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A Study on the Structural Behaviour of AFG Non-uniform Plates on Elastic Foundation: Static and Free Vibration Analysis

  • Hareram Lohar
  • Anirban Mitra
  • Sarmila Sahoo
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)

Abstract

Plate on elastic foundation is an important realization of actual boundary condition of structures. The present paper studies the effect of the stiffness value of the elastic foundation on large deflection and free vibration of AFG non-uniform plate. Governing set of equation of the system is obtained by energy principle through variational method. Derived nonlinear equations are handled through utilizing an iterative method, which is direct substitution method. The effects of the elastic foundations are represented through load versus maximum deflection plot, deflected shape plot and backbone curves. The effects of the edged boundary conditions and non-uniformity of the plate shape are also highlighted.

Keywords

Large deflection Free vibration AFG plate Elastic foundation Minimum potential energy principle Hamilton principle Direct substitution method 

Nomenclature

\(A_{0}\)

Cross-sectional area of the plate at root side

\(a\)

Length of the plate

\(b\)

Breadth of the plate

\(d\)

Unknown coefficients

\(E_{0}\)

Elastic modulus of the plate material at root side

\(\left\{ f \right\}\)

Load vector

\(I_{0}\)

Moment of inertia of the plate at root side

\(D\)

Flexural rigidity of plate

\(\left[ K \right]\)

Stiffness matrix

\(\left[ M \right]\)

Mass matrix

\(nf,nw,nv\)

Number of constituent functions for w, u and v, respectively

\(\upalpha_{i}\)

Set of orthogonal functions for u

\(\upbeta_{i}\)

Set of orthogonal functions for v

\(\phi_{i}\)

Set of orthogonal functions for w

\(K_{f}\)

Foundation stiffness

\(\omega_{1}\)

Fundamental linear frequency

\(nfg\)

Number of Gauss points

\(q\)

External uniformly distributed load

\(t_{0}\)

Thickness of the plate at root side

\(T\)

Kinetic energy of the system

\(u\)

Displacement field in x-axis

\(U\)

Strain energy stored in the system

\(v\)

Displacement field in y-axis

\(V\)

Potential energy of the external forces

\(w\)

Displacement field in z-axis

\(\delta\)

Variational operator

\(\mu\)

Poisson’s ratio

\(\rho_{0}\)

Density of the plate material at (ξ = 0)

\(\tau\)

Time coordinate

\(\xi , \eta\)

Normalized axial coordinates

\(\alpha\)

Taper parameter

\(\omega_{nf}\)

Nonlinear frequency

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia
  2. 2.Department of Civil EngineeringHeritage Institute of TechnologyKolkataIndia

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