What is "neutral axis"

Neutral \Neu"tral\, a. [L. neutralis, fr. neuter. See Neuter.]

1. Not engaged on either side; not taking part with or assisting either of two or more contending parties; neuter; indifferent.

   The heart can not possibly remain neutral, but constantly takes part one way or the other.
   --Shaftesbury.

2. Neither good nor bad; of medium quality; middling; not decided or pronounced.

   Some things good, and some things ill, do seem, And neutral some, in her fantastic eye.
   --Sir J. Davies.

3. (Biol.) Neuter. See Neuter, a., 3.

4. (Chem.) Having neither acid nor basic properties; unable to turn red litmus blue or blue litmus red; -- said of certain salts or other compounds. Contrasted with acid, and alkaline.

   Neutral axis, Neutral surface (Mech.), that line or plane, in a beam under transverse pressure, at which the fibers are neither stretched nor compressed, or where the longitudinal stress is zero. See Axis.

   Neutral equilibrium (Mech.), the kind of equilibrium of a body so placed that when moved slighty it neither tends to return to its former position not depart more widely from it, as a perfect sphere or cylinder on a horizontal plane.

   Neutral salt (Chem.), a salt formed by the complete replacement of the hydrogen in an acid or base; in the former case by a positive or basic, in the latter by a negative or acid, element or radical.

   Neutral tint, a bluish gray pigment, used in water colors, made by mixing indigo or other blue some warm color. the shades vary greatly.

   Neutral vowel, the vowel element having an obscure and indefinite quality, such as is commonly taken by the vowel in many unaccented syllables. It is regarded by some as identical with the [u^] in up, and is called also the natural vowel, as unformed by art and effort; it is also called the indefinite vowel. It is symbolized in some phonetic alphabets by the schwa ([schwa]). See Guide to Pronunciation, [sect] 17.

Axis \Ax"is\, n.; pl. Axes. [L. axis axis, axle. See Axle.] A straight line, real or imaginary, passing through a body, on which it revolves, or may be supposed to revolve; a line passing through a body or system around which the parts are symmetrically arranged. 2. (Math.) A straight line with respect to which the different parts of a magnitude are symmetrically arranged; as, the axis of a cylinder, i. e., the axis of a cone, that is, the straight line joining the vertex and the center of the base; the axis of a circle, any straight line passing through the center. 3. (Bot.) The stem; the central part, or longitudinal support, on which organs or parts are arranged; the central line of any body. --Gray. 4. (Anat.)

1. The second vertebra of the neck, or vertebra dentata.

2. Also used of the body only of the vertebra, which is prolonged anteriorly within the foramen of the first vertebra or atlas, so as to form the odontoid process or peg which serves as a pivot for the atlas and head to turn upon.

   5. (Crystallog.) One of several imaginary lines, assumed in describing the position of the planes by which a crystal is bounded.

   6. (Fine Arts) The primary or secondary central line of any design.

   Anticlinal axis (Geol.), a line or ridge from which the strata slope downward on the two opposite sides.

   Synclinal axis, a line from which the strata slope upward in opposite directions, so as to form a valley.

   Axis cylinder (Anat.), the neuraxis or essential, central substance of a nerve fiber; -- called also axis band, axial fiber, and cylinder axis.

   Axis in peritrochio, the wheel and axle, one of the mechanical powers.

   Axis of a curve (Geom.), a straight line which bisects a system of parallel chords of a curve; called a principal axis, when cutting them at right angles, in which case it divides the curve into two symmetrical portions, as in the parabola, which has one such axis, the ellipse, which has two, or the circle, which has an infinite number. The two axes of the ellipse are the major axis and the minor axis, and the two axes of the hyperbola are the transverse axis and the conjugate axis.

   Axis of a lens, the straight line passing through its center and perpendicular to its surfaces.

   Axis of a microscope or Axis of a telescope, the straight line with which coincide the axes of the several lenses which compose it.

   Axes of co["o]rdinates in a plane, two straight lines intersecting each other, to which points are referred for the purpose of determining their relative position: they are either rectangular or oblique.

   Axes of co["o]rdinates in space, the three straight lines in which the co["o]rdinate planes intersect each other.

   Axis of a balance, that line about which it turns.

   Axis of oscillation, of a pendulum, a right line passing through the center about which it vibrates, and perpendicular to the plane of vibration.

   Axis of polarization, the central line around which the prismatic rings or curves are arranged.
   --Brewster.

   Axis of revolution (Descriptive Geom.), a straight line about which some line or plane is revolved, so that the several points of the line or plane shall describe circles with their centers in the fixed line, and their planes perpendicular to it, the line describing a surface of revolution, and the plane a solid of revolution.

   Axis of symmetry (Geom.), any line in a plane figure which divides the figure into two such parts that one part, when folded over along the axis, shall coincide with the other part.

   Axis of the equator, ecliptic, horizon (or other circle considered with reference to the sphere on which it lies), the diameter of the sphere which is perpendicular to the plane of the circle.
   --Hutton.

   Axis of the Ionic capital (Arch.), a line passing perpendicularly through the middle of the eye of the volute.

   Neutral axis (Mech.), the line of demarcation between the horizontal elastic forces of tension and compression, exerted by the fibers in any cross section of a girder.

   Optic axis of a crystal, the direction in which a ray of transmitted light suffers no double refraction. All crystals, not of the isometric system, are either uniaxial or biaxial.

   Optic axis, Visual axis (Opt.), the straight line passing through the center of the pupil, and perpendicular to the surface of the eye.

   Radical axis of two circles (Geom.), the straight line perpendicular to the line joining their centers and such that the tangents from any point of it to the two circles shall be equal to each other.

   Spiral axis (Arch.), the axis of a twisted column drawn spirally in order to trace the circumvolutions without.

   Axis of abscissas and Axis of ordinates. See Abscissa.

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. 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:

γ = γ = τ = τ = 0

Where γ is the shear strain and τ is the shear stress

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, 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.

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

Since the bending is uniform and pure, there is therefore at a distance y from the neutral axis with the inherent property of having no strain:

$\epsilon_x(y)=\frac{L(y)-L}{L} = \frac{\theta\,(\rho\, - y) - \theta \rho \,}{\theta \rho \,} = \frac{-y\theta}{\rho \theta} = \frac{-y}{\rho}$

Therefore the longitudinal normal strain ε varies linearly with the distance y from the neutral surface. Denoting ε as the maximum strain in the beam (at a distance c from the neutral axis), it becomes clear that:

$\epsilon_m = \frac{c}{\rho}$

Therefore, we can solve for ρ, and find that:

$\rho = \frac{c}{\epsilon_m}$

Substituting this back into the original expression, we find that:

$\epsilon_x(y) = \frac {-\epsilon_my}{c}$

Due to Hooke's Law, the stress in the beam is proportional to the strain by E, the modulus of Elasticity:

σ = Eε 

Therefore:

$E\epsilon_x(y) = \frac {-E\epsilon_my}{c}$

$\sigma_x(y) = \frac {-\sigma_my}{c}$

From statics, 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.

∫σdA = 0

Therefore:

$\int \frac {-\sigma_my}{c} dA = 0$

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:

∫ydA = 0

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);

τ = (T * Q) ÷ (w * I)

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.