Deformations in a Transverse Cross Section

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Mechanical Engineering
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JoVE Core Mechanical Engineering
Deformations in a Transverse Cross Section

Vidéo suivante20.6: Bending of Material: Problem Solving

When a material is subjected to uniaxial stress, the material undergoes elongation or contraction in the direction of the applied stress.

At the same time, deformation occurs in the transverse direction to the applied stress, governed by Poisson's ratio.

The deformation in the transverse direction will result either in expansion or contraction above and below the neutral surface.

The expansion and contraction in the vertical transverse direction compensate each other and vanish.

For the horizontal transverse direction, expansion and contraction will result in various horizontal lines of the section, which are bent into circular arcs.

The radius of the curvature of the neutral surface can be expressed as the ratio of the radius of curvature due to bending to the Poisson's ratio of the material.

This radius of curvature corresponds to the circle whose center is situated on the other side of the radius of curvature due to bending.

The reciprocal of the radius of curvature of the neutral surface represents the curvature of the transverse section and is known as the anticlastic curvature.

Deformations in a Transverse Cross Section

When a material is subjected to uniaxial stress, it elongates or contracts in the direction of the applied force, and also undergoes changes in the perpendicular directions. This behavior is crucial for understanding how materials behave under stress and is governed by mechanical properties such as Poisson's ratio v, which measures the ratio of transverse strain to axial strain.

As the material stretches, it expands or contracts in orthogonal directions to the load. This phenomenon varies across different sections of the material.

For instance, in the vertical transverse direction, expansions and contractions cancel each other out on what is known as the neutral surface, where there is no longitudinal stress.

However, the material behaves differently in the horizontal transverse direction; the sections bend into circular arcs due to the varied expansions and contractions across the material's thickness. The circular arcs are centered on a point O', with a radius of curvature r'. The radius of curvature of the beam due to bending is centered on a point O, with a radius of curvature r. Bending in both directions is related to the material's Poisson's ratio. The radius of curvature of these arcs, particularly noticeable away from the neutral surface, is inversely proportional to Poisson's ratio.

The curvature of the transverse section of the material, known as anticlastic curvature, reveals how the material bends in directions orthogonal to the primary direction of bending.