Review of solution methodologies for structural analysis of composites

This paper presents a unified approach for predicting the free and forced (steady-state and transient) vibration analyses of annular sector and annular plates with various combinations of classical and non-classical boundary supports. In spite of the types of the boundary restraints and the shapes of the plates, the admissible displacement function is described as a modified trigonometric series expansion, and four sine terms are introduced to overcome all the relevant discontinuities or jumps of elastic boundary conditions. Mathematically, the unification of various boundary value problems for annular sector and annular plates is physically realized by setting a set of coupling springs to ensure appropriate continuity conditions along the radial edges of concern. Numerous examples are presented for the free vibration analyses of annular sector and annular plates with different boundary restraints. With regard to the forced vibration analysis, annular sector and annular plates with different external excitations are examined. The accuracy, convergence and numerical robustness of the current approach are extensively demonstrated and verified through numerical examples which involve plates with various shapes and boundary conditions.

In the present study, buckling and free vibration analyses of annular thin sector plate made of functionally graded materials (FGMs) resting on visco-elastic Pasternak foundation, subjected to external radial, circumferential and shear in-plane loads is investigated. Material properties are assumed to vary along the thickness according to an power law with Poisson\'s ratio held constant. First, based on the classical plate theory (CPT), the governing equation of motion is derived using Hamilton\'s principle and then is solved using the generalized differential quadrature method (GDQM). Numerical results are compared to those available in the literature to validate the convergence and accuracy of the present approach. Finally, the effects of power-law exponent, ratio of radii, thickness of the plate, sector angle, and coefficients of foundation on the fundamental and higher natural frequencies of transverse vibration and critical buckling loads are considered for various boundary conditions. Also, vibration and buckling mode shapes of functionally graded (FG) sector plate have been shown in this research. One of the important obtained results from this work show that ratio of the frequency of FG annular sector plate to the corresponding values of homogeneous plate are independent from boundary conditions and frequency number.

Differential quadrature element method for static analysis of Reissner–Mindlin polar plates

In this article, the differential quadrature method (DQM) is used to study the free vibration of functionally graded (FG) thin annular sector plates. The material properties of the FG-plate are assumed to vary continuously through the thickness, according to the power-law distribution. The governing differential equations of motion are derived based on the classical plate theory and solved numerically using DQM. The natural frequencies of thin FG annular sector plates under various combinations of clamped, free, and simply supported boundary conditions are presented for the first time. To ensure the accuracy of the method, the natural frequencies of a pure metallic plate are calculated and compared with those existing in the literature for the homogeneous plate. In this case, the result shows very good agreement. For the FG-plates, the effects of boundary conditions, volume fraction exponent, and variation of Poisson's ratio on the free vibrational behaviour of the plate are studied.

Geometrical nonlinear free vibration responses of FG-CNT reinforced composite annular sector plates integrated with piezoelectric layers

A new mathematical model of porous functionally graded (PFG) conical sector micro/nanoplates with temperature-dependent properties was developed based on the modified couple stress theory. The variational iteration method was employed to solve the nonlinear differential equations describing the bending of flexible conical (annular) sector plates under thermomechanical loading. The proposed method yielded an almost exact solution while requiring much less computational time compared to the finite difference and finite element methods.

This paper presents three-dimensional elasticity solutions for an annular sector plate made of transversely isotropic functionally graded material (FGM) subjected to concentrated forces \(\left( X,Y,0 \right) \) or couples \(\left( M_X,M_Y,M_Z\right) \) applied at one of its radial edges. The elastic coefficients can vary arbitrarily through the plate thickness. The analysis was based on the assumed forms of displacements for bending of an FGM plate (Mian and Spencer in J Mech Phys Solids 4:2283–2295, 1998), in which the four analytical functions were constructed properly. Appropriate boundary conditions and end conditions similar to those in the classic plate theory were employed to determine the unknown constants contained in the analytical functions so as to accomplish the analysis. When the material coefficients are all constant, the obtained analytical solutions can be degenerated into those for a homogeneous transversely isotropic annular sector plate, which have never been reported before. The solutions may be further reduced to those for a homogeneous isotropic annular sector plate, among which the ones for concentrated couples \(\left( M_X ,M_Y,0 \right) \) are also new to the literature.

