Fact-checked by Grok 2 weeks ago

Neutral axis

In beam theory within the mechanics of materials, the neutral axis is defined as the longitudinal plane or line in a beam's cross-section where the normal stress and strain due to bending are zero, serving as the boundary between regions of tensile stress below it and compressive stress above it during pure bending.^[1]^[2]^[3] For homogeneous beams subjected to elastic bending, the neutral axis passes through the centroid of the cross-section, ensuring symmetry in the deformation where fibers along this axis neither elongate nor shorten.^[1]^[4] In contrast, for composite or non-homogeneous beams—such as those made of reinforced concrete or wood with varying material properties—the neutral axis shifts to the centroid of the transformed section, accounting for differences in modulus of elasticity between materials to maintain equilibrium.^[5]^[2] The position and properties of the neutral axis are fundamental to calculating bending stresses using the flexure formula \sigma = \frac{M y}{I}, where \sigma is the normal stress, M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia about the neutral axis; this enables engineers to predict failure modes and design safe structures like bridges and buildings.^[1]^[6] Additionally, in shear loading, the maximum transverse shear stress often occurs at the neutral axis for rectangular sections, influencing overall beam strength analysis.^[7]

Fundamentals

Definition

In structural engineering, the neutral axis is defined as the line, axis, or plane within the cross-section of a beam or structural element that experiences zero normal stress during bending deformation.^[1] This locus separates regions of tensile and compressive stresses, where longitudinal fibers above the neutral axis undergo compression and those below experience tension, or vice versa depending on the bending direction.^[2] Bending deformation in a beam arises from applied moments that cause the longitudinal fibers to either elongate or shorten relative to their original length, resulting in a linear distribution of normal strain and stress symmetric about the neutral axis.^[8] Fibers farther from the neutral axis exhibit greater strain magnitudes, with the distribution assuming a planar cross-section remains plane after deformation, as per the fundamental assumptions of beam theory.^[1] Geometrically, in the cross-section of a beam under pure bending, the neutral axis is oriented perpendicular to the plane of bending and passes through the centroid of the section for homogeneous materials.^[9] In elastic theory for isotropic materials, this neutral axis precisely coincides with the centroidal axis, ensuring balanced stress distribution without net axial force.^[10]

Historical Development

The concept of the neutral axis emerged from early efforts to explain beam deformation and failure. In 1638, Galileo Galilei, in his seminal work Dialogues Concerning Two New Sciences, analyzed the breaking of cantilever beams under transverse loading, modeling failure as rotation about a fulcrum near the fixed support; however, his approach assumed uniform tensile stress across the section without identifying a distinct locus of zero longitudinal stress.^[11] Building on this, Edme Mariotte in 1686 proposed a triangular stress distribution with a central neutral axis, and Antoine Parent in 1713 correctly derived the elastic section modulus assuming a central neutral axis and linear stress distribution.^[11] Further advancements came with the Euler-Bernoulli beam theory, formulated around 1750 through collaborations between Leonhard Euler and Daniel Bernoulli, which implicitly incorporated the neutral axis via assumptions of linear strain variation through the beam depth and plane sections remaining plane after deformation, enabling predictions of curvature and deflection in elastic beams.^[12] In 1826, Claude-Louis Navier, in his Résumé des Leçons, explicitly positioned the neutral axis at the centroid of homogeneous cross-sections under pure bending, linking it to the beam's moment of inertia.^[13] Refinements continued in the mid-19th century with Adhémar Jean Claude Barré de Saint-Venant's 1856 publication in Liouville's Journal de Mathématiques Pures et Appliquées, providing the exact solution for bending stresses in prismatic bars and confirming the neutral axis as the plane where longitudinal stresses vanish, passing through the centroid for symmetric sections.^[14] This work resolved inconsistencies in earlier models by integrating equilibrium and compatibility conditions, marking a pivotal advancement in solid mechanics. Concurrently, 19th-century developments in elasticity theory by Navier and others refined the framework. Twentieth-century extensions addressed nonlinear behaviors, particularly in plastic regimes. Ludwig Prandtl and Erich Reuss advanced the theory in the 1920s–1930s with the Prandtl-Reuss equations, describing incremental plastic flow in metals and enabling analysis of the shifting plastic neutral axis, where compressive and tensile plastic zones balance during elasto-plastic bending.^[15] These contributions, rooted in associated flow rules and yield criteria, expanded the neutral axis concept beyond elastic limits, influencing modern structural design for ductile materials. By the early 1900s, the term "neutral axis" had become standard in engineering literature, often illustrated via Mohr's circle to visualize stress transformations across beam sections.^[16]

