Simple bending or pure bending
Consider a simply supported beam AB of length L subjected to moment M0 at its ends as shown in figures.
Consider an element of length dx at distance x from hinged support A. Due to applied moment, the beam will sag in a manner.
It can be seen in figure the element dx will be converted into an curve. Let the normal tangents at the end point of element intersect a point O . This point O is known as centre of curvature. The distance Ox1 and Ox2 is known as radius of curvature. Let the angle made at centre of curvature by both the normal is denoted by dθ.
Curvature, C is defined as reciprocal of radius of curvature R i.e. C =1/R
As from figure ds = Rdθ
R = ds/dθ
Here, ds is length of arch x1x2
As, curvature is reciprocal of radius of curvature, so, from eq. (i)
Curvature, C = dθ/ds
For small deflection, as is usually case with beam, ds = dx
Hence, curvature,
C = dθ/dx
These equations are used to calculate normal or bending stress and strains in beam. Bending moment diagram for beam.
As clear from bending moment diagram, the beam is free from shear force. This type of bending in which there is no shear force is called as simple bending or pure bending. Bending in presence of shear force is known as non-uniform bending.
In pure bending, radius of curvature is constant at different sections of beam. For example, as the beam in figure is the case of pure bending, radius of curvature is R at section x3 also. As circular of curvature is constant, deflection curve of beam is circular.
Another case of pure bending can be a simply supported beam of length L subjected to concentrated loads as shown in figure.
Its shear force diagram and bending moment diagram is shown in figure above.
As from shear force diagram, it is cleared that there is no shear force in region CD of beam. So in region CD, pure bending will occur due to which beam will be converted into an arc of circle.
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