This paper presents a static analysis of functionally graded plates (FGPs) by using a new first shear deformation theory (FSDT). This theory contains only four unknowns, with is even less than the classical FSDT. In this paper a simply supported FG square sandwich plate is subjected to a bi-sinusoidal load. The governing equations for static bending analysis are derived by employing the principle of virtual works. These equations are then solved via Navier-type, closed form solutions. The accuracy of the present theory is ascertained by comparing it with various available solutions in the literature.

The closed-form solution of a generalized hybrid type quasi-3D higher order shear deformation theory (HSDT) for the bending analysis of functionally graded shells is presented. From the generalized quasi-3D HSDT (which involves the shear strain functions “f(ζ)” and “g(ζ)” and therefore their parameters to be selected “m” and “n”, respectively), infinite six unknowns' hybrid shear deformation theories with thickness stretching effect included, can be derived and solved in a closed-from. The generalized governing equations are also “m” and “n” parameter dependent. Navier-type closed-form solution is obtained for functionally graded shells subjected to transverse load for simply supported boundary conditions. Numerical results of new optimized hybrid type quasi-3D HSDTs are compared with the first order shear deformation theory (FSDT), and other quasi-3D HSDTs. The...

This paper presents a simplified first order shear deformation theory (FSDT) for laminated composite and sandwich plates. Unlike the existing FSDT, the present one has a novel displacement field which include undetermined integral terms and contains only four unknowns. Equations of motion and boundary conditions are derived from the Hamilton’s principle. Navier-type analytical solution is obtained in closed form and by solving the eigenvalue equation. The comparison of the present results with the available elasticity solutions and the results computed independently using the FSDTs available in the literature shows that this theory predicts the fundamental frequencies with good accurately. It can be concluded that the proposed theory is accurate and simple in solving the dynamic behavior of single and sandwich laminated composite plates.

The purpose of this paper is to study the vibrational behavior of advanced composite plates by using a novel first shear deformation theory (FSDT). This theory contains only four unknowns, with is even less than the classical FSDT. The governing equations are derived by employing the Hamilton's principles and solved via Navier's solution. The present results were validiated by comparing it with the 3D, classical FSDT and other solutions available in the literature. Shear correction factor apper to be unfovarable in some cases (case dependent). Finally, authors recommend further study of this new manner to model the displacement field.

This paper presents a static analysis of functionally graded (FG) single and sandwich beams by using a simple and efficient 4-unknown quasi-3D hybrid type theory, which includes both shear deformation and thickness stretching effects. The governing equations and boundary conditions are derived by employing the principle of virtual works. Navier-type closed-form solution is obtained for several beams. New hybrid type shear strain shape functions for the inplane and transverse displacement were introduced in general manner to model the displacement field of beams. Numerical results of the present compact quasi-3D theory are compared with other quasi-3D higher order shear deformation theories (HSDTs).

The mechanical behavior of advanced composites can be modeled mathematically through unknown variables and Shear Strain Thickness Functions (SSTFs). Such SSTFs can be of polynomial or non-polynomial nature and some parameters of non-polynomial SSTFs can be optimized to get optimal results. In this paper, these parameters are called “r” and “s” and they are the argument of the trigonometric SSTFs introduced within the Carrera Unified Formulation (CUF). The Equivalent Single Layer (ESL) governing equations are obtained by employing the Principle of Virtual Displacement (PVD) and are solved using Navier method solution. Furthermore, trigonometric expansion with Murakami theory was implemented in order to reproduce the Zig-Zag effects which are important for multilayer structures. Several combinations of optimization parameters are evaluated and selected by different criteria of aver...

The mechanical behavior of advanced composites can be modeled mathematically through unknown variables and Shear Strain Thickness Functions (SSTFs). Such SSTFs can be of polynomial or non-polynomial nature and some parameters of non-polynomial SSTFs can be optimized to get optimal results. In this paper, these parameters are called “r” and “s” and they are the argument of the trigonometric SSTFs introduced within the Carrera Unified Formulation (CUF). The Equivalent Single Layer (ESL) governing equations are obtained by employing the Principle of Virtual Displacement (PVD) and are solved using Navier method solution. Furthermore, trigonometric expansion with Murakami theory was implemented in order to reproduce the Zig-Zag effects which are important for multilayer structures. Several combinations of optimization parameters are evaluated and selected by different criteria of aver...