Theoretical Framework

Derivation in Beam Bending

The derivation of the neutral axis in beam bending relies on the Euler-Bernoulli beam theory, which makes several key assumptions to simplify the analysis of slender beams under transverse loading. These include: (1) plane cross-sections perpendicular to the beam's neutral axis remain plane and perpendicular to the deformed axis after bending; (2) deformations are small, allowing linear approximations; and (3) the material is linear elastic, isotropic, and homogeneous, following Hooke's law with negligible transverse shear strains and stresses.^[17]^[18]^[19] Consider a beam segment subjected to pure bending, where the neutral axis deforms into a curve with radius of curvature \rho. The curvature is defined as \kappa = 1/\rho. Under the plane sections assumption, the longitudinal strain \varepsilon_x varies linearly with the distance y from the neutral axis, given by \varepsilon_x = -y / \rho = -y \kappa, where the negative sign indicates compression above and tension below the neutral axis for positive curvature. This linear strain distribution arises because fibers at distance y elongate or shorten proportionally to their offset from the neutral surface.^[18]^[19] Assuming Hooke's law, the normal stress \sigma_x is \sigma_x = E \varepsilon_x = -E y / \rho, where E is the Young's modulus. The neutral axis, defined as the locus where \sigma_x = 0, naturally emerges at y = 0. To ensure equilibrium under pure bending (no net axial force), the position of the neutral axis is determined by the condition \int_A \sigma_x \, dA = 0, which implies \int_A y \, dA = 0; thus, it passes through the centroid of the cross-section.^[17]^[18] The internal bending moment M about the neutral axis is obtained by integrating the stress distribution:

M = \int_A \sigma_x y \, dA = \int_A \left( - \frac{E y}{\rho} \right) y \, dA = -\frac{E}{\rho} \int_A y^2 \, dA.

Here, \int_A y^2 \, dA = I is the second moment of area (moment of inertia) of the cross-section about the neutral axis. This yields the moment-curvature relation M = -E I / \rho, or equivalently, \sigma_x = -M y / I. For small deformations, \kappa \approx d^2 v / dx^2, where v(x) is the transverse deflection, leading to M = E I \, d^2 v / dx^2. These equations form the foundation for predicting stress and deflection in beams.^[17]^[18]

Location and Properties

In homogeneous beams subjected to pure bending, the neutral axis passes through the geometric centroid of the cross-section, ensuring zero net axial force across the section.^[4] The position of this centroid, denoted as \bar{y}, is calculated using the first moment of area formula:

\bar{y} = \frac{\int y \, dA}{\int dA} = \frac{\int y \, dA}{A},

where y is the distance from a reference axis, dA is an elemental area, and A is the total cross-sectional area.^[20] For a rectangular cross-section of height h, the neutral axis is located at mid-height, \bar{y} = h/2, simplifying stress calculations in symmetric loading.^[20] In non-homogeneous beams, where the modulus of elasticity E varies across the cross-section (e.g., in composite materials), the neutral axis shifts from the geometric centroid toward the region of higher stiffness to balance the resultant force.^[20] The location y_n is determined by the weighted centroid:

y_n = \frac{\int E y \, dA}{\int E \, dA},

reflecting the influence of material properties on strain compatibility.^[20] This approach, often implemented via the transformed section method, adjusts areas by the modular ratio n = E_i / E_j for discrete materials. Key properties of the neutral axis include its stationarity under pure bending, where it remains fixed relative to the cross-section as curvature develops uniformly.^[21] However, under combined bending and axial loading, the neutral axis shifts parallel to itself, altering the stress distribution.^[21] Additionally, the neutral axis is always perpendicular to the plane of bending, aligning with the direction of zero longitudinal strain.^[4] In standard I-beams with symmetric flanges, the neutral axis aligns with the centroid of the web, typically at the mid-height of the section for balanced geometry.^[6]

