Name: plastic bending and the neutral axis
Uploaded: 2024-05-23T12:59:43.000Z
Duration: 1 min 14 s
Channel: JoVE

Pour la flexion pure, lorsque la contrainte maximale dépasse la limite d'élasticité du matériau de l'élément, celui-ci subit une déformation plastique.
La déformation en tout point de l’élément est exprimée en termes de déformation maximale. Pour les déformations plastiques, l’axe neutre de l’élément ne doit pas passer par le centroïde de l’élément.
Ici, l’axe neutre de la barre est localisé à l’aide d’une méthode itérative jusqu’à ce que la courbe de distribution des contraintes soit formée.
Pour les éléments ayant un plan de symétrie vertical et horizontal et la même relation contrainte-déformation pour les deux axes, l’axe neutre coïncide avec l’axe horizontal de symétrie. La courbe de distribution des contraintes peut être tracée à l’aide d’une valeur de contrainte maximale spécifiée.
Ici, le moment de flexion d'un tel élément est donné en fonction de la largeur de l'élément et de la contrainte à la distance y de l'axe neutre. La valeur ultime du moment de flexion qui provoque la défaillance d’un élément peut être déterminée expérimentalement.
Ce moment de flexion ultime donne la contrainte maximale correspondante, connue sous le nom de module de rupture.

déformations plastiques

Il est essentiel de comprendre comment les éléments structurels se comportent sous une déformation plastique lorsque la contrainte de flexion dépasse la limite d'élasticité du matériau. Cet état de déformation modifie de façon permanente la forme de l'élément, contrairement au comportement élastique linéaire observé avant la plastification. La déformation en tout point de la barre est exprimée en termes de déformation maximale. Notamment, l’axe neutre, qui coïncide avec le centroïde lors de la flexion élastique, s’éloigne du centroïde dans des conditions plastiques.

La localisation de l'axe neutre dans cet état modifié implique une méthode itérative, dans laquelle la position supposée de l'axe est ajustée jusqu'à ce que la courbe de répartition des contraintes se stabilise. Ce processus est particulièrement simple dans les éléments symétriques par rapport aux plans vertical et horizontal, avec des réponses contrainte-déformation identiques le long de ces axes, permettant à l'axe neutre de s'aligner avec le plan horizontal de symétrie.

La répartition des contraintes pour de tels éléments symétriques peut être cartographiée à l'aide des valeurs de contrainte maximale en fonction de la distance par rapport à l'axe neutre. L'expérimentation détermine le moment de flexion ultime, c'est-à-dire le moment—maximum qu'un élément peut supporter avant la rupture. Ce moment est corrélé au module de rupture R_u, qui est proportionnel au M_u, qui est le moment de flexion auquel l'élément échoue. M_u est une valeur critique représentant la résistance ultime du matériau sous contraintes de flexion. Comprendre ces principes permet de concevoir des structures capables de supporter des charges plus élevées sans défaillance catastrophique, améliorant ainsi la sécurité et l'efficacité.

Déformations plastiques

Pour la flexion pure, lorsque la contrainte maximale dépasse la limite d'élasticité du matériau de l'élément, celui-ci subit une déformation plastique.
La tension à n’importe quel moment ...

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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 an iterative method. The assumed axis position is adjusted until the stress distribution curve becomes stable.

For members that are symmetric about both vertical and horizontal planes, this process is simpler. If the stress-strain response is the same along both axes, the neutral axis can align with the horizontal plane of symmetry. The stress distribution in these members can then be mapped using maximum stress values as a function of distance from the neutral axis.

The video also examines the ultimate bending moment, or the maximum moment a member can resist before failure. Experiments are used to determine this value. It is related to the modulus of rupture, Ru, which is proportional to Mu, the bending moment at failure. Mu is a key measure of a material’s ultimate strength under bending stress, and it helps engineers design safer structures that can carry higher loads without catastrophic failure.

Plastic Bending and the Neutral Axis

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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 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

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Composite Beam Stress and Neutral Axis

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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

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...

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Finding the Plastic Neutral Axis in Bending

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Eccentric Load Stress in Structural Members

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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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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

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Curved Beam Stress and Neutral Axis

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key terms and concepts

Learn how plastic bending shifts the neutral axis, changes stress distribution, and helps determine the ultimate bending moment before failure.

Plastic Bending and the Neutral Axis | JoVE K12

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Plastic Bending and the Neutral Axis

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