Frequency response of rotating two-directional functionally graded GPL-reinforced conical shells on elastic foundation

: This paper analyzes the nonlinear buckling and post-buckling characteristics of the porous eccentrically stiffened functionally graded sandwich truncated conical shells resting on the Pasternak elastic foundation subjected to axial compressive loads. The core layer is made of a porous material (metal foam) characterized by a porosity coefficient which influences the physical properties of the shells in the form of a harmonic function in the shell’s thickness direction. The physical properties of the functionally graded (FG) coatings and stiffeners depend on the volume fractions of the constituents which play the role of the exponent in the exponential function of the thickness direction coordinate axis. The classical shell theory and the smeared stiffeners technique are applied to derive the governing equations taking the von Kármán geometrical nonlinearity into account. Based on the displacement approach, the explicit expressions of the critical buckling load and the post-buckling load-deflection curves for the sandwich truncated conical shells with simply supported edge conditions are obtained by applying the Galerkin method. The effects of material properties, core layer thickness, number of stiffeners, dimensional parameters, semi vertex angle and elastic foundation on buckling and post-buckling behaviors of the shell are investigated. The obtained results are validated by comparing with those in the literature.

Download Full-text

Geometrically Non-Linear Frequency Response of Axially Functionally Graded Beams Resting on Elastic Foundation Under Harmonic Excitation

International Journal of Manufacturing Materials and Mechanical Engineering ◽

10.4018/ijmmme.2018070103 ◽

2018 ◽

Vol 8 (3) ◽

pp. 23-43

Author(s):

Hareram Lohar ◽

Anirban Mitra ◽

Sarmila Sahoo

Keyword(s):

Frequency Response ◽

Elastic Foundation ◽

Mathematical Formulation ◽

Excitation Amplitude ◽

Harmonic Excitation ◽

Functionally Graded ◽

System Of Nonlinear Equations ◽

Response Curves ◽

Displacement Based ◽

Forced Vibration Analysis

This article presents geometrically nonlinear forced vibration analysis of an axially functionally graded (AFG) non-uniform beam resting on an elastic foundation. The mathematical formulation is displacement based and derivation of governing equations is accomplished following Hamilton's principle. The foundation has been mathematically incorporated into the analysis as a set of linear springs. According to the basic assumption of the present method force equilibrium condition is satisfied at a maximum excitation amplitude value. Thus, the dynamic problem is equivalently represented as a static one, which is solved by following a numerical implementation of the Broyden method. It is a method that utilizes the Jacobian matrix and subsequent correction of the initial Jacobian to solve a system of nonlinear equations. The large amplitude dynamic behaviour of the system in terms of non-dimensional frequency response curves is validated against established results and new results are furnished for a parabolic tapered AFG beam on a linear elastic foundation.

Download Full-text

Nonlinear response of axially functionally graded Timoshenko beams on elastic foundation under harmonic excitation

Curved and Layered Structures ◽

10.1515/cls-2019-0008 ◽

2019 ◽

Vol 6 (1) ◽

pp. 90-104

Author(s):

Hareram Lohar ◽

Anirban Mitra ◽

Sarmila Sahoo

Keyword(s):

Frequency Response ◽

Elastic Foundation ◽

Free Vibration Analysis ◽

Longitudinal Axis ◽

Harmonic Excitation ◽

Functionally Graded ◽

Amplitude Curve ◽

Response Curves ◽

Functional Variation ◽

Beams On Elastic Foundation

AbstractForced vibration of non-uniform axially functionally graded (AFG) Timoshenko beam on elastic foundation is performed under harmonic excitation. A linear elastic foundation is considered with three different classical boundary conditions. AFG materials are an advanced class of materials that have potential for application in various engineering fields. In the present work, variation of material properties along the longitudinal axis of the beam are considered according to power-law forms. Five values of material gradation parameter provides different functional variation and their effect on the frequency response of the system is studied. The present approximate method is displacement based and Von-Karman type of geometric nonlinearity is considered with rotational component to incorporate transverse shear. Hamilton’s principle is used to derive nonlinear set of governing equation and Broyden method is implemented to solve the nonlinear equations numerically. The results are successfully validated with previously published article. Frequency vs. amplitude curve corresponding to different combinations of system parameters are presented and are capable of serving as benchmark results. A separate free vibration analysis is undertaken to include backbone curves with the frequency response curves in the non-dimensional plane.

Download Full-text

Effect of Boundary Conditions and Taper Patterns on Geometrically Nonlinear Frequency Response of Axially Graded Beams on Elastic Foundation

Handbook of Research on Advancements in Manufacturing, Materials, and Mechanical Engineering - Advances in Mechatronics and Mechanical Engineering ◽

10.4018/978-1-7998-4939-1.ch006 ◽

2021 ◽

pp. 110-140

Author(s):

Hareram Lohar ◽

Anirban Mitra ◽

Sarmila Sahoo

Keyword(s):

Boundary Conditions ◽

Frequency Response ◽

Elastic Foundation ◽

Forced Vibration ◽

Mathematical Formulation ◽

Functionally Graded ◽

Response Curves ◽

Nonlinear Frequency Response ◽

Nonlinear Forced Vibration ◽

Displacement Based

Forced vibration analysis is performed on a tapered axially functionally graded beam resting on elastic foundation under externally applied harmonic excitations to present the effect of boundary conditions and taper patterns on the frequency response. The elastic foundation is modelled in the present analysis as Winkler foundation. A displacement based semi-analytical method is adopted for mathematical formulation and the derivation of governing equations is carried out following Hamilton's principle. Von Karman nonlinear strain-displacement relation employed to incorporate geometric nonlinearity. Broyden method is adopted to solve the nonlinear set of equations. Frequency response curves are plotted in non-dimensional frequency-amplitude plane to represent nonlinear forced vibration characteristic of the system. New benchmark results are also provided for different combination of system parameters (i.e., excitation amplitudes, foundation stiffness values, material models, taper patterns, and flexural boundary conditions). Operational deflection shapes (ODS) are also presented.

