Skip to main content
SpringerLink
Log in

A nonlinear dynamical theory of non-classical plates

  • Published:
Journal of Shanghai University (English Edition)

Abstract

In this paper, a new nonlinear formulation of plates, including shear and rotatory inertia and transverse normal stress effects, is developed by means of general assumptions, of which the von Karman-type formulation and some thick plate theories are special cases. To keep the formulation fairly general, the problem addressed in this paper simultaneously includes: the effects of shear deformation according to the geometric deformation similarity of the cross-section, the rotatory inertia, and the transverse normal stress. The three-dimensional compatible equations are applied to derive the basic equations. Numerical results are given for linear and non-linear analysis of plates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

h :

thickness of the plate

u, v, w :

displacements at any point (x, y, z) in the x-, y-, and z-direction, respectively

u 0, v0, w0 :

The middle-surface displacement components in the x-, y-, and z-direction, respectively

ϕ x, ϕy :

shear rotations in addition to the usual flexural rotations

f(z) :

deformation distribution function of thickness

q(x, y, t) :

transverse distributed loading

π :

mass density

D :

flexural rigidity of the plate, D=Eh 3 / 12(1 − v 2)

E :

Young’s modulus

G :

modulus of elasticity in shear

V :

Poission’s ratio

J :

inertia moment

P(z) :

displacement distribution function of shear

B(z) :

function of transverse normal deformation

β x, βy :

angle functions of rotation at x- and y-directions

β,β:

influence coefficients of shear and transverse normal stress, respectively

References

  1. Bhimaraddi, A. Nonlinear dynamics of in-plane loaded imperfect rectangular plates. Journal of Applied Mechan. cs, 59, 893–901 (1992)

    MATH  Google Scholar 

  2. Celep, Z. Vibrations of initially imperfect circular plates including the shear and rotatory inertia effects. Journal of Sound and Vibration, 85, 513–523 (1985)

    Article  Google Scholar 

  3. Chia, C. Y. and Sathyamoorthy, M. Nonlinear vibration of circular plates with transverse shear and rotatory inertia. Journal of Sound and Vibration, 78, 131–137 (1981)

    Article  Google Scholar 

  4. Dumir, P. C. and Singal, L. Nonlinear analysis of thick circular plates. Journal of Engineering Mechanics, 112, 260–272 (1986)

    Google Scholar 

  5. Kant, T. and Kommineni, J. R. Finite element geometrically nonlinear analysis of fibre reinforced composite and sandwich laminates based on a higher-order theory. Computers and Structures, 45, 511–520 (1992)

    Article  MATH  Google Scholar 

  6. Leung, A. Y. T. An unconstrained third-order plate theory. Computers and Structures, 40, 871–875 (1991)

    Article  MATH  Google Scholar 

  7. Li, G. Dynamics of Engineering Structures. Chinese Railway Press, 1988, (in Chinese)

  8. Liu, C. F. and Reddy, J. N. Nonlinear vibration of laminated rectangular plates using a shear deformable finite element. Nonlinear Analysis and NDE of Composite Material Vessels and Components, ASME-PVP, 115, 35–42 (1986)

    Google Scholar 

  9. Reddy, J. N. and Chao, W. C. Nonlinear oscillations of laminated anisotropic rectangular plates. Journal of Applied Mechanics, 49, 396–402 (1982)

    Google Scholar 

  10. Reddy, J. N. Refined nonlinear theory of plates with transverse shear deformation. International Journal of Solids and Structures, 20, (3), 881–896 (1984)

    Article  MATH  Google Scholar 

  11. Sathyamoorthy, M. and Prasad, M. E. Nonlinear vibration of elliptical plates by a multiple mode approach. Developments in Mechanics, 11, Proc. 17th midwestern mechanics conference, Ann Arbor MI, 153–154 (1981)

  12. Sathyamoorthy, M. Nonlinear vibration analysis of plates: review and survey of current developments. Applied mechanies Review, 40, 1553–15561 (1988)

    Article  Google Scholar 

  13. Sivakumaran, K. S. and Chia, C. Y. Large-amplitude oscillations of un-symmetrically laminated anisotropic rectangular plates including shear, rotatory inertia, and transverse normal stress. Journal of Applied Mechanics, 52, 536–542 (1985)

    Article  MATH  Google Scholar 

  14. Ye, Zhiming and Jing, Guangsheng. On the plane stress state in elasticity. Mechanics and Practice, 7, 16–22, (1987)

    Google Scholar 

  15. Ye, Zhiming. A new finite element formulation for planar deformation analysis. Int. J. Numerical Methods in Engineering, 40 (1997).

Download references

Author information

Authors and Affiliations

  1. School of Architectural Eng. Shanghai University, 149 Yanchang Road, 200072, Shanghai

    Ye Zhiming Ph. D. (Prof.)

  2. Shanghai Institute of Applied Mathematics & Mechanics, China

    Ye Zhiming Ph. D. (Prof.)

Authors
  1. Ye Zhiming Ph. D.
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This Work was supported by the Natural Science Fundation of Shanghai for Returning Scholars

About this article

Cite this article

Ye, Z. A nonlinear dynamical theory of non-classical plates. J. of Shanghai Univ. 1, 28–35 (1997). https://doi.org/10.1007/s11741-997-0040-2

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11741-997-0040-2

Key words

Search

Navigation