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Pure bending

Pure bending refers to a state of deformation in prismatic beams subjected to equal and opposite couples applied in the same longitudinal plane, resulting in a constant bending moment and the absence of transverse shear forces.^[1] Under this loading, the beam assumes a uniform curvature, bending into an arc of a circle with a large radius compared to its cross-sectional dimensions, while plane cross-sections remain plane and perpendicular to the longitudinal axis.^[2] A key feature is the neutral axis, which passes through the centroid of the cross-section and experiences zero normal stress and strain, with tensile stresses developing on one side and compressive stresses on the other, varying linearly with distance from the neutral axis.^[3] The normal stress distribution is given by the formula \sigma = -\frac{My}{I}, where M is the bending moment, y is the perpendicular distance from the neutral axis, and I is the moment of inertia of the cross-section.^[1] This idealized condition, though rarely encountered exactly in practice, forms the basis for analyzing bending stresses in beams and is derived from assumptions including linear elastic material behavior and small deformations.^[4]

Fundamentals

Definition and Conditions

Pure bending is a specific loading condition in beam theory where a segment of a beam experiences a constant bending moment along its length accompanied by zero shear force, resulting in uniform curvature without transverse shear deformation. This idealized state simplifies the analysis of flexural behavior by focusing solely on the effects of the applied moment.^[1] For pure bending to occur, the beam must be prismatic, meaning it is initially straight with a constant cross-section throughout the segment of interest, and the loading must generate equal and opposite couples that produce a uniform moment distribution. A common experimental setup achieving this is the four-point bending test, where two outer supports and two inner loading points create a region of constant moment and zero shear between the inner points. Additionally, transverse shear effects must be negligible, which is typically valid for slender beams where the length-to-depth ratio is sufficiently large.^[5]^[6] This configuration of pure bending is essential in solid mechanics for isolating pure flexural responses, enabling precise theoretical and experimental studies of material behavior under bending without complications from shear influences. In such cases, the beam deforms into an arc of a circle with uniform curvature.^[7]

Distinction from General Bending

In beam theory, general bending refers to the deformation of a structural member subjected to both transverse shear forces and bending moments that may vary along its length, resulting in non-uniform curvature and a combination of normal and shear stresses throughout the cross-section.^[8]^[5] This scenario is common in real-world applications, such as simply supported beams under distributed loads, where the shear force influences the overall stress distribution and deflection profile.^[3] Pure bending, by contrast, occurs under the specific conditions of zero shear force and a constant bending moment, leading to uniform curvature and a purely uniaxial stress state dominated by normal stresses alone.^[8]^[9] In general bending, the presence of shear deformation and moment variation complicates the analysis, as shear stresses must be superimposed on normal stresses, requiring consideration of higher-order effects like those in Timoshenko beam theory.^[5]^[8] The implications for structural analysis are significant: pure bending allows for simplified, exact analytical solutions based on Euler-Bernoulli assumptions, facilitating straightforward computation of stresses and strains in regions of constant moment, such as the central span of a four-point bend test.^[9]^[5] General bending, however, often necessitates numerical methods, finite element analysis, or superposition of shear and flexural effects to accurately predict behavior, especially in beams with significant shear contributions or complex loading.^[8]^[3]

Kinematics

Deformation Geometry

In pure bending, a prismatic beam with an initially straight longitudinal axis and symmetric cross-section deforms under equal and opposite end moments that produce no shear force along its length. This results in a uniform curvature, transforming the beam's axis into a circular arc with a constant radius of curvature \rho. The deformed shape assumes that the beam segment bends into part of a circle centered at a point offset from the neutral axis, maintaining the prismatic nature of the cross-sections while altering their orientation.^[8] The key geometric feature of this deformation is the rotation of cross-sections, where originally plane sections perpendicular to the beam's axis remain plane and rotate to stay perpendicular to the deformed neutral axis. This rotation occurs without distortion of the cross-sectional shape, preserving the beam's transverse dimensions while the longitudinal fibers adjust their lengths relative to the curvature. The neutral axis, defined as the original centerline passing through the centroid of the cross-section, undergoes no net extension or contraction and serves as the reference for the circular path with radius \rho.^[10]^[11] Fibers located above the neutral axis shorten (compress) due to the concave curvature, while those below lengthen (extend) symmetrically to the same degree, creating a mirrored deformation pattern about the neutral plane. This symmetric fiber behavior ensures that the neutral axis remains the locus of zero longitudinal deformation, with the extent of compression and extension varying linearly with distance from this axis in the deformed configuration. The overall geometry thus describes a smooth, arc-like bending without transverse shear effects, characteristic of pure bending conditions.^[8]^[10]

