Name: bending stress and the neutral axis
Uploaded: 2024-05-23T12:59:43.000Z
Duration: 1 min 16 s
Channel: JoVE

Рассмотрим изгибающий момент, при котором симметричный стержень испытывает напряжение в пределах предела упругости. Продольное напряжение можно выразить с помощью закона Гука.
Вспомним выражение для продольной деформации в терминах максимальной деформации на расстоянии 'c' от нейтральной поверхности.
Умножение его на модуль упругости и подстановка его в уравнение напряжения показывает, что нормальное напряжение изменяется линейно с расстоянием от нейтральной поверхности.
Теперь вспомните выражения для суммы компонент силы и моментов. Замена напряжения в уравнении силы указывает на то, что в пределах упругости нейтральная ось проходит через центроид сечения.
Подставляя напряжение в уравнение момента и упрощая его, получаем выражение, содержащее интеграл, равный моменту инерции поперечного сечения относительно центроидальной оси, перпендикулярной плоскости пары.
Последним упрощенным выражением является формула упругого изгиба для максимального напряжения. Для произвольного расстояния 'y' от нейтральной поверхности эта формула дает напряжение изгиба, вызванное изгибом стержня.

Напряжение при изгибе

Flexural Stress

При анализе изгиба симметричных элементов очень важно понимать, как распределяются напряжения под действием изгибающих моментов. Это распределение напряжений эффективно описывается с помощью фундаментальной механики и принципов материаловедения, в частности закона Гука для упругих материалов.

Закон Гука гласит, что в пределах упругости материала напряжение прямо пропорционально деформации. В элементе, испытывающем изгибающий момент, деформация в любой точке зависит от его расстояния от нейтральной оси, центрального слоя, который не испытывает продольной деформации. Деформация изменяется линейно от нуля на нейтральной оси до максимума на крайних волокнах элемента.

Из этого следует, что продольное напряжение в любой точке также линейно меняется с расстоянием от нейтральной оси. Интегрирование этого линейного изменения по площади поперечного сечения элемента, где напряжение равно нулю на нейтральной оси, подтверждает, что эта ось совпадает с центроидом поперечного сечения.

Этот процесс интегрирования также определяет выражение для изгибающего момента, а также определяет момент инерции сечения. Этот расчет дополнительно устанавливает взаимосвязь между изгибающим моментом и максимальным напряжением в самой дальней точке от нейтральной оси и дает изгибное напряжение, вызванное изгибом элемента.

<img alt="Equation 1" src="/files/ftp_upload/15627/15627_Equation_1.svg" style="text-align: center; height: 28px;">

Рассмотрим изгибающий момент, при котором симметричный стержень испытывает напряжение в пределах предела упругости. Продольное напряжение может быть выражено с помощью ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Bending

Гибка

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

bending stress and the neutral axis

Bending stress in a symmetric member depends on how the material responds to a bending moment. Within the elastic limit, Hooke's Law applies, so stress is proportional to strain. That relationship helps describe how the member stretches and compresses as it bends.

In a bent member, strain depends on distance from the neutral axis. The neutral axis is the central layer that has no longitudinal strain. Strain is zero at that axis and increases linearly toward the outermost fibers.

Because stress follows strain in elastic materials, longitudinal stress also changes linearly with distance from the neutral axis. When this linear stress pattern is integrated across the cross-sectional area, the neutral axis is shown to match the centroid of the section. The same integration also leads to the bending moment expression and defines the moment of inertia of the section.

These relationships connect bending moment to the maximum stress at the farthest point from the neutral axis. They explain how flexural stress develops in a member under bending. This makes the neutral axis, centroid, and moment of inertia key ideas for analyzing bending in symmetric members.

