) :

Area of the cutout and laminated shell panel

t :

Time

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  1. Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela, 769008, Odisha, India

    Hukum Chand Dewangan & Subrata Kumar Panda

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Dewangan, H.C., Panda, S.K. Numerical thermoelastic eigenfrequency prediction of damaged layered shell panel with concentric/eccentric cutout and corrugated (TD/TID) properties. Engineering with Computers 38, 2009–2025 (2022). https://doi.org/10.1007/s00366-020-01199-1

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