Longitudinal Strain Distribution

In pure bending, the longitudinal strain distribution across the beam's cross-section is linear and varies with the distance from the neutral axis, arising directly from the geometric deformation of the beam under curvature. This strain field is purely kinematic, determined by the change in length of longitudinal fibers as the beam bends into an arc. The neutral axis, located at the centroid of the cross-section, experiences zero strain, serving as the reference for measuring deformations in fibers above and below it.^[5]^[8] The kinematic relation for the normal longitudinal strain \epsilon_x at a distance y from the neutral axis is given by

\epsilon_x = -\frac{y}{\rho},

where \rho is the radius of curvature of the neutral axis. This equation quantifies the strain as the relative elongation or shortening of fibers, with the linear variation ensuring that strain increases proportionally from zero at the neutral axis to maximum values at the extreme fibers of the beam height.^[9]^[8] The derivation stems from the geometry of deformation, assuming plane sections remain plane and perpendicular to the longitudinal axis after bending. Consider a small segment of the beam subtending an angle \Delta \phi at the center of curvature. The undeformed length of a fiber at distance y from the neutral axis is \Delta s = \rho \Delta \phi. After deformation, this fiber follows an arc of radius \rho - y, yielding a deformed length \Delta s' = (\rho - y) \Delta \phi. The longitudinal strain is then the limit of the relative length change:

\epsilon_x = \lim_{\Delta s \to 0} \frac{\Delta s' - \Delta s}{\Delta s} = \frac{(\rho - y) \Delta \phi - \rho \Delta \phi}{\rho \Delta \phi} = -\frac{y}{\rho}.

This geometric approach confirms the linear strain profile, with no strain alteration along the neutral axis where y = 0.^[5]^[8] The sign convention in the relation \epsilon_x = -y / \rho follows standard beam theory, where y is positive upward from the neutral axis and \rho > 0 for concave-upward curvature (positive bending). Thus, fibers above the neutral axis (y > 0) experience compressive strain (\epsilon_x < 0), while those below (y < 0) undergo tensile strain (\epsilon_x > 0). This distribution holds symmetrically for the opposite curvature when \rho < 0.^[9]^[5]

Assumptions

Core Assumptions

The theory of pure bending relies on several foundational assumptions that simplify the analysis of beam deformation under constant bending moment. These assumptions enable the derivation of key relationships between applied moments, internal stresses, and strains, while neglecting complexities such as shear effects and material nonlinearities.^[9]^[12] A primary assumption is that the beam is initially straight, possesses a constant cross-section along its length, and is composed of a homogeneous material with isotropic properties. This ensures uniformity in the beam's geometry and material response, allowing for consistent stress and strain distributions without variations due to initial imperfections or non-uniformity.^[9]^[13] The homogeneity implies that the material's elastic properties, such as Young's modulus, are the same throughout the beam, facilitating the application of linear elasticity principles.^[14] Central to the theory is the Bernoulli-Euler hypothesis, which posits that plane cross-sections perpendicular to the longitudinal axis before deformation remain plane and normal to the deformed axis after bending. This kinematic assumption implies that deformation occurs primarily through rotation of cross-sections without distortion in their plane, leading to a linear distribution of longitudinal strains across the section height.^[12]^[15] It underpins the neglect of shear deformation, assuming that transverse shear strains are negligible compared to bending strains, particularly in slender beams.^[16] The material is further assumed to behave linearly elastically, obeying Hooke's law within the elastic limit, with transverse strains being negligible and no significant shear deformation contributing to the overall response. This allows stresses to be directly proportional to strains via the material's modulus of elasticity, simplifying the stress analysis to uniaxial conditions along the beam length.^[14]^[17] Young's modulus is considered identical in tension and compression, ensuring symmetric material response.^[14] Finally, pure bending is characterized by deformation occurring in a single plane, with the bending moment remaining constant along the beam segment and transverse shear forces being zero. This condition isolates the effects of moment-induced curvature, excluding influences from varying loads or shear that could alter the stress state.^[9]^[8] The radius of curvature is assumed large relative to the cross-sectional dimensions, reinforcing the applicability to gentle curvatures.^[14]

