Name: curved beam stress and neutral axis
Uploaded: 2024-05-23T12:59:44.000Z
Duration: 1 min 16 s
Channel: JoVE

断面が均一な曲線部材の場合、断面に沿ったひずみ方程式は、中立軸からの距離によって非線形に変化することを示しています。
フックの法則を適用すると、湾曲した部材に生成される応力が得られます。ひずみと同様に、応力も非線形に変化し、中立軸からの距離に対する応力のプロットは双曲線になります。
ここで、セクションに作用するすべての基本力は、曲げカップルと静的に同等であり、横方向のz軸上のその合計はモーメントの方程式を与えます。
応力の値を代入し、方程式を単純化すると、曲率中心Cから中立面までの距離を求める関係が得られます。これは、曲げの下の湾曲した部材の中立面がそのセクションの重心を通過しないことを示しています。
応力の観点から瞬間の式を書き換えて積分を実行すると、湾曲した部材の中立サーフェスは、その形状に関係なく、常に重心と曲率半径の間に配置されることがわかります。

曲面部材の曲げ: 中立面

Bending of Curved Members - Neutral Surface

曲線梁では、直線梁とは異なり、梁の曲率により断面全体の応力分布が均一ではありません。この不均一性は、応力がゼロである中立軸が断面の重心と一致していないために発生します。湾曲した梁では、ひずみは中立軸からの距離の関数として断面に沿って変化します。

前のレッスンで説明した湾曲した部材について考えてみましょう。材料の弾性限界内で応力をひずみに関係付けるフックの法則によれば、応力も非線形に変化し、結果として中立軸から双曲線状の応力分布が生じます。湾曲した梁の曲げモーメントは、式 1 に示すように、梁の断面全体にわたるこれらの応力分布を積分することによって計算されます。

<img alt="Equation 1" src="/files/ftp_upload/15641/15641_Equation_1.svg" style="height: 36px;">

どの断面にも作用する基本的な力を合計すると、モーメントに相当する曲げ偶力が生成されます。この応力の累積効果によりモーメント方程式が生成され、これは荷重下での梁の挙動を決定するために不可欠です。分析の結果、縦方向の応力がゼロである中立面は重心と一致せず、曲率の中心に近づくことがわかります。梁の形状に関係なく、中立軸は常に重心と曲率半径の間に位置します。

断面が均一な湾曲した部材の場合、断面に沿ったひずみ方程式は、...

bending-of-curved-members-neutral-surface

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Bending

曲げ

KEY TERMS AND CONCEPTS

主要用語と概念

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 non-linearly and forms a hyperbolic distribution from the neutral axis.

The bending moment in a curved beam is found by integrating the stress over the cross-section. The small forces acting on the section combine to form a bending couple that matches the moment. This moment equation is important for predicting how the beam responds under load.

The neutral surface, where longitudinal stress is zero, shifts toward the center of curvature. It does not stay at the centroid of the section. For any beam shape, the neutral axis lies between the centroid and the radius of curvature.

Curved Beam Stress and 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...

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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...

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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...

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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...

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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...

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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.

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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...

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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...

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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...

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Bending Stages of Elastoplastic Members

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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...

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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...

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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...

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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...

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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...

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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,...

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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

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

Learn how curvature changes stress in curved beams, why the neutral axis shifts, and how bending moment is found from the cross-section.

Curved Beam Stress and Neutral Axis | JoVE K12

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mechanical engineering

Curved Beam Stress and Neutral Axis

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