In
solid mechanics
Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ...
, pure bending (also known as the theory of simple bending) is a condition of
stress where a
bending moment
In solid mechanics, a bending moment is the Reaction (physics), reaction induced in a structural element when an external force or Moment of force, moment is applied to the element, causing the element to bending, bend. The most common or simplest ...
is applied to a
beam
Beam may refer to:
Streams of particles or energy
*Light beam, or beam of light, a directional projection of light energy
**Laser beam
*Radio beam
*Particle beam, a stream of charged or neutral particles
**Charged particle beam, a spatially lo ...
without the simultaneous presence of
axial,
shear, or
torsional forces.
Pure bending occurs only under a constant bending moment () since the shear force (), 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 around a main axis (a chosen directed line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpend ...
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 (MOE)) is a quantity that describes an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it.
Definition
The elastic modu ...
is same in tension and compression.
#The
transverse sections
Transverse may refer to:
*Transverse engine, an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle
*Transverse flute, a flute that is held horizontally
* Transverse force (or ''Euler force''), the tangen ...
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