Applications in Structural Elements

Straight Beams

In straight beams subjected to bending, the neutral axis serves as the reference plane where normal stress is zero, dividing the cross-section into compressive and tensile regions above and below it, respectively. This concept is central to Euler-Bernoulli beam theory, which assumes plane sections remain plane and perpendicular to the neutral axis after deformation.^[1]^[22] The bending stress distribution is linear across the cross-section, varying directly with the distance from the neutral axis. The normal stress \sigma at any point is given by \sigma = \frac{My}{I}, where M is the bending moment, y is the perpendicular distance from the neutral axis, and I is the second moment of area about the neutral axis. Consequently, the maximum stress \sigma_{\max} occurs at the extreme fibers, calculated as \sigma_{\max} = \frac{Mc}{I}, with c denoting the distance from the neutral axis to the outermost fiber. This formula enables engineers to predict failure risks by ensuring \sigma_{\max} remains below the material's yield strength.^[23]^[24]^[1] Shear stress in straight beams interacts with the neutral axis, influencing the overall stress state. The transverse shear stress \tau follows a parabolic distribution through the depth, derived from equilibrium considerations as \tau = \frac{VQ}{It}, where V is the shear force, Q is the first moment of area about the neutral axis for the portion above the point of interest, and t is the width at that location. In rectangular sections, \tau is maximum at the neutral axis and zero at the top and bottom surfaces, highlighting the neutral axis as a critical zone for shear failure in deep beams.^[7]^[25]^[26] In beam design, the neutral axis informs sizing to control stresses within allowable limits. The elastic section modulus Z = \frac{I}{c} simplifies maximum stress computation to \sigma_{\max} = \frac{M}{Z}, allowing rapid assessment of required cross-sectional dimensions for a given moment capacity. Designers select beam profiles (e.g., I-beams) to optimize Z relative to weight, ensuring the neutral axis aligns with the centroid for symmetric sections to minimize eccentricity effects.^[27]^[8] A representative example is a cantilever beam of length L with a concentrated end load P, fixed at one end. The bending moment diagram is triangular, peaking at M_{\max} = [PL](/page/PL) at the fixed support, where the neutral axis experiences zero normal stress but the surrounding fibers reach \sigma_{\max} = \frac{[PL](/page/PL)c}{I}. This illustrates how the neutral axis traces the locus of zero strain along the beam length, guiding stress analysis for support design and deflection control.^[28]^[24]

Curved Beams and Arches

In curved beams, the presence of initial curvature causes the neutral axis to deviate from the centroidal axis, shifting toward the center of curvature due to the hyperbolic distribution of normal stresses. This shift arises because the strain varies nonlinearly with radial distance, unlike the linear variation in straight beams where the neutral axis passes through the centroid. The Winkler-Bach theory provides the foundational approach for analyzing this behavior in curved beams subjected to bending in the plane of curvature, assuming plane sections remain plane and radial stresses are negligible.^[29]^[30] The radius to the neutral axis, R_n, is determined from the condition that the integral of normal stress over the cross-section is zero, yielding

R_n = \frac{A}{\int_A \frac{dA}{r}},

where A is the cross-sectional area and r is the radial distance from the center of curvature to a differential area element dA. This formula shows that R_n is generally less than the centroidal radius R = \frac{1}{A} \int_A r \, dA, confirming the inward shift. The eccentricity e is defined as e = R - R_n, representing the distance between the centroidal and neutral axes.^[29]^[31] The resulting normal stress \sigma in a curved beam under pure bending moment M (neglecting axial force) is given by

\sigma = \frac{M (r - R_n)}{A e r},

which highlights the nonlinear stress variation: stresses are higher on the inner fibers (smaller r) and lower on the outer fibers compared to straight beam theory. This formula, derived from equilibrium and compatibility, is widely used for design in applications like crane hooks and ring segments.^[31]^[30] In arches, the neutral axis position plays a critical role in determining the internal force distribution, particularly through its influence on the thrust line—the locus of points where the resultant compressive force acts. When the thrust line aligns closely with the neutral axis, bending moments are minimized, leading to primarily axial compression; deviations introduce significant bending moments that must be resisted by the structure. For circular arches under uniform vertical loading, the neutral axis lies inside the centroidal axis throughout the span, resulting in elevated stresses on the inner (intrados) fibers due to the combined effects of curvature and load eccentricity. This inward shift exacerbates inner fiber compression, often requiring thicker sections or reinforcement at the intrados to prevent failure.^[32]^[30]

