# Neutral axis

Neutral axis
Beam with neutral axis (x).

The neutral axis is an axis in the cross section of a beam 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:

γxy = γzx = τxy = τxz = 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 $\epsilon_x$ varies linearly with the distance y from the neutral surface. Denoting $\epsilon_m$ 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:

$\sigma_x = E\epsilon_x\,$

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 stress across the cross section must be 0.

$\int \sigma_x dA = 0$

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:

$\int y dA = 0$

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);

$\tau = (T * Q) \div (w * I)$

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.

## Arches

Arches 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.

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### Look at other dictionaries:

• 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. [1913 Webster] The heart can not possibly remain… …   The Collaborative International Dictionary of English

• Neutral axis — 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… …   The Collaborative International Dictionary of English

• neutral axis — neutralioji ašis statusas T sritis automatika atitikmenys: angl. neutral axis vok. neutrale Achse, f rus. нейтральная ось, f pranc. axe neutre, m …   Automatikos terminų žodynas

• neutral axis — neutralioji ašis statusas T sritis fizika atitikmenys: angl. neutral axis vok. neutrale Achse, f rus. нейтральная ось, f pranc. axe neutre, m …   Fizikos terminų žodynas

• neutral axis — an imaginary line in the cross section of a beam, shaft, or the like, along which no stresses occur. [1835 45] * * * …   Universalium

• neutral axis — Смотри нейтральная ось в сопротивлении материалов …   Энциклопедический словарь по металлургии

• neutral axis — noun : the line in a beam or other member subjected to a bending action in which the fibers are neither stretched nor compressed or where the longitudinal stress is zero …   Useful english dictionary

• 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. [1913 Webster] The heart can not possibly remain neutral, but… …   The Collaborative International Dictionary of English

• Neutral equilibrium — 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. [1913 Webster] The heart can not possibly remain… …   The Collaborative International Dictionary of English

• Neutral salt — 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. [1913 Webster] The heart can not possibly remain… …   The Collaborative International Dictionary of English