Name: neutral axis and strain in bending
Uploaded: 2024-05-23T12:59:43.000Z
Duration: 1 min 18 s
Channel: JoVE

Consider the deformations of a symmetric prismatic member subjected to equal and opposite couples.
As the member bends uniformly, the lines on the wider faces of the members transform into a circle of constant curvature centered at point C.
By dividing the member into tiny cubic elements, it becomes apparent that the only non-zero stress component is the normal one, leading to uniaxial stress at any given point.
So, a neutral surface exists parallel to the member&#39;s upper and lower faces where longitudinal components of strain and stress are zero.
The deformation of an arc, located at a distance y above the neutral surface, can be expressed as the difference in the length of this arc and that of the neutral surface.
Expressing these lengths in terms of the radius and the angle subtended and dividing the deformation by the length of the neutral arc shows that the longitudinal normal strain varies linearly with the distance y from the neutral surface.
Assuming a positive bending, the negative sign indicates the beam&#39;s upward concavity.

deformations in a symmetric member in bending

When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
When the member is segmented into tiny cubic elements, it is observed that the primary stress experienced within the member is normal stress, leading to uniaxial stress conditions at any point. This arrangement reveals the existence of a neutral surface, where both the strain and stress longitudinal components are zero. This surface runs parallel to the upper and lower faces of the member, and the distance from the neutral surface to point C is &#961;
To explore the deformation of this member, consider an arc at a distance y from the neutral surface. The deformation is the difference in lengths from point C between the arc at y (L&#39;) and the neutral surface arc (L). Dividing the deformation &#948; = L&#39; - L by the length of the neutral arc shows that the longitudinal normal strain varies linearly with the distance from the neutral surface. By applying Hooke&#39;s Law, which relates stress and strain in elastic materials, the stress can be determined at any point based on its distance from the neutral surface.

Deformations in a Symmetric Member in Bending

Consider the deformations of a symmetric prismatic member subjected to equal and opposite couples.
As the member bends uniformly, the lines on the wider ...

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

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JoVE Core

Mechanical Engineering

Unit 1

Bending

KEY TERMS AND CONCEPTS

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 surface, where the longitudinal strain and the longitudinal stress are both zero. This surface stays parallel to the upper and lower faces of the member, and the distance from the neutral surface to Point C is ρ.

To describe the deformation, consider an arc at a distance y from the neutral surface. The deformation is the difference between the arc length at y, L', and the neutral arc length, L. When δ = L' - L is divided by the length of the neutral arc, the longitudinal normal strain is shown to change linearly with distance from the neutral surface. Hooke's Law then links that strain to stress, so the stress at any point can be found from its position relative to the neutral surface.

Neutral Axis and Strain in Bending

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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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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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key terms and concepts

Learn how bending creates a neutral surface, linear strain, and normal stress in a symmetric member. Clear high school engineering science content.

Neutral Axis and Strain in Bending | JoVE K12

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Neutral Axis and Strain in Bending

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