Computational formulation for periodic vibration of geometrically nonlinear structures—part 1: Theoretical background

Computational formulation for periodic vibration of geometrically nonlinear structures—part 2: Numerical strategy and examples

The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part II: iterative and explicit analytical solution for non-linear coupled transverse and in-plane vibrations

Non-linear forced vibrations of plates by an asymptotic–numerical method

Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates

Finite element methods for nonlinear eigenvalue problems and the postbuckling behavior of elastic plates

The problem of free vibration of non-linear structures is considered initially. It is shown that this problem can be represented as a non-linear eigenvalue problem. Variational principles for non-linear eigenvalue problems are defined. These variational principles are implemented with finite element models to define numerical approximations for the free vibration problem. The solution of these approximate equations provides a set of non-linear modal vectors and natural frequencies which vary with the amplitude of the solution. The non-linear eigenvalue parameters can be used in modal expansion approximations for the non-linear transient or steady state response of structural systems. To demonstrate the proposed techniques the free vibration and steady state vibration characteristics of a geometrically non-linear circular plate are determined.

Free and steady state vibration of non‐linear structures using a finite element–non‐linear eigenvalue technique

A series of finite element algorithms for the calculation of the post-buckling behaviour of structures are proposed. The algorithms represent a finite element implementation of a variational principle for nonlinear eigenvalue problems. Incremental amplitude solution schemes are defined. Numerical results are presented for the post-buckling behaviour of specific structures with linear and nonlinear pre-buckling states.

Ghassan M. Dib

Papers

Finite element methods for nonlinear eigenvalue problems and the postbuckling behavior of elastic plates

Free and steady state vibration of non‐linear structures using a finite element–non‐linear eigenvalue technique

Post‐buckling behaviour of structures using a finite element–nonlinear eigenvalue technique