Advanced Concepts

Composite and Reinforced Sections

In composite sections, such as those consisting of concrete and steel reinforcement, the neutral axis location accounts for the differing moduli of elasticity of the materials involved. The transformed section method converts the dissimilar materials into an equivalent section of a single material, typically concrete, to simplify analysis under elastic bending assumptions. This approach multiplies the area of the stiffer material (e.g., steel) by the modular ratio n = E_s / E_c, where E_s is the modulus of elasticity of steel and E_c is that of concrete, effectively widening the steel area in the transformed diagram while preserving the geometry of the concrete.^[33] The neutral axis then coincides with the centroid of this transformed section, enabling standard centroidal calculations for moment of inertia and stress distribution.^[33] The precise location of the neutral axis y_n, measured from a reference point, satisfies the equilibrium condition derived from zero net axial force: \int E y \, dA = 0, where the integration is over the entire cross-section and E varies by material.^[34] For reinforced concrete beams, this often requires an iterative solution, particularly in under-reinforced sections where tensile concrete is neglected below the neutral axis (cracked section assumption), transforming only the compression concrete and the n-multiplied steel area.^[33] The modular ratio n typically ranges from 6 to 10 depending on concrete strength, with E_s \approx 200 GPa and E_c varying from 20 to 40 GPa.^[33] Strain compatibility in these sections assumes a linear strain distribution across the depth, consistent with the plane sections remaining plane hypothesis in beam theory.^[34] Consequently, strains \epsilon = -y / \rho (with \rho as the radius of curvature) are continuous, but stresses \sigma = E \epsilon become discontinuous at material interfaces due to the modulus difference, with steel experiencing higher stresses than surrounding concrete at the same strain level.^[34] A representative example is the T-beam in reinforced concrete, where the wide flange aids compression resistance. The neutral axis position depends on the reinforcement ratio \rho = A_s / (b_w d), with b_w as web width and d as effective depth. If \rho is low, the neutral axis lies within the flange, treating the section as rectangular with flange width b; for higher \rho, it shifts into the web, requiring separate centroid calculation for the transformed flange (concrete area) and web (concrete plus n A_s).^[35]

Plastic Neutral Axis

In plastic analysis of beams, the neutral axis plays a crucial role during the elastic-plastic transition and in the fully plastic state. Initially, under elastic loading, the neutral axis is located at the centroid of the cross-section. As the applied moment increases and outer fibers begin to yield, the neutral axis shifts to accommodate the nonlinear stress distribution, effectively reducing the contributing elastic core of the section while maintaining equilibrium of internal forces.^[36] In the fully plastic state, the plastic neutral axis is positioned such that it divides the cross-section into two equal areas—one in tension and one in compression—each stressed to the yield strength \sigma_y. For symmetric sections, this location coincides with the geometric centroid. In a rectangular section of depth d, the plastic neutral axis lies at mid-depth, ensuring equal areas above and below. The plastic moment capacity M_p is then given by M_p = \sigma_y Z_p, where Z_p is the plastic section modulus; for a rectangular section, Z_p = \frac{A}{2} \times \frac{d}{2}, with A as the total cross-sectional area.^[36] For I-sections, the plastic neutral axis is determined to equalize the tensile and compressive areas, maximizing the moment capacity by fully utilizing the material up to \sigma_y. This positioning often places the axis within the web, depending on the flange and web proportions. The shape factor f = \frac{Z_p}{Z_e} > 1, where Z_e is the elastic section modulus, quantifies the reserve strength beyond yielding; for typical wide-flange I-beams, f \approx 1.1 to $1.2, indicating a modest increase in capacity compared to elastic limits.^[36]^[37]

