Elastic Beam Theory

TTo derive the deflection, we look at initially straight beam that is elastically deformed by loads applied perpendicular to beam x-axis and lying in the x-v plane of symmetry.

·       Due to the loading, the beam deforms under shear and bending.

·       If beam L>>d, greatest deformation will be caused by bending.

·       When Moment deforms, the angle between the cross-sections becomes dθ.

·       The arc dx that represents a portion of the elastic curve intersects the neutral axis.

·       The radius of curvature for this arc is defined as the distance ρ, which is measured from center of curvature O to dx.

·       Any arc on the element other than dx is subjected to the normal strain.

·       The strain in arc ds located at position y from the neutral axis is

·       If the material is homogenous and behaves in a linear manner, then Hooks law applies

·       The flexural formulae also applies

Combining these equations, we have

Here, ρ is the radius of the curvature at a specific point on the elastic curve, M is the internal moment in the beam at the point where ρ is to be determined, E is the modulus of the elasticity, I is the beam moment of inertia computed about the neutral axis.

EI is the flexural rigidity;

·       This equation represents a non linear second differential element.

·       V=f(x) gives the exact shape of the elastic curve.

·       The slope of the elastic curve for most structures is very small.

·       Using small deflection theory, we assume dv/dx~~0.