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Free Vibration of Thermally Stressed Angle-Ply Laminated Composite Using First-Order Shear Deformation Theory Model with Assumed Natural Shear Strain

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Abstract

A four-nodded rectangular finite element based on first-order shear deformation theory is used to investigate the dynamic behavior of thermally stressed angle-ply laminated composite plates. Total potential energy and Hamilton’s principles have been used to formulate stiffness, geometric and mass matrices. Assumed natural strain method has been employed to avoid potential shear locking. Convergence of the first natural frequency and critical temperature rise has been checked out through a set of examples whose results compare well with 3D solution and other finite elements models from the literature. The effects of side-to-thickness ratio, anisotropy degree and fibers orientation angle, on the first natural frequency and critical temperature have also been investigated. Furthermore, the free vibration of thermally stressed angle-ply laminated composite plates has been investigated for different side-to-thickness ratios and fiber orientations. The results have shown that the first natural frequency decreases linearly with temperature rise, which is in accordance with those from the literature.

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Abbreviations

\(A_{ij} , B_{ij} , D_{ij}\) and \(H_{ij}\) :

Elasticity matrices of the laminate

\(E_{1}^{\left( k \right)}\), \(E_{2}^{\left( k \right)}\) :

Young modulus of the material in the 1 and 2 directions of the kth layer

\(G_{12}^{\left( k \right)}\), \(G_{23}^{\left( k \right)}\) and \(G_{13}^{\left( k \right)}\) :

Shear modulus in the 1–2, 2–3 and 1–3 planes of the kth layer

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

Curvature vector

\(\left[ {K_{e} } \right]\) :

The elementary stiffness matrix

\(\left[ {K_{g}^{e} } \right]\) :

The elementary geometrical matrix

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

The inertia matrix

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

The mass matrix

\(M_{x} , M_{y}\) and \(M_{xy}\) :

Moment resultants

\(N_{x} , N_{y }\) and \(N_{xy}\) :

Normal stress resultants

\(N_{tx} , N_{ty}\) and \(N_{txy}\) :

Normal stress resultants due to temperature rise

\(N_{\alpha } \left( {x,y} \right)\) :

The bilinear Lagrange shape functions associated with node α

\(P_{i} \left( y \right)\) and \(Q_{i} \left( x \right)\) :

Interpolation functions

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

Elementary displacement vector

\(Q_{ij}^{\left( k \right)^\circ }\) :

The reduced stiffness components in local coordinates system of the kth layer

\(Q_{xz} , Q_{yz}\) :

Transverse shear stress resultants

\(\bar{Q}_{11}^{\left( k \right)}\) :

The reduced stiffness components in laminate coordinates system of the kth layer

T:

Time parameter

\(T\) :

The kinetic energy

\(u, v\) :

In-plane displacement vector components

\(u_{0} , v_{0}\) :

In-plane displacement vector components at the mid-plane of the laminate \(\left( {z = 0} \right)\)

\(U\) :

The strain potential energy

\(w\) :

Out-of-plane displacement vector component

\(W\) :

The external forces work

\(x, y\) and \(z\) :

Coordinates of point within the plate

\(\alpha_{1}^{\left( k \right)}\), \(\alpha_{2}^{\left( k \right)}\) :

Coefficients of thermal expansion of the kth layer in local coordinates system

\(\alpha_{x}^{\left( k \right)} , \alpha_{y}^{\left( k \right)}\) and \(\alpha_{xy}^{\left( k \right)}\) :

Coefficients of thermal expansion of the kth layer in laminate coordinates system

\(\gamma_{12}^{\left( k \right)}\) :

In-plane shear strain of the kth layer in local coordinates system

\(\gamma_{13}^{\left( k \right)}\), \(\gamma_{23}^{\left( k \right)}\) :

Transverse shear strain of the kth layer in local coordinates system

\(\left\{ {\gamma_{s} } \right\}\) :

Transverse shear strain vector

\(\bar{\gamma }_{xz}^{A} ,\bar{\gamma }_{yz}^{A}\) :

The sampling points

\(\delta^{\alpha }\) :

Displacement vector component associated with node α

\(\delta \left( {x,y} \right)\) :

The displacement vector component of a given point M(x, y) within an element

\(\Delta T\) :

The temperature rise

\(\Delta T_{\text{cr}}\) :

Critical temperature rise

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

Strain vector

\(\left\{ {\varepsilon^{0} } \right\}\) :

Membrane strain vector

\(\varepsilon_{1}^{\left( k \right)} , \varepsilon_{2}^{\left( k \right)}\) :

Membrane strain of the kth layer in local coordinates system

\(\varepsilon_{x}^{\left( k \right)} , \varepsilon_{y}^{\left( k \right)}\) :

Membrane strain of the kth layer in laminate coordinates system

θ :

Fibers orientation angle with respect to laminate coordinates system

\(\lambda\) :

Loading factor

\(\lambda_{\text{cr}}\) :

Critical loading factor

\(\nu_{12}^{\left( k \right)}\), \(\nu_{21}^{\left( k \right)}\) :

Poisson’s ratios

\(\pi\) :

The total potential energy

\(\sigma_{1}^{K} , \sigma_{2}^{K}\) :

Normal stresses of the kth layer in local coordinates system

\(\sigma_{x}^{\left( k \right)}\), \(\sigma_{y}^{\left( k \right)}\) :

Normal stresses of the kth layer in laminate coordinates system

\(\tau_{13}^{\left( k \right)}\), \(\tau_{23}^{\left( k \right)}\) :

Transverse shear stresses of the kth layer in local coordinates system

\(\tau_{xy}^{k}\) :

In-plane shear stress of the kth layer in local coordinates system

\(\tau_{xy}^{\left( k \right)}\) :

In-plane shear stress of the kth layer in laminate coordinates system

\(\tau_{xz}^{\left( k \right)}\), \(\tau_{xz}^{\left( k \right)}\) :

Transverse shear stresses of the kth layer in laminate coordinates system

\(\varphi_{x}\) :

Angle of rotation with respect to the y-axis

\(\varphi_{y}\) :

Angle of rotation with respect to the x-axis

\(\omega\) :

The natural frequency

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Authors and Affiliations

  1. Energetic and Material Engineering Laboratory (LGEM), University of Biskra, BP 145, RP, Biskra, 07000, Algeria

    Abdelouahab Tati, Souhia Bouadjadja & Yassine Bada

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  1. Abdelouahab Tati
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Tati, A., Bouadjadja, S. & Bada, Y. Free Vibration of Thermally Stressed Angle-Ply Laminated Composite Using First-Order Shear Deformation Theory Model with Assumed Natural Shear Strain. J. Inst. Eng. India Ser. C 100, 937–947 (2019). https://doi.org/10.1007/s40032-018-0484-0

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