Limitations and Validity

The pure bending theory, grounded in the Euler-Bernoulli beam framework, applies primarily to slender beams where the length significantly exceeds the cross-sectional depth (typically L/h > 10), ensuring negligible transverse shear effects relative to bending. It is valid under conditions of small deformations, where strains remain infinitesimal and rotations are minimal, aligning with linear elasticity principles, and for homogeneous, elastic materials operating below the yield stress to maintain Hooke's law proportionality.^[5]^[18] Key limitations emerge when these conditions are not met. In short beams or those with deep sections, transverse shear deformations become significant, rendering the assumption of plane sections remaining perpendicular to the longitudinal axis invalid and requiring alternative models like Timoshenko theory for accuracy. Plastic deformation or large strains beyond the elastic limit violate the theory's linear stress-strain relationship, as the material response deviates from elasticity. Additionally, in composite or non-homogeneous materials with varying moduli across the cross-section, the neutral axis shifts away from the geometric centroid, complicating the uniform strain distribution assumption and leading to erroneous stress calculations.^[5]^[18]^[19] Violating these validity boundaries results in practical consequences, such as overestimation of beam stiffness due to ignored shear contributions, which underpredicts actual deflections and can mislead design assessments. Inaccurate stress predictions from mislocated neutral axes or nonlinear behavior heighten failure risks, potentially causing premature yielding or fracture in engineering applications like structural components or machine elements. These issues underscore the need to verify applicability through slenderness ratios and material properties before relying on pure bending analysis.^[20]^[5]

Stress and Strength Analysis

Normal Stress Derivation

In pure bending, the normal stress distribution arises from the kinematic strain field through the material's constitutive response and equilibrium considerations. The longitudinal strain ε_x varies linearly with the distance y from the neutral axis, given by ε_x = -y / ρ, where ρ is the radius of curvature of the bent beam.^[8]^[9] Under the assumption of linear elastic material behavior, Hooke's law relates stress and strain as σ_x = E ε_x, where E is the Young's modulus. Substituting the strain expression yields the normal stress σ_x = -E y / ρ. This indicates a linear variation of normal stress across the cross-section, with tensile stress (positive σ_x) below the neutral axis for positive curvature and compressive stress (negative σ_x) above it, reaching zero at the neutral axis (y = 0) and achieving maximum magnitude at the outermost fibers.^[3]^[8] To ensure equilibrium, the internal bending moment M must be balanced by the moment resultant of the stress distribution over the cross-section. This balance is expressed as M = - ∫ σ_x y , dA, integrated across the entire area A of the beam's cross-section. Substituting the derived stress gives M = - ∫ (-E y / ρ) y , dA = (E / ρ) ∫ y^2 , dA, confirming that the linear stress profile sustains the applied moment without net axial force, as the stresses are antisymmetric about the neutral axis.^[9]^[8]

Flexural Formula

The flexural formula provides the relationship between the normal stress in a beam under pure bending, the applied bending moment, and the geometric properties of the beam's cross-section. It is expressed as

\sigma_x = -\frac{M y}{I},

where \sigma_x is the normal stress at a point in the cross-section, M is the internal bending moment, y is the perpendicular distance from the neutral axis to the point, and I is the second moment of area (moment of inertia) of the cross-section about the neutral axis.^[21]^[5] This formula arises from the Euler-Bernoulli beam theory and assumes linear elastic material behavior, with the stress distribution being linear across the cross-section and zero at the neutral axis.^[21] To derive the flexural formula, start from the kinematic relation for longitudinal strain in pure bending, \epsilon_x = -y / \rho, where \rho is the radius of curvature of the deformed beam axis. Applying Hooke's law for uniaxial stress, \sigma_x = E \epsilon_x = -E y / \rho, where E is the Young's modulus.^[5] The relationship between the bending moment and curvature is given by the Euler-Bernoulli equation, M = E I / \rho.^[21] Eliminating \rho from these equations yields \sigma_x = -M y / I, completing the derivation and linking the stress directly to the moment without explicit reference to curvature.^[5]^[21] For practical analysis, the maximum normal stress \sigma_{\max} occurs at the outermost fiber, where y = y_{\max} (the distance from the neutral axis to the farthest point in the cross-section). This leads to \sigma_{\max} = M / Z, with the section modulus defined as Z = I / y_{\max}.^[22]^[9] In symmetric cross-sections, such as rectangular or circular beams, the neutral axis passes through the centroid, ensuring balanced tensile and compressive stresses.^[21] The section modulus Z thus serves as a key geometric parameter for assessing the bending capacity of a section under a given moment.^[22]

