Pure bending ( Theory of simple bending) is a condition of stress where a
bending moment
In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mo ...
is applied to a beam without the simultaneous presence of
axial,
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
, or
torsional forces.
Pure bending occurs only under a constant bending moment (M) since the shear force (V), which is equal to
, has to be equal to zero. In reality, a state of pure bending does not practically exist, because such a state needs an absolutely weightless member. The state of pure bending is an approximation made to derive formulas.
Kinematics of pure bending
#In pure bending the axial lines bend to form
circumferential lines and
transverse lines remain straight and become
radial line
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
s.
#Axial lines that do not extend or contract form a neutral surface.
Assumptions made in the theory of Pure Bending
#The material of the beam is homogeneous
1 and isotropic
2.
#The value of Young's
Modulus of Elasticity
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
is same in tension and compression.
#The
transverse sections which were plane before bending, remain plane after bending also.
#The beam is initially straight and all longitudinal filaments bend into circular arcs with a common centre of curvature.
#The radius of curvature is large as compared to the dimensions of the cross-section.
#Each layer of the beam is free to expand or contract, independently of the layer, above or below it.
Notes:
1 Homogeneous means the material is of same kind throughout.
2 Isotropic means that the elastic properties in all directions are equal.
References
*E P Popov; Sammurthy Nagarajan; Z A Lu.
Mechanics of Material. Englewood Cliffs, N.J. : Prentice-Hall, ©1976, p. 119, "Pure Bending of Beams", {{ISBN, 978-0-13-571356-3
Force
Solid mechanics
Structural system