The neutral axis is an axis in the cross section of a
beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric
centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
. All fibers on one side of the neutral axis are in a state of
tension, while those on the opposite side are in
compression.
Since the beam is undergoing uniform bending, a plane on the beam remains plane. That is:
Where
is the
shear strain and
is the
shear stress
Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
There is a compressive (negative) strain at the top of the beam, and a tensile (positive) strain at the bottom of the beam. Therefore by the
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two impor ...
, there must be some point in between the top and the bottom that has no strain, since the strain in a beam is a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
.
Let L be the original length of the beam (
span)
ε(y) is the strain as a function of coordinate on the face of the beam.
σ(y) is the stress as a function of coordinate on the face of the beam.
ρ is the
radius of curvature of the beam at its neutral axis.
θ is the
bend angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
Since the bending is
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
and pure, there is therefore at a distance y from the neutral axis with the inherent property of having no strain:
Therefore the longitudinal normal strain
varies linearly with the distance y from the neutral surface. Denoting
as the maximum strain in the beam (at a distance c from the neutral axis), it becomes clear that:
Therefore, we can solve for ρ, and find that:
Substituting this back into the original expression, we find that:
Due to
Hooke's Law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of t ...
, the stress in the beam is proportional to the strain by E, the
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 ...
:
Therefore:
From
statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ...
, a moment (i.e.
pure bending) consists of equal and opposite forces. Therefore, the total amount of force across the cross section must be 0.
Therefore:
Since y denotes the distance from the neutral axis to any point on the face, it is the only variable that changes with respect to dA. Therefore:
Therefore the
first moment of the cross section about its neutral axis must be zero. Therefore the neutral axis lies on the centroid of the cross section.
Note that the neutral axis does not change in length when under bending. It may seem counterintuitive at first, but this is because there are no bending stresses in the neutral axis. However, there are shear stresses (τ) in the neutral axis, zero in the middle of the span but increasing towards the supports, as can be seen in this function (Jourawski's formula);
:
where
T = shear force
Q =
first moment of area of the section above/below the neutral axis
w = width of the beam
I =
second moment of area of the beam
This definition is suitable for the so-called long beams, i.e. its length is much larger than the other two dimensions.
Arches
Arch
An arch is a vertical curved structure that spans an elevated space and may or may not support the weight above it, or in case of a horizontal arch like an arch dam, the hydrostatic pressure against it.
Arches may be synonymous with vau ...
es also have a neutral axis if they are made of stone; stone is an inelastic medium, and has little strength in tension. Therefore as the loading on the arch changes the neutral axis moves- if the neutral axis leaves the stonework, then the arch will fail.
This theory (also known as the
thrust line method) was proposed by
Thomas Young and developed by
Isambard Kingdom Brunel
Isambard Kingdom Brunel (; 9 April 1806 – 15 September 1859) was a British civil engineer who is considered "one of the most ingenious and prolific figures in engineering history," "one of the 19th-century engineering giants," and "on ...
.
See also
*
Neutral plane
*
Second moment of inertia
Boilermaking
Solid mechanics