Abstract
We present a parameterized family of finite-difference schemes to analyze the energy properties for linearly elastic constant-coefficient Timoshenko systems considering shear deformation and rotatory inertia. We derive numerical energies showing the positivity, and the energy conservation property and we show how to avoid a numerical anomaly known as locking phenomenon on shear force. Our method of proof relies on discrete multiplier techniques.
Highlights
The importance of beam theories is well known in the world of engineering and mathematics
We present a parameterized family of finite-difference schemes to analyze the energy properties for linearly elastic constantcoefficient Timoshenko systems considering shear deformation and rotatory inertia
The governing equations of Timoshenko beams are considered an improvement over the Euler-Bernoulli and Rayleigh beams since shear deformation is taken into account
Summary
The importance of beam theories is well known in the world of engineering and mathematics. Taking into account (2) and (1), Timoshenko [2] established the following differential equations that incorporate the effects of transverse shear in the cross section of the beam of length L: ρ1φtt − κ(φx + ψ)x = 0, in (0, L) × (0, T) , (3). It is well known that standard Galerkin finite element method using equal-order piecewise linear approximations for the rotations ψ and displacement φ yields locking phenomenon This means that this method produces unsatisfactory numerical results when the thickness is very small. This paper is mainly concerned with theoretical analysis of finite-difference schemes applied to Timoshenko equations (3)-(4) aiming to identify the locking numbers.
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