We introduce a compact and unified shear deformation theory for plates with elasto-plastic behavior. We formulate the kinematics of the two-dimensional structure in a compact and unified manner using the Carrera Unified Formulation. This formulation allows for generalized expansions of the primary variables and through-the-thickness functions. We obtain the governing equations using the principle of virtual work and a finite element discretization. We solve the nonlinear equations using a Newton–Raphson linearization scheme, and linearize the constitutive equations using the algorithmic tangent moduli. We consider the J2 flow theory of plasticity, and use a backwards Euler scheme to update the stresses. We analyze the convergence, and compare the effectiveness of the Mixed Interpolation of Tensorial Components technique in contrasting the shear locking phenomenon in the nonlinear regim...

This paper presents an exact solution for doubly-curved shells subjected to thermal loads. The solution is based on equilibrium equations. The through-the-thickness shell temperature is calculated using the Fourier’s heat conduction equation. The governing equations for displacement and temperature are solved using Navier method, which is valid for shell panels with constant curvature and simply-supported edges. The governing equations for temperature and displacements are in term of the thickness variable and they are solved using the differential quadrature method (DQM). The structures are discretized by each layer applying the Chebyshev-Gauss-Lobatto grid distribution. Lagrange interpolation polynomials are used as basis functions. The inter-laminar continuity of transverse stresses and displacements is imposed. The out-of-plane zero-stresses condition are imposed at the top and bot...

A quasi-three-dimensional solution for the bending analysis of simply supported orthotropic piezoelectric shallow shell panels subjected to thermal, mechanical and electrical potential is introduced. The mechanical governing equations are derived in term of three-dimensional equilibrium relations and the classical Maxwell’s equations. The trough-the-thickness temperature is modeled by the Fourier’s heat conduction equation. The coupled partial differential equations are solved by Navier closed form solutions. The trough-the-thickness profile for electrical potential, temperature profile and displacements is obtained by using a quasi-exact method so-called the differential quadrature method (DQM). Chebyshev polynomials of the third kind are used as the basis functions and the grid thickness domain discretization for the DQM. The interlaminar conditions for transverse stresses, tempera...

This paper presents an exact solution for the static analysis of magneto-electro-elastic simply supported shallow shells panels. The mechanical equations are derived via equilibrium elasticity relations. The electrical and magnetic governing equations are obtained by electrostatic and magnetostatic equilibrium relations. The shell displacements, electrical and magnetic potential functions are solved analytically by the Navier closed form solutions. The governing equations formulated in terms of thickness coordinate are solved semi-analytically by using the differential quadrature method. The Lagrange polynomials are employed as basis functions. The equations are discretized per each layer by the Chebyshev-Gauss-Lobatto grid distribution. The continuity conditions in the adjacent layers for mechanical displacement, transverse dielectric displacement, electric and magnetic scalar function ...

The present mathematical model for complex shells is given in the framework of Carrera unified formulation. The mechanical, electrical, and magnetic equations are derived in terms of the principle of virtual displacement, Maxwell’s equations and Gauss equations. Fourier’s heat conduction equation is used. The governing equations are discretized by the Chebyshev–Gauss–Lobatto and solved with the differential quadrature method. The three-dimensional (3D) equilibrium for mechanical, electrical, and magnetic equations are employed for recovering the transverse stresses, electrical displacement and magnetic induction. Finally, quasi-3D solutions for cycloidal shell of revolution and a funnel panel are introduced in this paper.

A numerical solution for the three-dimensional static analysis of functionally graded shells with constant curvature is presented. The solution is based on three-elasticity equations written in orthogonal curvilinear coordinates which are valid for spherical, cylindrical shell panels and rectangular plates. The equations in term of the mid-surface variables are solved using a summation of harmonics in term of Navier method which is valid only for simply supported structures. The equations in term of the thickness direction are solved numerically by the Differential Quadrature method (DQM) which permitted to easily calculate the approximate derivative of a function using a weighting sum of the functions evaluated in a certain grid. The layers of the structure are discretized separately by the Chebyshev-Gauss-Lobatto grid and Lagrange interpolation polynomials are considered as the basis f...

