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Nonlinear bending models for beams and plates.

¹Department of Mathematics , Louisiana State University , Baton Rouge, LA 70803, USA.

Proceedings. Mathematical, Physical, and Engineering Sciences

|October 9, 2014

Summary

This summary is machine-generated.

A novel nonlinear model accurately predicts large beam deflections using an integral condition, aligning with elastica model results. This approach also extends to dynamic and 2D plate scenarios, offering a robust framework for structural analysis.

Keywords:

elastica plate bending singular integral equation

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Area of Science:

Mechanical Engineering
Applied Mathematics
Continuum Mechanics

Background:

Large deflections in beams and plates present complex nonlinear behavior.
Existing models like the Euler-Bernoulli beam theory often simplify or neglect nonlinear effects.
Accurate modeling is crucial for predicting structural integrity under extreme loads.

Purpose of the Study:

To propose a new nonlinear model for large beam deflections.
To ensure the inextensibility condition is met, maintaining original beam length.
To explore dynamic and 2D generalizations for broader applicability.

Main Methods:

Development of a nonlinear integral condition coupled with the Euler-Bernoulli boundary value problem.
Numerical validation against the established elastica model.
Reduction of a 2D Kirchhoff plate problem to a singular integral equation.
Application of the collocation method using Jacobi polynomials for series-form solutions.

Main Results:

The proposed model accurately simulates large beam deflections.
Numerical results show good agreement with the elastica model.
A series-form solution was derived for the 2D plate problem, handling singularities.
The inextensibility condition was successfully incorporated.

Conclusions:

The new nonlinear model provides an accurate and efficient method for analyzing large beam deflections.
The model's framework is adaptable for dynamic and two-dimensional plate analyses.
The methodology offers a robust approach to solving complex structural mechanics problems with inextensibility constraints.