References

[1]
Mechanics of Materials: Bending – Normal Stress - Boston University
Just like torsion, in pure bending there is an axis within the material where the stress and strain are zero. This is referred to as the neutral axis. And, just ...
[2]
[PDF] Revisiting the neutral axis in wood beams - Oregon State University
Nov 25, 2011 · Introduction. The neutral axis (NA) for beams is defined as a longitudinal plane in the cross section that does not experience any stress and ...
[3]
[PDF] STRESSES IN BEAMS
Nov 21, 2000 · At the transition between the compressive and tensile regions, the stress becomes zero; this is the neutral axis of the beam.
[4]
Section II.2
Neutral Axis: When a homogeneous beam is subjected to elastic bending, the neutral axis (NA) will pass through the centroid of its cross section, but the ...
[5]
Lecture Notes - MST.edu
The neutral axis of bending is at the centroid of the transformed section and flexure stresses are calculated with the flexure stress formula. One final step is ...
[6]
[PDF] 10. Beams: Flexural and shear stresses x
neutral axis of the beam in its undeformed state. The intersection of the plane of bending and neutral surface is known as the beam axis. The deformation of ...
[7]
[PDF] Shear Stresses in Beams
The maximum shear stress occurs at the neutral axis and is zero at both the top and bottom surface of the beam. For a rectangular cross section, the maximum ...<|control11|><|separator|>
[8]
[PDF] 3. beams: strain, stress, deflections
The intersection of the neutral surface with the any cross section is the neutral axis of the section. Strain. Although strain is not usually required for ...Missing: structural | Show results with:structural
[9]
[PDF] ME 354, MECHANICS OF MATERIALS LABORATORY STRAINS ...
The location in the beam where the stress is zero is called the neutral axis (N/A) coincides with the centroid of the beam's cross section in straight beams.
[10]
[PDF] Wood Material Science and Engineering Location of the neutral axis ...
Nov 20, 2010 · It is commonly accepted in the analysis of wood beams that the neutral axis coincides with the centroid of the beam. Although this assumption is ...
[11]
The history of the theory of beam bending – Part 1
Feb 27, 2008 · The theory of the flexural strength and stiffness of beams is now attributed to Bernoulli and Euler, but developed over almost 400 years.Missing: Saint- Venant
[12]
[PDF] Euler-Bernoulli Beams - Professor Terje Haukaas
Aug 21, 2023 · The Euler-Bernoulli beam theory was established around 1750 with contributions from. Leonard Euler and Daniel Bernoulli.
[13]
https://www.cambridge.org/core/books/elements-of-the-theory-of-structures/simple-plastic-bending-of-beams/AE8BE9906609B368AA5650DBE4FCD507
[14]
The simple plastic bending of beams (Chapter 12)
... Navier stated explicitly that as a consequence the neutral axis must pass through the centre of gravity of the cross-section. However, even Navier was not ...
[15]
“Historical Introduction” from A Treatise on the Mathematical Theory ...
Jan 1, 2007 · The first problem to which Saint-Venant applied his method was that of the torsion of prisms, the theory of which he gave in the famous memoir ...
[16]
(PDF) A Short Introduction to the Theory of Plasticity - ResearchGate
Apr 22, 2020 · PDF | A short introduction to the theory of plasticity for students. | Find, read and cite all the research you need on ResearchGate.
[17]
Historical development of the beam bending equation M equals fS
Jul 18, 2011 · Mariotte applied Galileo's model of a failure hinge to the problem of a cantilever beam but modified the mathematics to place the neutral axis ...Missing: Cauchy | Show results with:Cauchy
[18]
Euler Bernoulli Beam - Engineering at Alberta Courses
There are three basic assumptions for an Euler Bernoulli beam that will be used to derive the equations. These are: Plane sections perpendicular to the neutral ...
[19]
[PDF] 7.4 The Elementary Beam Theory
The intersection of the longitudinal plane of symmetry and the neutral surface is called the axis of the beam, and the deformed axis is called the deflection ...Missing: Cauchy history
[20]
1.5: Euler-Bernoulli Hypothesis - Engineering LibreTexts