Applications

Design Implications

In beam design, pure bending theory guides the selection of cross-sections to ensure structural integrity under applied moments. For strength considerations, engineers verify that the maximum normal stress does not exceed the allowable stress by applying the flexural formula, where the peak stress occurs at the extreme fibers: σ_max = M y_max / I ≤ σ_allow. Here, M is the maximum bending moment, y_max is the distance from the neutral axis to the outermost fiber, I is the moment of inertia, and σ_allow is typically derived from the material's yield strength F_y divided by a factor of safety (FS), such as σ_allow = F_y / FS for general materials or 0.66 F_y for compact steel sections under allowable stress design (ASD). This approach allows designers to choose sections with sufficient section modulus S = I / y_max to resist the moment, prioritizing resistance to yielding or fracture at critical locations like midspan in simply supported beams.^[23]^[24] Stiffness requirements in pure bending focus on limiting deflections to maintain serviceability, using the moment-curvature relationship to relate applied moments to beam deformation. The curvature is given by 1/ρ = M / (E I), where ρ is the radius of curvature and E is the modulus of elasticity; this informs deflection calculations, ensuring maximum deflection δ_max does not exceed limits like L/360 for floors (L being the span length). Designers select sections with adequate flexural rigidity E I to control excessive sagging or vibration, particularly in long-span applications where user comfort or functional clearances are concerns.^[8]^[24] Design factors incorporate load amplification and material properties to account for uncertainties, enhancing safety in pure bending applications. Load factors, such as 1.7 for dead plus live loads in plastic design or 1.3 when including wind or seismic effects, scale service loads to ultimate conditions for strength checks. Material allowables vary by code; for steel, compact sections permit higher utilization (e.g., 0.66 F_y) compared to noncompact ones (down to 0.60 F_y), while factors like unbraced length L_b influence lateral torsional buckling reductions. The pure bending approximation suffices in regions of constant moment and negligible shear (V ≈ 0), such as central spans in four-point loaded beams or long-span members where shear stresses are small relative to bending stresses (order h/L, with h as depth and L as span). This simplifies analysis for prismatic beams under symmetric loading, but requires validation against shear-dominated cases near supports.^[23]^[24]^[8]

Illustrative Examples

To demonstrate the application of pure bending analysis, consider a rectangular beam subjected to a constant bending moment of M = 10 kNm. The beam has a width b = 50 mm ($0.05 m) and height h = 100 mm ($0.1 m). The moment of inertia I about the neutral axis is calculated as

I = \frac{b h^3}{12} = \frac{0.05 \times (0.1)^3}{12} = 4.167 \times 10^{-6} \, \mathrm{m}^4.

The maximum distance from the neutral axis is y_{\max} = h/2 = 0.05 m. The maximum normal stress \sigma_{\max} is then given by the flexural formula

\sigma_{\max} = \frac{M y_{\max}}{I} = \frac{10 \times 10^3 \times 0.05}{4.167 \times 10^{-6}} = 120 \, \mathrm{MPa}.

This stress occurs at the outer fibers of the beam.^[25] In a second example, determine the required section modulus Z for a steel beam to support a bending moment of M = 50 kNm without yielding. The steel has a modulus of elasticity E = 200 GPa and a yield stress \sigma_{\mathrm{y}} = 250 MPa, corresponding to the yield strength for ASTM A36 steel. The section modulus is

Z = \frac{M}{\sigma_{\mathrm{y}}} = \frac{50 \times 10^3}{250 \times 10^6} = 2 \times 10^{-4} \, \mathrm{m}^3 = 200 \, \mathrm{cm}^3.

This value ensures the maximum stress equals the yield stress under the given bending moment in pure bending. The modulus E confirms elastic behavior, as the corresponding maximum strain \epsilon_{\max} = \sigma_{\max}/E = 250 \times 10^6 / 200 \times 10^9 = 0.00125 is well within the elastic range for steel.^[26]^[25] These examples assume the beam is sufficiently slender to validate pure bending conditions, such as a span-to-depth ratio L/h > 10, which minimizes shear effects and ensures plane sections remain plane. For the first example, with h = 0.1 m, a span L > 1 m satisfies this; similarly for the second, selecting a section with equivalent depth yielding Z = 200 cm³ (e.g., a rectangular beam of approximate dimensions b = 100 mm, h \approx 120 mm) would require L > 1.2 m. Under these conditions, the calculated stresses accurately represent the uniaxial tensile and compressive states without significant deviations from linear elastic theory.^[27]

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