Bending Stress and the Neutral Axis

bending

Pure Bending in Beams and Structures

Pure bending describes how a beam bends under equal and opposite moments. It is a key idea in structural mechanics. Engineers use it to study how prismatic members, such as beams, deform under symmetrical loads without direct forces on the member. In pure bending, the beam experiences bending without axial force or shear force. This makes it possible to predict the stress pattern across the cross-section. The bending stress depends on the bending moment, the moment of inertia of the...

symmetric member in bending

Bending Stress in a Symmetric Member

Bending stress in a symmetric member comes from a prismatic member that is loaded by two couples acting in opposite directions. A couple is made of two equal and opposite forces. It creates a moment, but it does not create a resultant force. To keep the member in equilibrium, internal forces appear at any section cut. These internal forces must balance the external couples. They are resolved into normal stress and shear stress components. Normal stress acts perpendicular to the...

deformations in a symmetric member in bending

deformations-in-a-symmetric-member-in-bending

Neutral Axis and Strain in Bending

A symmetric prismatic member in bending shows how neutral surfaces, strain, and stress work together. When equal and opposite couples bend the member, the straight lines on its wider faces become circular arcs centered at Point C. That curved shape helps show how deformation changes across the section. If the member is divided into tiny cubic elements, the main stress inside it is normal stress. This creates uniaxial stress conditions at any point. The bending pattern also reveals a neutral...

flexural stress

Bending stress in a symmetric member depends on how the material responds to a bending moment. Within the elastic limit, Hooke's Law applies, so stress is proportional to strain. That relationship helps describe how the member stretches and compresses as it bends. In a bent member, strain depends on distance from the neutral axis. The neutral axis is the central layer that has no longitudinal strain. Strain is zero at that axis and increases linearly toward the outermost fibers. Because...

deformations in a transverse cross section

deformations-in-a-transverse-cross-section

Poisson Effects in Beam Bending

Poisson effects in beam bending describe how a material changes shape in directions perpendicular to the applied force. When a material is under uniaxial stress, it stretches or contracts along the load and also changes in the crosswise directions. Poisson's ratio, written as v, measures the ratio of transverse strain to axial strain. As the material stretches, it may expand or contract in directions orthogonal to the load. These changes are not the same in every part of the material. In the...

bending of material: problem solving

Pipe Wall Thickness and Bending Strength

Pipe wall thickness and bending strength are compared by finding the ratio of the maximum bending moments for two metal pipes. Both pipes can withstand a maximum stress of 100 MPa. Each pipe has the same outer radius of 1.8 cm, but Pipe A has an inner radius of 1.5 cm and Pipe B has an inner radius of 1 cm. The first step is to calculate the moment of inertia for each pipe. The moment of inertia depends on the pipe dimensions and is found using the given equation for a hollow circular section.

bending of members made of several materials

bending-of-members-made-of-several-materials

Composite Beam Stress and Neutral Axis

Composite beam stress depends on the elastic properties of each material in the member. When a structural member is made of two materials with the same cross-sectional area, each material responds differently under load because its modulus of elasticity is different. To analyze this behavior, engineers use the transformed section concept, which compares the two materials in a single equivalent section. Hooke’s Law links stress and strain in each material. Stress is proportional to strain, but...

stress concentrations

Stress at Holes and Notches in Bending

Stress at holes and notches in bending shows how a material can carry more load in one small area than in the rest of the cross-section. For a symmetric member under pure bending, the stress is usually treated as evenly spread across the cross-section. That simple assumption works only when the cross-section is uniform and has no irregularities. When the shape changes, the stress pattern changes too. Notches, holes, and other discontinuities can raise the local stress above the nominal stress...

plastic deformations

Plastic Bending and the Neutral Axis

Plastic bending changes the shape of a structural member after the bending stress goes beyond the material’s yield strength. Before yielding, the member behaves in a linear elastic way. After yielding, the deformation is permanent. The strain at any point in the member is described in terms of the maximum strain. During elastic bending, the neutral axis, where strain is zero, matches the centroid. Under plastic conditions, that axis shifts away from the centroid. Finding its new position uses...

members made of elastoplastic material

Bending Stages of Elastoplastic Members

Elastoplastic members under bending show a clear change in stress as the load increases. For a rectangular cross-section, the first response is elastic. The stress follows Hooke's Law and forms a linear pattern across the section. In this early stage, stress grows from the neutral axis to the outer fibers. The outer fibers reach the elastic limit first because they carry the highest stress. The inner part of the section stays elastic while the bending moment remains in this range. When the...