Download Full-text

Non-linear forced vibration analysis of higher-order shear-deformable functionally graded material beam in thermal environment subjected to harmonic excitation and resting on non-linear elastic foundation

The Journal of Strain Analysis for Engineering Design ◽

10.1177/0309324718782230 ◽

2018 ◽

Vol 53 (6) ◽

pp. 446-462 ◽

Cited By ~ 2

Author(s):

Amlan Paul ◽

Debabrata Das

Keyword(s):

Frequency Response ◽

Elastic Foundation ◽

Functionally Graded Material ◽

Thermal Loading ◽

Thermal Environment ◽

Harmonic Excitation ◽

Functionally Graded ◽

Non Linear ◽

Graded Material ◽

Shear Deformable

Geometrically non-linear forced vibration analysis of higher-order shear-deformable functionally graded material beam under harmonic excitation and supported on three-parameter non-linear elastic foundation is presented. The beam is immovably clamped and is considered to be under static thermal loading due to uniform temperature rise. Reddy’s third-order shear-deformable beam theory in conjunction with von Kármán geometric non-linearity is considered to derive the governing equations employing Hamilton’s principle, and Ritz method is followed for approximating the displacement and rotation fields. A numerical algorithm based on iterative substitution method and Broyden’s method is proposed to predict the stable regions of frequency-response behavior. The frequency-response curves are presented in normalized plane for variations of load-amplitude, elastic foundation parameters, temperature rise, gradation index and functionally graded material composition, and their effects are discussed in detail. It is found that the load-amplitude, elastic foundation parameters, thermal loading and some of the functionally graded material compositions significantly affect the frequency response; whereas, the effect of gradation index is found to be relatively small. A comparative frequency-response curve between Voigt model and Mori–Tanaka scheme of functionally graded material modeling is presented, and it shows negligible difference between these two models. The present problem under thermal environment is studied for the first time through this work, and the proposed model and the numerical algorithm provide a simplified approach to study the non-linear frequency-response behavior.

Download Full-text

Vibration characteristics of rotating truncated conical shells reinforced with agglomerated carbon nanotubes

Journal of Vibration and Control ◽

10.1177/10775463211000499 ◽

2021 ◽

pp. 107754632110004

Author(s):

Hassan Afshari ◽

Hossein Amirabadi

Keyword(s):

Carbon Nanotubes ◽

Differential Quadrature Method ◽

Shear Deformation Theory ◽

Free Vibration Analysis ◽

Functionally Graded ◽

Governing Equations ◽

Conical Shells ◽

First Order ◽

Effective Mechanical Properties ◽

Comprehensive Study

In this article, a comprehensive study is conducted on the free vibration analysis of rotating truncated conical shells reinforced with functionally graded agglomerated carbon nanotubes The shell is modeled based on the first-order shear deformation theory, and effective mechanical properties are calculated based on the Eshelby–Mori–Tanaka scheme along with the rule of mixture. By considering centrifugal and Coriolis accelerations and initial hoop tension, the set of governing equations is derived using Hamilton’s principle and is solved numerically using the differential quadrature method Convergence and accuracy of the presented model are confirmed and the effects of different parameters on the forward and backward frequencies of the rotating carbon nanotube-reinforced truncated conical shells are investigated.

Download Full-text

Buckling Behavior of FG-CNT Reinforced Composite Conical Shells Subjected to a Combined Loading

Nanomaterials ◽

10.3390/nano10030419 ◽

2020 ◽

Vol 10 (3) ◽

pp. 419 ◽

Cited By ~ 3

Author(s):

Abdullah H. Sofiyev ◽

Francesco Tornabene ◽

Rossana Dimitri ◽

Nuri Kuruoglu

Keyword(s):

Volume Fraction ◽

Shear Deformation Theory ◽

Structural Response ◽

Systematic Investigation ◽

Functionally Graded ◽

Combined Loading ◽

Proportional Loading ◽

Conical Shells ◽

Reinforced Composite ◽

Buckling Behavior

The buckling behavior of functionally graded carbon nanotube reinforced composite conical shells (FG-CNTRC-CSs) is here investigated by means of the first order shear deformation theory (FSDT), under a combined axial/lateral or axial/hydrostatic loading condition. Two types of CNTRC-CSs are considered herein, namely, a uniform distribution or a functionally graded (FG) distribution of reinforcement, with a linear variation of the mechanical properties throughout the thickness. The basic equations of the problem are here derived and solved in a closed form, using the Galerkin procedure, to determine the critical combined loading for the selected structure. First, we check for the reliability of the proposed formulation and the accuracy of results with respect to the available literature. It follows a systematic investigation aimed at checking the sensitivity of the structural response to the geometry, the proportional loading parameter, the type of distribution, and volume fraction of CNTs.

Download Full-text