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.
Highlights
This article introduces the method for computing lateral deflections of plane beams undergoing symmetric bending
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
Summary
Equation to different application obviously representing beam bending. One of the most common types of structural components is a beam, recommended more in Civil and Mechanical Engineering. Beams resist against transverse loads through a bending action, which creates compressive longitudinal stresses on one side of a beam and tensile stress on the other side With these two combinations between compressive longitudinal stress and tensile stress, an internal bending moment starts to occur. 4) Strain energy: Transverse shear and axial forces are ignored, while only internal strain energy from another object accounts for the bending moment deformations. Transverse shear and axial force are ignored, while only internal strain energy from another object accounts for the bending moment deformations. Beam motion: We associate beam motion as the loading on a x, y plane beam is structured in to two dimensional displacement field u ( x, y) and v ( x, y) u and v are respected as the axial and transverse displacement components with respect to a beam point. Airplane wings, diving boards, and stabilizers are prime examples of cantilever beams
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