plastic deformations of members with a single plane of symmetry

plastic-deformations-of-members-with-a-single-plane-of-symmetry

Finding the Plastic Neutral Axis in Bending

Plastic bending in a member with one plane of symmetry depends on the plastic neutral axis and the stress pattern across the cross-section. In this state, the stress is uniform above and below the neutral axis. The region above the axis is in compression, so it carries negative stress. The region below the axis is in tension, so it carries positive stress. The plastic neutral axis does not usually pass through the centroid of the cross-section. This shift can happen when the section is not...

residual stresses in bending

Bending, Yielding, and Residual Stress

Bending in elastoplastic members shows how a material responds during loading and unloading. At first, the member behaves elastically, so stress is proportional to strain and follows Hooke's Law. This stage continues until the bending moment becomes large enough to pass the yield strength. Once yield strength is exceeded, plastic deformation begins. Plastic deformation means the material does not fully return to its original shape. It leaves permanent strain and deformation even after the load...

eccentric axial loading in a plane of symmetry

eccentric-axial-loading-in-a-plane-of-symmetry

Eccentric Load Stress in Structural Members

Eccentric axial loading happens when an axial load is applied away from the centroidal axis of a structural member. In engineering, this can happen when parts are not directly aligned because of design or function. The load then creates both a direct axial force and a bending effect. The offset load produces a moment, or turning effect, that acts like a couple and must be balanced to keep the member in equilibrium. The internal stress across the cross-section is therefore a mix of uniform...

unsymmetric bending

Bending About Multiple Axes

Unsymmetrical bending happens when a bending moment does not line up with a member’s principal axis. This creates a stress pattern and deflection shape that are more complex than in symmetrical bending. These effects matter when engineers design structures for different loading conditions. In unsymmetrical bending, the neutral axis is the line where stress is zero. It does not always match the geometric axes of the cross-section. Its orientation depends on the relationship between the applied...

unsymmetric bending - angle of neutral axis

unsymmetric-bending-angle-of-neutral-axis

Unsymmetric Bending and Neutral Axis

Unsymmetric bending happens when a beam or other structural member bends in a plane that does not match its principal axes. It often appears when a load is applied at a non-ideal angle, which makes stress analysis more complex. When a bending moment is applied at an angle θ to the vertical axis of a symmetrical member, it can be split into components along the member’s principal centroidal axes. The stress from each component is found separately. Then the results are combined using the...

general case of eccentric axial loading

Unsymmetrical Bending and Neutral Axis

Unsymmetrical bending happens when the bending moment on a structural member does not line up with its principal axis. This creates stress and deflection patterns that differ from symmetrical bending. Understanding this case helps explain how structures respond to different loading conditions. The member in this situation is acted on by equal and opposite forces. The forces are placed along a line that does not pass through the neutral axis. The neutral axis is the line where stress is zero,...

bending of curved members - strain analysis

bending-of-curved-members-strain-analysis

Curved Beam Bending and the Neutral Axis

Curved beams and arches bend in a way that changes both their shape and their strain pattern. When a curved member that is symmetric about the y-axis is loaded by equal and opposite forces, its circular shape deforms and the center of curvature moves from C to C'. The new curve is tighter than the original one. A key idea in this bending analysis is the neutral axis. The neutral axis is a hypothetical line inside the material whose length does not change during bending. It does not experience...

bending of curved members - neutral surface

bending-of-curved-members-neutral-surface

Curved Beam Stress and Neutral Axis

Curved beams have a non-uniform stress pattern across the cross-section because of their curvature. The stress is not spread evenly the way it is in a straight beam. The neutral axis, or the line where stress is zero, does not pass through the centroid of the section. In a curved member, strain changes with distance from the neutral axis. Hooke’s law still links stress and strain within the elastic limit of the material. Because the strain changes in a non-linear way, the stress also changes...

key terms and concepts

Explore how bending stress varies across a symmetric member, why the neutral axis has zero strain, and how moment of inertia links to maximum stress.

Bending Stress and the Neutral Axis | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Bending Stress and the Neutral Axis

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