An original work is presented for the free vibration analysis of thin functionally graded material (FGM) annular sectorial plates with arbitrary boundary supports using the weak form quadrature element method (QEM). Novel C 1 quadrature annular sectorial plate element is developed based on the physically neutral surface thin plate theory in cylindrical coordinate system. Three displacement functions are used to avoid the introduction of the mixed second order derivative degree of freedom at corner points. The derived explicit formulations with a varying node number can be applied for any type of node. A way to define the elastic spring constants is proposed to ensure the independent of the non-dimensional frequency parameter on the power-law exponent. The convergence and accuracy of the present model are validated by comparing the obtained results with those available in literatures. It is demonstrated that the proposed model is simple, straightforward and has good performance. A wide range of numerical results are reported to demonstrate the effect of the sector angle, ratio of inner/outer radii, and boundary conditions on the vibration behavior of the thin annular sectorial plate. Some tabulated results should be new which may serve as a reference for verifying numerical solutions obtained by other methods.

The nonlinear steady state large amplitude forced vibration response of a laminated composite annular sector plate is presented. The nonlinear governing equation of motion of the laminated composite annular sector plate has been obtained using kinematics of first-order shear deformation theory (FSDT) and employing Hamilton’s principle. The governing equations of motion have been solved in a time domain using a modified shooting method and arc-length/pseudo-arc length continuation technique. The influence of the boundary condition, sector angle, and annularity ratio on the linear as well as nonlinear steady state forced vibration response has been investigated. The strain/stress variation across the thickness of the annular sector plate is presented to explain the reason for a decrease/increase in hardening nonlinear behaviour. The periodic variation of the non-linear steady state stress has also been obtained to throw light into the factors influencing the unequal stress half cycles and multiple cyclic stress reversals, which is detrimental to the fatigue design of laminated composite annular sectorial plates. The frequency spectra of the steady state stress reveals large even and odd higher harmonic contributions for different cases due to changes in the restoring force dynamics. The modal interaction/exchange during a cycle is demonstrated using a deformed configuration of the laminated annular sector plate.

In this paper, static, dynamic and natural frequency responses of composite annular sector plate with carbon nanotubes (CNTs) reinforcements resting on viscoelastic foundation are investigated. The carbon nanotubes are considered to have uniform or functionally graded pattern in the plate thickness. Kelvin-voight model is used to model the viscoelastic foundation. To model the problem, Hamilton principle based on first shear deformation plate theory and finite element method are applied. The mechanical properties of annular sector plate composed of CNTs and polymer matrix are evaluated by using the rule of mixtures. The numerical results are obtained for investigating the effect of various factors such as distribution and volume fraction of CNTs, boundary conditions, stiffness and damping coefficients of viscoelastic foundation and sector angles on natural frequency, static and dynamic transient responses of the plate.

Natural frequency analysis of anisotropic functionally graded material (FGM) annular sector plates on Winkler elastic foundations based on three-dimensional theory of elasticity was investigated. The three-dimensional graded finite element formulation was derived based on the principle of minimum potential energy and the Rayleigh-Ritz method. For an orthotropic FGM, the material properties were assumed to have in-plane polar orthotropy and transverse heterogeneity according to an exponential law, whereas the mass density was assumed to be constant. For an isotropic FGM, material properties varied continuously through the thickness direction according to a power-law distribution, whereas Poisson’s ratio was set to be constant. The effects of material gradient exponents, different sector angles, different thickness ratio, Winkler parameter and two different boundary conditions on the natural frequencies and mode shapes of FGM annular sector plates have been investigated. Numerical solution was compared with the result of an FGM annular circular plate, which showed good agreement.

This paper deals with frequency optimization of symmetrically laminated angle-ply annular sector plates. The design objective is the maximization of the fundamental frequency and the design variable is the fiber orientation in the layers. The first order shear deformation theory is used for the finite element solution of laminates. Nine-node Lagrangian rectangular plate element which have five degrees of freedom per node is used for the finite element solution of the laminated annular sector plates. The modified feasible direction method is used for the optimization routine. For this purpose, a program based on FORTRAN is used. The numerical analysis is carried out to investigate the effects of annularity, boundary conditions and sector angle on the optimal designs and the results are shown in tabular and graphical forms. Finally, the obtained results are compared. It can be shown that the parameters have a key role on the optimum design of laminated annular sector plates.

On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates

Thermal buckling of FG graphene platelet reinforced composite annular sector plates

A new fluid–solid interface algorithm for simulating fluid structure problems in FGM plates