Elastic Beam Theory

· To 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.

Elastic Beam Theory

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Elastic Beam Theory

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