This paper presents Best Theory Diagrams (BTDs) constructed from various non-polynomial theories for the static analysis of thick and thin symmetric and asymmetric cross-ply laminated plates. The BTD is a curve that provides the minimum number of unknown variables necessary for a fixed error or vice versa. The plate theories that belong to the BTD have been obtained by means of the Axiomatic/Asymptotic Method (AAM). The different plate theories reported are implemented by using the Carrera Unified Formulation (CUF). Navier-type solutions have been obtained for the case of simply- supported plates loaded by a bisinuisoidal transverse pressure with different length-to-thickness ratios. The BTDs built from non-polynomials functions are compared with BTDs using Maclaurin expansion. The results suggest that the plate models obtained from the BTD using non-polynomial terms can improve the accu...

This paper presents an exact solution for doubly-curved shells subjected to thermal loads. The solution is based on equilibrium equations. The through-the-thickness shell temperature is calculated using the Fourier’s heat conduction equation. The governing equations for displacement and temperature are solved using Navier method, which is valid for shell panels with constant curvature and simply-supported edges.

This paper was written in the context of the project: “Desarrollo de materiales avanzados para el diseño de nuevos productos y servicios tecnológicos para la minería Peruana” founded by CONCYTEC (FONDECYT and GRUPO BANCO MUNDIAL) under the contract number N° 032-2019-FONDECYT-BM-INC.INV. The authors of this manuscript appreciate the financial support from the Peruvian Government.

Se presenta una solución numérica para el análisis estático tridimensional de láminas graduadas funcionalmente con curvatura constante. La solución se basa en tres ecuaciones de elasticidad escritas en coordenadas curvilíneas ortogonales que son válidas para paneles de cubierta esféricos, cilíndricos y placas rectangulares. Las ecuaciones en términos de las variables de la superficie media se resuelven mediante una suma de armónicos en términos del método de Navier, que es válido solo para estructuras simplemente apoyadas. Las ecuaciones en función de la dirección del espesor se resuelven numéricamente por el método de Cuadratura Diferencial (DQM) que permitió calcular fácilmente la derivada aproximada de una función utilizando una suma ponderada de las funciones evaluadas en una determinada grilla. Las capas de la estructura se discretizan por separado mediante la ...

This paper presents Best Theory Diagrams (BTDs) constructed from non-polynomial terms to identify best shell theories for bending analysis of cross-ply single skin and sandwich shell panels. This structure presents a constant radii of curvature. The shell theories are constructed using Axiomatic/Asymptotic Method (AAM). The different shell theories are described using the Carrera's Unified Formulation. The governing equations are derived from the Principle of Virtual Displacement (PVD). Navier-Type closed form solution is used for solving the bending problem of simply supported doubly curved shell panels subjected to bi-sinusoidal transverse pressure. The BTDs built from non-polynomial functions are compared with Maclaurin expansions. Spherical shell panels with different layer-configurations are investigated. The results demonstrated that the shell models obtained from the BTD using non...

A quasi-three-dimensional solution for the bending analysis of simply supported orthotropic piezoelectric shallow shell panels subjected to thermal, mechanical and electrical potential is introduced. The mechanical governing equations are derived in term of three-dimensional equilibrium relations and the classical Maxwell's equations. The trough-the-thickness temperature is modeled by the Fourier's heat conduction equation. The coupled partial differential equations are solved by Navier closed form solutions. The trough-the-thickness profile for electrical potential, temperature profile and displacements is obtained by using a quasi-exact method so-called the differential quadrature method (DQM). Chebyshev polynomials of the third kind are used as the basis functions and the grid thickness domain discretization for the DQM. The interlaminar conditions for transverse stresses, temperature...

This paper outlines the conceptual and numerical design process of a comminution equipment centered on particle breakdown through impact. The process is divided into four stages, starting with the generation of device concepts achieved by developing a needs matrix for an optimal machine. Subsequently, in the second stage, various equipment shape proposals were introduced and tested, with the selection of an optimal proposal determined through performance comparisons. For comparison purposes, simulations utilizing the discrete element method (DEM) were conducted, considering analyses of accumulated power from collisions and particle breakage. Once the optimized prototype was identified, a breakage simulation was conducted to measure the device's reduction ratio. In the third stage, the machine elements of the device were calculated. Finally, in the fourth stage, a series of simulations ut...