A beam is said to be in a state of pure bending when it is subjected to only bending moment and the shear force is zero. Because the shear force, V = 0 and recalling the expression dM/dx = V = 0, this implies that pure bending refers to flexure of a beam under a constant bending moment. In contrast, nonuniform bending refers to flexure in the presence of shear forces, i.e. the bending moment changes along the axis of the beam.
In deriving expressions for the pure bending of beams, the following assumptions are made:
(1) The beam is prismatic and has an axial plane of symmetry, the x-y plane.
(2) All applied loads act in the plane of symmetry, i.e. the x-y plane.
(3) Bending occurs only in the x-y plane, i.e. the deflection curve lies in the x-y plane
(4) Material of the beam is homogeneous.
(5) Material obeys Hooke's Law with Young's modulus of elasticity in in tension and compression being the same.
(6) Plane section before bending remain plane after bending.
Curvature of Beams
The effect of the transverse load on a beam is to cause the beam to deflect. The deflected shape of the longitudinal axis of the beam is known as the deflection curve. If normals are drawn to the tangents to the deflection curve at two different cross-sections of the beam, these normals will intersect at a point known as the centre of curvature for the deflection curve. The distance from the centre of curvature to the deflection curve is called the radius of curvature, ?. The reciprocal of the radius of curvature is defined as the curvature, K, i.e.
Consider the cantilever beam subjected to a point load at the end. Consider two x-sections initially at a distance dx prior to the application of the load. After the application of the load, the distance ?? the curve between the normals to the tangents to the deflection curve at the two section is ds. The angle dt is the small angle between the normals to the deflection curve at the two sections. From geometry, we have that rdt = ds which is approximately equal to dx.
Also observe that if ? is the angle between the normals at the ends of the beam,
Sign Convention for Curvature
If the coordinate system is chosen such that the x-axis is positive to the right and the y-axis is positive upwards, then the curvature is positive if the deflection curve is concave upwards. The curvature is negative if the deflection curve is concave downwards. Note that positive curvature is produced by positive bending moments and negative curvature by negative bending moments. However, observe that positive bending moments produce negative deflections while negative bending moments produce positive deflections.
Longitudinal Strains in Beams
Because the bending moment is constant (uniform) in a beam subjected to pure bending, the bending deformation will also be uniform and the deflection curve will be in the form of a circular arc. Consider the beam element shown below which is bent into a circular arc by the couples M0.
As a result of the bending deformation, fibres on the convex side are elongated (i.e. in tension) while those on the concave side are shortened slightly (i.e. in compression). Somewhere between the top and bottom of the beam, there is a layer of fibres which remain unchanged in length. This surface is called the neutral surface of the beam. The intersection of the neutral surface with the longitudinal/axial plane of symmetry (i.e. the x-y plane) is called the neutral axis of the beam. Its intersection with the plane of any x-section is called the neutral axis of that cross-section.
To obtain the normal strain, consider a longitudinal fiber ef that is located at a distance y from the neytral surface. Prior to the bending of the beam, ef = gh. After bending, however, ef shortens to ef'. Because gh lies on the neutral surface, it's length is unchanged. Hence,
Original length of ef = gh= rdq
Final length of ef = ef'= (r-y)dq
Change in length = final length - original length = ef' -ef
= (r-y)dq - rdq = -ydq
Normal Strain
This shows that the normal strains in the beam are proportional to the curvature k and vary linearly with the distance y from the neutral surface. Note that the expression above applies to beams with a positive curvature. For such beams, all fibres below the neutral surface have a positive y value and hence will have compressive strains. However, fibres above the neutral surface will have a negative y value and hence it will have normal tensile strains.