Question

Describe the mathematical models of the boundary value problem to evaluate the elastic deflection of the beam based on Euler-Bernoulli and Timoshenko theories with finite difference discretisation (for numerical integration and differentiation)

Answer 1

One dimensional mathematical models of structural beams are constructed on the basis of beam theories because beams are actually three dimensional bodies,all models necessarily involve some form of approximation to the underlying physics.The simplest and best known models for straight,prismatic beams are based on the Bernoulli- Euler beam theory.The Timoshenko model incorporates a first order kinematic correction for transverse shear effect.This model assumed additional importance in dynamic and vibration.Under transverse loading one of the top surfaces shortens while the other elongates.We can clearly see it in the above picture.Thpicture.Therefore a neutral surface that undergoes no axial strain exists between the top and the bottom.The intersection of this surface with each cross section defines the neutral axis of that cross section.If the beam is homogeneous the neutral axis will pass through the centroid of the cross section.

The motion under loading of a plane beam member in the x,y plane is described by the two dimensional displacement field u(x,y) v(x,y) where u and v are the axial and transverse displacement components, respectively,of an arbitrary beam material point.The motion in the z direction,which is primarily due to poisson's ratio effects,is of no interest.The normality assumption of the Bernoulli- user model can be represented mathematically as

u(x,y)= -y dv(x)/dx = -yv' = -y?, v(x,y)=v(x)