TITLE:
On Applications of Generalized Functions in the Discontinuous Beam Bending Differential Equations
AUTHORS:
Dimplekumar Chalishajar, Austin States, Brad Lipscomb
KEYWORDS:
Mechanics of Solids, Discontinuities in a Beam Bending Differential Equations, Generalized Functions, Jump Discontinuities
JOURNAL NAME:
Applied Mathematics,
Vol.7 No.16,
October
25,
2016
ABSTRACT: This paper discusses the mathematical
modeling for the mechanics of solid using the distribution theory of Schwartz
to the beam bending differential Equations. This problem is solved by the use
of generalized functions, among which is the well known Dirac delta function.
The governing differential Equation is Euler-Bernoulli beams with jump
discontinuities on displacements and rotations. Also, the governing
differential Equations of a Timoshenko beam with jump discontinuities in slope,
deflection, flexural stiffness, and shear stiffness are obtained in the space
of generalized functions. The operator of one of the governing differential
Equations changes so that for both Equations the Dirac Delta function and its
first distributional derivative appear in the new force terms as we present the
same in a Euler-Bernoulli beam. Examples are provided to illustrate the
abstract theory. This research is useful to Mechanical Engineering, Ocean
Engineering, Civil Engineering, and Aerospace Engineering.