Mar 24, 2021 · The Euler-Bernoulli hypothesis gives rise to an elegant theory of infinitesimal strains in beams with arbitrary cross-sections and loading in two out-of-plane ...Missing: historical Galileo Saint- Venant<|control11|><|separator|>
[21]
[PDF] 9 Stresses: Beams in Bending
We have positioned our x,y reference coordinate frame with the x axis coincident with this neutral axis. We first define a radius of curvature of the deformed ...
[22]
Monitoring Neutral Axis Position Using Monthly Sample Residuals ...
In symmetrical pure bending, the N.A. passes through the area centroid of the cross-section of the beam. When axial loads are applied, the N.A. location is ...
[23]
Pure Bending - Engineering Mechanics
The surface descrived by the set of lines that do not extend or contract is called the neutral surface. Lines on one side of the neutral surface extend and on ...Missing: definition | Show results with:definition
[24]
[PDF] Flexural Stresses In Beams (Derivation of Bending Stress Equation)
The strain is proportional to the distance from the neutral axis and, as shown below is the greatest at the top and bottom of the beam. Stress – Strain ...
[25]
[PDF] STRESSES IN BEAMS - MIT
Nov 21, 2000 · The stress is then given by Eqn. 7, which requires that we know the location of the neutral axis (since y and I are measured from there). The ...
[26]
[PDF] Steel Beam Shear Stress
We see that the shear stresses are maximum at the neutral axis, are zero at the top and bottom of the beam, and that they vary parabolically through the ...
[27]
Shear Stresses in a Rectangular Beam - Lecture Notes
Shear strain distribution is parabolic like the shear stress distribution. max. shear strain at neutral axis; zero shear strain at top and bottom surfaces.
[28]
[PDF] SUMMARY OF TOPICS & FORMULAE - NJIT
Jan 18, 2008 · Neutral axis (NA) ... I = Moment of Inertia of the beam cross section about the neutral axis (NA), v = distance of the point from NA ...<|control11|><|separator|>
[29]
[PDF] Chapter 8 DEFLECTION OF BEAMS BY INTEGRATION
Chapter 8 uses Eq. (8.1) to find the deflection (y) of a beam at a point (x) by relating it to the elastic curve, which is the equation of the curve into which ...Missing: homogeneous | Show results with:homogeneous
[30]
[PDF] Bending of Curved Beams – Strength of Materials Approach
neutral axis, as for a straight beam, and that the only significant stress ... y dA I y R. = →. = +. ∫. ∫ and we recover the straight beam flexure ...
[31]
Chapter 10 Bending of Curved Beams
The resulting relation between the stress, moment and the deflection is called as Winkler-Bach formula. Then, using the two dimensional elasticity formulation, ...
[32]
5.14 Curved Beam Formula | Bending of Beams - InformIT
Aug 1, 2019 · Winkler's curved beam formula extends straight beam theory, with the neutral axis not coinciding with the centroidal axis. It has two forms, ...
[33]
[PDF] Arches, rings and fuselage frames - VTechWorks
That is, the centroidal axis does not coincide with the neutral axis of the cross section for the curved bar. The neutral axis of the curved bar is located ...
[34]
[PDF] 4.9.2. Cracked Sections
Find the location of neutral axis (First Moment of Area = 0). 15”. 8”. 12”. (8y) y. 2 = 8(15 − y) × 15 − y. 2. + (12 − y)(7.92) y = 7.78 in. (n − 1)As ...Missing: homogeneous | Show results with:homogeneous<|control11|><|separator|>
[35]
[PDF] 2.001 - MECHANICS AND MATERIALS I Lecture 11/21/2006 Prof ...
Nov 21, 2006 · So the neutral axis is in the center of the beam. What if E2 > E1 in: a is the distance to neutral axis. 3. Page ...Missing: ∫ | Show results with:∫
[36]
[PDF] Reinforced Concrete Beam
Neutral Axis. • The NA passes through the centroid, the balance point ... Itr of the Transformed Section. • Find Itr. • Solve for Itr. (. ) (. ) 3. 3. 2. ( 1).Missing: "structural | Show results with:"structural
[37]
[PDF] Chapter 2. Design of Beams – Flexure and Shear 2.1 Section force ...
Z = plastic section modulus from the Properties section of the AISC manual. S = elastic section modulus, also from the Properties section of the AISC manual.
[38]
[PDF] Steel Design
Instability from Plastic. Hinges. Shape Factor: The ratio of the plastic moment to the elastic moment at yield: k = 3/2 for a rectangle k ≈ 1.1 for an I beam.