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)


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.


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



Cross-sectional area of the plate at root side


Length of the plate


Breadth of the plate


Unknown coefficients


Elastic modulus of the plate material at root side

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

Load vector


Moment of inertia of the plate at root side


Flexural rigidity of plate

\(\left[ K \right]\)

Stiffness matrix

\(\left[ M \right]\)

Mass matrix


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


Set of orthogonal functions for u


Set of orthogonal functions for v


Set of orthogonal functions for w


Foundation stiffness


Fundamental linear frequency


Number of Gauss points


External uniformly distributed load


Thickness of the plate at root side


Kinetic energy of the system


Displacement field in x-axis


Strain energy stored in the system


Displacement field in y-axis


Potential energy of the external forces


Displacement field in z-axis


Variational operator


Poisson’s ratio


Density of the plate material at (ξ = 0)


Time coordinate

\(\xi , \eta\)

Normalized axial coordinates


Taper parameter


Nonlinear frequency


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© 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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